15895:
15353:
15890:{\displaystyle {\begin{aligned}\det(A)&={\frac {1}{2}}\left(\left(\operatorname {tr} (A)\right)^{2}-\operatorname {tr} \left(A^{2}\right)\right),\\\det(A)&={\frac {1}{6}}\left(\left(\operatorname {tr} (A)\right)^{3}-3\operatorname {tr} (A)~\operatorname {tr} \left(A^{2}\right)+2\operatorname {tr} \left(A^{3}\right)\right),\\\det(A)&={\frac {1}{24}}\left(\left(\operatorname {tr} (A)\right)^{4}-6\operatorname {tr} \left(A^{2}\right)\left(\operatorname {tr} (A)\right)^{2}+3\left(\operatorname {tr} \left(A^{2}\right)\right)^{2}+8\operatorname {tr} \left(A^{3}\right)~\operatorname {tr} (A)-6\operatorname {tr} \left(A^{4}\right)\right).\end{aligned}}}
22260:
27823:
25286:
21166:
1520:
10979:
10422:
9287:
1956:
10562:
10005:
28087:
8913:
26411:
20394:
3500:
1950:
10974:{\displaystyle {\begin{aligned}\det {\begin{pmatrix}A&B\\C&D\end{pmatrix}}&=\det(D)\det {\begin{pmatrix}A&B\\C&D\end{pmatrix}}\underbrace {\det {\begin{pmatrix}I_{m}&0\\-D^{-1}C&D^{-1}\end{pmatrix}}} _{=\,\det(D^{-1})\,=\,(\det D)^{-1}}\\&=\det(D)\det {\begin{pmatrix}A-BD^{-1}C&BD^{-1}\\0&I_{n}\end{pmatrix}}\\&=\det(D)\det(A-BD^{-1}C).\end{aligned}}}
10417:{\displaystyle {\begin{aligned}\det {\begin{pmatrix}A&B\\C&D\end{pmatrix}}&=\det(A)\det {\begin{pmatrix}A&B\\C&D\end{pmatrix}}\underbrace {\det {\begin{pmatrix}A^{-1}&-A^{-1}B\\0&I_{n}\end{pmatrix}}} _{=\,\det(A^{-1})\,=\,(\det A)^{-1}}\\&=\det(A)\det {\begin{pmatrix}I_{m}&0\\CA^{-1}&D-CA^{-1}B\end{pmatrix}}\\&=\det(A)\det(D-CA^{-1}B),\end{aligned}}}
19567:
18799:
26149:
2364:
8732:
14920:
9282:{\displaystyle {\begin{vmatrix}1&1&1&\cdots &1\\x_{1}&x_{2}&x_{3}&\cdots &x_{n}\\x_{1}^{2}&x_{2}^{2}&x_{3}^{2}&\cdots &x_{n}^{2}\\\vdots &\vdots &\vdots &\ddots &\vdots \\x_{1}^{n-1}&x_{2}^{n-1}&x_{3}^{n-1}&\cdots &x_{n}^{n-1}\end{vmatrix}}=\prod _{1\leq i<j\leq n}\left(x_{j}-x_{i}\right).}
2968:
3259:
16401:
20025:
4201:
23126:
1319:
18278:
17529:
1759:
5277:
6232:(without changing the determinant). For such a matrix, using the linearity in each column reduces to the identity matrix, in which case the stated formula holds by the very first characterizing property of determinants. Alternatively, this formula can also be deduced from the Leibniz formula, since the only permutation
24149:
determinant, but non-commutativity then leads to the loss of many fundamental properties of the determinant, such as the multiplicative property or that the determinant is unchanged under transposition of the matrix. Over non-commutative rings, there is no reasonable notion of a multilinear form (existence of a nonzero
19319:
18636:
3272:, an explicit formula involving sums of products of certain entries of the matrix. The determinant can also be characterized as the unique function depending on the entries of the matrix satisfying certain properties. This approach can also be used to compute determinants by simplifying the matrices in question.
2160:
26406:{\displaystyle {\begin{aligned}ab&=ab{\begin{vmatrix}1&0\\0&1\end{vmatrix}}=a{\begin{vmatrix}1&0\\0&b\end{vmatrix}}\\&={\begin{vmatrix}a&0\\0&b\end{vmatrix}}=b{\begin{vmatrix}a&0\\0&1\end{vmatrix}}=ba{\begin{vmatrix}1&0\\0&1\end{vmatrix}}=ba,\end{aligned}}}
21612:
17027:
8531:
25920:
The first use of the word "determinant" in the modern sense appeared in: Cauchy, Augustin-Louis "Memoire sur les fonctions qui ne peuvent obtenir que deux valeurs égales et des signes contraires par suite des transpositions operées entre les variables qu'elles renferment," which was first read at the
25109:
In addition to the complexity of the algorithm, further criteria can be used to compare algorithms. Especially for applications concerning matrices over rings, algorithms that compute the determinant without any divisions exist. (By contrast, Gauss elimination requires divisions.) One such algorithm,
24148:
For square matrices with entries in a non-commutative ring, there are various difficulties in defining determinants analogously to that for commutative rings. A meaning can be given to the
Leibniz formula provided that the order for the product is specified, and similarly for other definitions of the
14678:
3513:
is a mnemonic for the expanded form of this determinant: the sum of the products of three diagonal north-west to south-east lines of matrix elements, minus the sum of the products of three diagonal south-west to north-east lines of elements, when the copies of the first two columns of the matrix are
17199:
5422:
vector. These determinants are either 0 (by property 9) or else ±1 (by properties 1 and 12 below), so the linear combination gives the expression above in terms of the Levi-Civita symbol. While less technical in appearance, this characterization cannot entirely replace the
Leibniz formula
21219:
sends a small square (left, in red) to a distorted parallelogram (right, in red). The
Jacobian at a point gives the best linear approximation of the distorted parallelogram near that point (right, in translucent white), and the Jacobian determinant gives the ratio of the area of the approximating
2730:
3479:
from the third row third column. The signs are determined by how many transpositions of factors are necessary to arrange the factors in increasing order of their columns (given that the terms are arranged left-to-right in increasing row order): positive for an even number of transpositions and
3027:
16123:
5384:
If the determinant is defined using the
Leibniz formula as above, these three properties can be proved by direct inspection of that formula. Some authors also approach the determinant directly using these three properties: it can be shown that there is exactly one function that assigns to any
1509:
925:
20389:{\displaystyle W(f_{1},\ldots ,f_{n})(x)={\begin{vmatrix}f_{1}(x)&f_{2}(x)&\cdots &f_{n}(x)\\f_{1}'(x)&f_{2}'(x)&\cdots &f_{n}'(x)\\\vdots &\vdots &\ddots &\vdots \\f_{1}^{(n-1)}(x)&f_{2}^{(n-1)}(x)&\cdots &f_{n}^{(n-1)}(x)\end{vmatrix}}.}
3976:
25153:, in which several items can be repeated. The resulting sum has more terms than in the Leibniz rule, but in the process several of these products can be reused, making it more efficient than naively computing with the Leibniz rule. Algorithms can also be assessed according to their
22950:
18096:
9970:
18919:.) In this he used the word "determinant" in its present sense, summarized and simplified what was then known on the subject, improved the notation, and gave the multiplication theorem with a proof more satisfactory than Binet's. With him begins the theory in its generality.
6004:
column does not change the value of the determinant. This is a consequence of multilinearity and being alternative: by multilinearity the determinant changes by a multiple of the determinant of a matrix with two equal columns, which determinant is 0, since the determinant is
5980:
23402:
16112:
13653:
17382:
17731:
1945:{\displaystyle {\text{Signed area}}=|{\boldsymbol {u}}|\,|{\boldsymbol {v}}|\,\sin \,\theta =\left|{\boldsymbol {u}}^{\perp }\right|\,\left|{\boldsymbol {v}}\right|\,\cos \,\theta '={\begin{pmatrix}-b\\a\end{pmatrix}}\cdot {\begin{pmatrix}c\\d\end{pmatrix}}=ad-bc.}
18085:
4618:
4337:
762:
13032:
2593:
21022:
5016:
3961:
3458:
366:
17926:
5720:
23994:
For matrices with an infinite number of rows and columns, the above definitions of the determinant do not carry over directly. For example, in the
Leibniz formula, an infinite sum (all of whose terms are infinite products) would have to be calculated.
1106:
19562:{\displaystyle \det(A_{i})=\det {\begin{bmatrix}a_{1}&\ldots &b&\ldots &a_{n}\end{bmatrix}}=\sum _{j=1}^{n}x_{j}\det {\begin{bmatrix}a_{1}&\ldots &a_{i-1}&a_{j}&a_{i+1}&\ldots &a_{n}\end{bmatrix}}=x_{i}\det(A)}
26137:
In a non-commutative setting left-linearity (compatibility with left-multiplication by scalars) should be distinguished from right-linearity. Assuming linearity in the columns is taken to be left-linearity, one would have, for non-commuting scalars
18794:{\displaystyle {\begin{aligned}\nabla _{\mathbf {a} }\det(A)&=\mathbf {b} \times \mathbf {c} \\\nabla _{\mathbf {b} }\det(A)&=\mathbf {c} \times \mathbf {a} \\\nabla _{\mathbf {c} }\det(A)&=\mathbf {a} \times \mathbf {b} .\end{aligned}}}
25204:, is an exact-division method (so it does use division, but only in cases where these divisions can be performed without remainder) is of the same order, but the bit complexity is roughly the bit size of the original entries in the matrix times
17350:
11916:
13425:
7503:
precisely if its determinant is nonzero. This follows from the multiplicativity of the determinant and the formula for the inverse involving the adjugate matrix mentioned below. In this event, the determinant of the inverse matrix is given by
22389:
16838:
18407:
14293:
6388:
6797:
6702:
6607:
6513:
2359:{\displaystyle A{\begin{pmatrix}1\\0\\\vdots \\0\end{pmatrix}}=\mathbf {a} _{1},\quad A{\begin{pmatrix}0\\1\\\vdots \\0\end{pmatrix}}=\mathbf {a} _{2},\quad \ldots ,\quad A{\begin{pmatrix}0\\0\\\vdots \\1\end{pmatrix}}=\mathbf {a} _{n}.}
23746:
This definition proceeds by establishing the characteristic polynomial independently of the determinant, and defining the determinant as the lowest order term of this polynomial. This general definition recovers the determinant for the
21411:
11523:
8727:{\displaystyle {\begin{vmatrix}a&b&c\\d&e&f\\g&h&i\end{vmatrix}}=a{\begin{vmatrix}e&f\\h&i\end{vmatrix}}-b{\begin{vmatrix}d&f\\g&i\end{vmatrix}}+c{\begin{vmatrix}d&e\\g&h\end{vmatrix}}}
21471:
16861:
14915:{\displaystyle A_{k}:={\begin{bmatrix}a_{1,1}&a_{1,2}&\cdots &a_{1,k}\\a_{2,1}&a_{2,2}&\cdots &a_{2,k}\\\vdots &\vdots &\ddots &\vdots \\a_{k,1}&a_{k,2}&\cdots &a_{k,k}\end{bmatrix}}}
13919:
6265:
can be applied to bring any matrix into upper triangular form, and the steps in this algorithm affect the determinant in a controlled way. The following concrete example illustrates the computation of the determinant of the matrix
12630:
11382:
21816:
19691:
17044:
13246:
2963:{\displaystyle A={\begin{bmatrix}a_{1,1}&a_{1,2}&\cdots &a_{1,n}\\a_{2,1}&a_{2,2}&\cdots &a_{2,n}\\\vdots &\vdots &\ddots &\vdots \\a_{n,1}&a_{n,2}&\cdots &a_{n,n}\end{bmatrix}}.}
3254:{\displaystyle {\begin{vmatrix}a_{1,1}&a_{1,2}&\cdots &a_{1,n}\\a_{2,1}&a_{2,2}&\cdots &a_{2,n}\\\vdots &\vdots &\ddots &\vdots \\a_{n,1}&a_{n,2}&\cdots &a_{n,n}\end{vmatrix}}.}
16396:{\displaystyle \det(A)=\sum _{\begin{array}{c}k_{1},k_{2},\ldots ,k_{n}\geq 0\\k_{1}+2k_{2}+\cdots +nk_{n}=n\end{array}}\prod _{l=1}^{n}{\frac {(-1)^{k_{l}+1}}{l^{k_{l}}k_{l}!}}\operatorname {tr} \left(A^{l}\right)^{k_{l}},}
11157:
16634:
12825:
11777:
19221:
9585:
7758:
5725:
2470:
1350:
773:
13537:
12466:
6226:
8906:
Laplace expansion can be used iteratively for computing determinants, but this approach is inefficient for large matrices. However, it is useful for computing the determinants of highly symmetric matrix such as the
4196:{\displaystyle \det(A)={\begin{vmatrix}a_{1,1}\ldots a_{1,n}\\\vdots \qquad \vdots \\a_{n,1}\ldots a_{n,n}\end{vmatrix}}=\sum _{\sigma \in S_{n}}\operatorname {sgn}(\sigma )a_{1,\sigma (1)}\cdots a_{n,\sigma (n)}.}
1095:
23121:{\displaystyle {\begin{aligned}\bigwedge ^{n}T:\bigwedge ^{n}V&\rightarrow \bigwedge ^{n}V\\v_{1}\wedge v_{2}\wedge \dots \wedge v_{n}&\mapsto Tv_{1}\wedge Tv_{2}\wedge \dots \wedge Tv_{n}.\end{aligned}}}
12218:
8065:
201:
20527:, and hence has determinant of ±1 (since all the vectors have length 1). The determinant is +1 if and only if the basis has the same orientation. It is −1 if and only if the basis has the opposite orientation.
7601:
22716:
22173:
18813:. The determinant "determines" whether the system has a unique solution (which occurs precisely if the determinant is non-zero). In this sense, determinants were first used in the Chinese mathematics textbook
18273:{\displaystyle \det(A+\epsilon X)-\det(A)=\operatorname {tr} (\operatorname {adj} (A)X)\epsilon +O\left(\epsilon ^{2}\right)=\det(A)\operatorname {tr} \left(A^{-1}X\right)\epsilon +O\left(\epsilon ^{2}\right)}
9507:
9655:
8901:
8486:
24097:
21290:
14169:
9847:
1015:
639:
26154:
18592:
13499:
21217:
23278:
17524:{\displaystyle {\frac {n}{\operatorname {tr} \left(A^{-1}\right)}}\leq \det(A)^{\frac {1}{n}}\leq {\frac {1}{n}}\operatorname {tr} (A)\leq {\sqrt {{\frac {1}{n}}\operatorname {tr} \left(A^{2}\right)}}.}
15165:
15081:
11655:
24859:
24173:. For some classes of matrices with non-commutative elements, one can define the determinant and prove linear algebra theorems that are very similar to their commutative analogs. Examples include the
15940:
7244:
5377:
23914:
22033:
3697:
21156:
12288:
20826:
20626:
23799:
22532:
15321:
5548:
4800:
1635:, except that it is negative when the angle from the first to the second vector defining the parallelogram turns in a clockwise direction (which is opposite to the direction one would get for the
22955:
18641:
15358:
10567:
10010:
7151:
5021:
17633:
4397:
17957:
4459:
4216:
651:
19855:
12836:
7250:
This can be proven by inspecting the
Leibniz formula. This implies that in all the properties mentioned above, the word "column" can be replaced by "row" throughout. For example, viewing an
14495:
5272:{\displaystyle {\begin{aligned}|A|&={\big |}a_{1},\dots ,a_{j-1},r\cdot v+w,a_{j+1},\dots ,a_{n}|\\&=r\cdot |a_{1},\dots ,v,\dots a_{n}|+|a_{1},\dots ,w,\dots ,a_{n}|\end{aligned}}}
25100:
2475:
23610:
20933:
14013:
3760:
2654:
13961:
13732:
3839:
3305:
7996:
7953:
7910:
7824:
7687:
25198:
25147:
24906:
24463:
20681:
19733:
216:
19983:
17818:
17595:
16477:
22621:
18445:
17766:
9409:
7384:
24498:
14655:
6261:
These characterizing properties and their consequences listed above are both theoretically significant, but can also be used to compute determinants for concrete matrices. In fact,
5545:
Interchanging any pair of columns of a matrix multiplies its determinant by −1. This follows from the determinant being multilinear and alternating (properties 2 and 3 above):
5415:
a number that satisfies these three properties. This also shows that this more abstract approach to the determinant yields the same definition as the one using the
Leibniz formula.
1314:{\displaystyle {\begin{vmatrix}a&b+b'\\c&d+d'\end{vmatrix}}=a(d+d')-(b+b')c={\begin{vmatrix}a&b\\c&d\end{vmatrix}}+{\begin{vmatrix}a&b'\\c&d'\end{vmatrix}}.}
5494:
4883:
27383:
A history of mathematical notations: Including Vol. I. Notations in elementary mathematics; Vol. II. Notations mainly in higher mathematics, Reprint of the 1928 and 1929 originals
18511:
7607:
In particular, products and inverses of matrices with non-zero determinant (respectively, determinant one) still have this property. Thus, the set of such matrices (of fixed size
26708:
10485:
115:
The determinant is completely determined by the two following properties: the determinant of a product of matrices is the product of their determinants, and the determinant of a
8346:
8228:
24216:
9344:
6851:
4990:
4666:
3609:
24410:
by writing the matrix as a product of matrices whose determinants can be more easily computed. Such techniques are referred to as decomposition methods. Examples include the
23540:
23482:
23189:
23159:
22902:
13792:
4443:
24997:
23968:
22105:
23741:
22654:
17258:
12683:
24725:
22544:
of a complex matrix (which is also the determinant of its conjugate transpose) is the complex conjugate of its determinant, and for integer matrices: the reduction modulo
21940:
17617:
11786:
5418:
To see this it suffices to expand the determinant by multi-linearity in the columns into a (huge) linear combination of determinants of matrices in which each column is a
13257:
8092:
7851:
24245:
While the determinant can be computed directly using the
Leibniz rule this approach is extremely inefficient for large matrices, since that formula requires calculating
23270:
22305:
16710:
15901:
24315:
22206:
21979:
20852:
20756:
19915:
18540:
15249:
14080:
13532:
12508:
12365:
12331:
11186:
10531:
9835:
9809:
9783:
9757:
9689:
8761:
5520:
5409:
4704:
3557:
959:
22762:
22300:
21861:
18474:
18320:
14610:
14196:
7005:
6066:
25061:
24408:
21074:
19111:
13084:
9731:
8178:
7054:
6953:
6905:
6292:
6250:
3780:
3632:
3004:
2152:
25023:
24559:
23566:
19780:
7450:
6118:
6092:
23648:
23452:
19882:
19597:
19251:
18624:
11068:
6710:
6615:
6520:
6423:
4833:
3728:
24955:
21314:
20925:
20717:
19062:
13761:
21607:{\displaystyle \int _{f(U)}\phi (\mathbf {v} )\,d\mathbf {v} =\int _{U}\phi (f(\mathbf {u} ))\left|\det(\operatorname {D} f)(\mathbf {u} )\right|\,d\mathbf {u} .}
20443:
20009:
18312:
17022:{\displaystyle \det(I+A)=\sum _{k=0}^{\infty }{\frac {1}{k!}}\left(-\sum _{j=1}^{\infty }{\frac {(-1)^{j}}{j}}\operatorname {tr} \left(A^{j}\right)\right)^{k}\,,}
12041:
11241:
11215:
8523:
3829:
3806:
24771:
24748:
24341:
24266:
24165:
is commutative). Nevertheless, various notions of non-commutative determinant have been formulated that preserve some of the properties of determinants, notably
12005:
11982:
406:
26475:
Camarero, Cristóbal (2018-12-05). "Simple, Fast and
Practicable Algorithms for Cholesky, LU and QR Decomposition Using Fast Rectangular Matrix Multiplication".
25222:
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24665:
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24521:
24286:
23703:
23679:
23502:
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22250:
22226:
22077:
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19800:
19311:
19291:
19271:
19082:
17949:
17810:
17790:
17534:
These inequalities can be proved by expressing the traces and the determinant in terms of the eigenvalues. As such, they represent the well-known fact that the
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14403:
14383:
14363:
14343:
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11013:
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whose three factors are from the first, second and third columns respectively; this is an odd number of transpositions, so the term appears with negative sign.
2387:
1342:
18954:
The study of special forms of determinants has been the natural result of the completion of the general theory. Axisymmetric determinants have been studied by
12370:
6123:
13800:
25150:
17194:{\displaystyle \sum _{k=0}^{\infty }{\frac {1}{k!}}\left(-\sum _{j=1}^{\infty }{\frac {(-1)^{j}s^{j}}{j}}\operatorname {tr} \left(A^{j}\right)\right)^{k}\,,}
12516:
11289:
5997:
set), the determinant is 0. As a special case, this includes: if some column is such that all its entries are zero, then the determinant of that matrix is 0.
21676:
23407:
This definition agrees with the more concrete coordinate-dependent definition. This can be shown using the uniqueness of a multilinear alternating form on
19609:
12120:
26531:
13112:
27270:
23999:
provides different extensions of the determinant for such infinite-dimensional situations, which however only work for particular kinds of operators.
11076:
16513:
19018:. Of the textbooks on the subject Spottiswoode's was the first. In America, Hanus (1886), Weld (1893), and Muir/Metzler (1933) published treatises.
12691:
1504:{\displaystyle {\begin{vmatrix}r\cdot a&b\\r\cdot c&d\end{vmatrix}}=rad-brc=r(ad-bc)=r\cdot {\begin{vmatrix}a&b\\c&d\end{vmatrix}}.}
920:{\displaystyle \det {\begin{pmatrix}3&7\\1&-4\end{pmatrix}}={\begin{vmatrix}3&7\\1&{-4}\end{vmatrix}}=(3\cdot (-4))-(7\cdot 1)=-19.}
27552:
11679:
6228:
Indeed, such a matrix can be reduced, by appropriately adding multiples of the columns with fewer nonzero entries to those with more entries, to a
19123:
13534:
matrix algebras. For example, consider the complex numbers as a matrix algebra. The complex numbers have a representation as matrices of the form
9518:
7699:
1523:
The area of the parallelogram is the absolute value of the determinant of the matrix formed by the vectors representing the parallelogram's sides.
24238:, where for applications such as checking invertibility and finding eigenvalues the determinant has largely been supplanted by other techniques.
2407:
9294:
25839:
20491:, by using the square matrix whose columns are the given vectors. The determinant will be nonzero if and only if the sequence of vectors is a
3006:
etc. are, for many purposes, real or complex numbers. As discussed below, the determinant is also defined for matrices whose entries are in a
1624:
is in general a different parallelogram, but since the determinant is symmetric with respect to rows and columns, the area will be the same.)
18878:. He introduced the word "determinant" (Laplace had used "resultant"), though not in the present signification, but rather as applied to the
12223:
11586:
components, the formula allows quick calculation of the determinant of a matrix that differs from the identity matrix by a matrix of rank 1:
1023:
569:
1701:
between the vectors, which is simply base times height, the length of one vector times the perpendicular component of the other. Due to the
24366:. In this case, the determinant of the resulting row echelon form equals the determinant of the initial matrix. As a row echelon form is a
21897:
The above definition of the determinant using the
Leibniz rule holds works more generally when the entries of the matrix are elements of a
8004:
132:
26449:"... we mention that the determinant, though a convenient notion theoretically, rarely finds a useful role in numerical algorithms.", see
7510:
26677:
24343:. The Laplace expansion is similarly inefficient. Therefore, more involved techniques have been developed for calculating determinants.
22670:
22123:
12510:
matrices with complex entries, the determinant of the sum can be written in terms of determinants and traces in the following identity:
26954:
21946:
of real or complex numbers. Moreover, the characterization of the determinant as the unique alternating multilinear map that satisfies
9965:{\displaystyle \det {\begin{pmatrix}A&0\\C&D\end{pmatrix}}=\det(A)\det(D)=\det {\begin{pmatrix}A&B\\0&D\end{pmatrix}}.}
9417:
9600:
8793:
8378:
24016:
21242:
18815:
5975:{\displaystyle |a_{3},a_{1},a_{2},a_{4}\dots ,a_{n}|=-|a_{1},a_{3},a_{2},a_{4},\dots ,a_{n}|=|a_{1},a_{2},a_{3},a_{4},\dots ,a_{n}|.}
3463:
In this expression, each term has one factor from each row, all in different columns, arranged in increasing row order. For example,
27633:
23397:{\displaystyle \left(\bigwedge ^{n}T\right)\left(v_{1}\wedge \dots \wedge v_{n}\right)=\det(T)\cdot v_{1}\wedge \dots \wedge v_{n}.}
14115:
967:
591:
27681:
27081:
18545:
13436:
21172:
16107:{\displaystyle c_{n}=1;~~~c_{n-m}=-{\frac {1}{m}}\sum _{k=1}^{m}c_{n-m+k}\operatorname {tr} \left(A^{k}\right)~~(1\leq m\leq n)~.}
13648:{\displaystyle aI+b\mathbf {i} :=a{\begin{pmatrix}1&0\\0&1\end{pmatrix}}+b{\begin{pmatrix}0&-1\\1&0\end{pmatrix}}}
27614:
15096:
28014:
18819:(九章算術, Chinese scholars, around the 3rd century BCE). In Europe, solutions of linear systems of two equations were expressed by
15015:
11591:
9837:, respectively. The easiest such formula, which can be proven using either the Leibniz formula or a factorization involving the
7472:
is the identity matrix. The above-mentioned unique characterization of alternating multilinear maps therefore shows this claim.
2088:
474:
The above properties relating to rows (properties 2–4) may be replaced by the corresponding statements with respect to columns.
28072:
24799:
7194:
5296:
23811:
21987:
3637:
27335:
27295:
27250:
27195:
27115:
27105:
27050:
27000:
26666:
26434:
21079:
961:-matrices, and that continue to hold for determinants of larger matrices. They are as follows: first, the determinant of the
20781:
20581:
435:
Determinants can also be defined by some of their properties. Namely, the determinant is the unique function defined on the
23753:
22437:
17726:{\displaystyle {\frac {d\det(A)}{d\alpha }}=\operatorname {tr} \left(\operatorname {adj} (A){\frac {dA}{d\alpha }}\right).}
15264:
5442:
4736:
25644:
18911:
reduces to the multiplication theorem. On the same day (November 30, 1812) that Binet presented his paper to the Academy,
4680:
The determinant can be characterized by the following three key properties. To state these, it is convenient to regard an
25266:
25157:, i.e., how many bits of accuracy are needed to store intermediate values occurring in the computation. For example, the
18080:{\displaystyle {\frac {\partial \det(A)}{\partial A_{ij}}}=\operatorname {adj} (A)_{ji}=\det(A)\left(A^{-1}\right)_{ji}.}
7065:
4613:{\displaystyle \det(A)=\sum _{i_{1},i_{2},\ldots ,i_{n}}\varepsilon _{i_{1}\cdots i_{n}}a_{1,i_{1}}\!\cdots a_{n,i_{n}},}
4332:{\displaystyle \det(A)=\sum _{\sigma \in S_{n}}\left(\operatorname {sgn}(\sigma )\prod _{i=1}^{n}a_{i,\sigma (i)}\right)}
1963:
is the absolute value of the determinant of the matrix formed by the columns constructed from the vectors r1, r2, and r3.
757:{\displaystyle \det {\begin{pmatrix}a&b\\c&d\end{pmatrix}}={\begin{vmatrix}a&b\\c&d\end{vmatrix}}=ad-bc.}
27499:
25947:
18886:. Gauss also arrived at the notion of reciprocal (inverse) determinants, and came very near the multiplication theorem.
13027:{\displaystyle A_{11}A_{22}+B_{11}A_{22}+A_{11}B_{22}+B_{11}B_{22}-A_{12}A_{21}-B_{12}A_{21}-A_{12}B_{21}-B_{12}B_{21}.}
4349:
25103:
24123:
20445:
are linearly independent. If it can be shown that the Wronskian is zero everywhere on an interval then, in the case of
11392:
10985:
17:
15905:
3808:
if the permutation can be obtained with an even number of transpositions (exchanges of two entries); otherwise, it is
2588:{\displaystyle P=\left\{c_{1}\mathbf {a} _{1}+\cdots +c_{n}\mathbf {a} _{n}\mid 0\leq c_{i}\leq 1\ \forall i\right\}.}
516:), although other methods of solution are computationally much more efficient. Determinants are used for defining the
27486:
27456:
27390:
27372:
27232:
27214:
27150:
27095:
27068:
27032:
26974:
26831:
26774:
26754:
26554:
21017:{\displaystyle \operatorname {volume} (f(S))={\sqrt {\det \left(A^{\textsf {T}}A\right)}}\operatorname {volume} (S).}
19805:
1344:(i.e., all entries in that column are multiplied by that number), the determinant is also multiplied by that number:
26848:
Fisikopoulos, Vissarion; Peñaranda, Luis (2016), "Faster geometric algorithms via dynamic determinant computation",
18935:
for 1841 he specially treats this subject, as well as the class of alternating functions which Sylvester has called
14434:
3956:{\displaystyle A={\begin{bmatrix}a_{1,1}\ldots a_{1,n}\\\vdots \qquad \vdots \\a_{n,1}\ldots a_{n,n}\end{bmatrix}},}
3453:{\displaystyle {\begin{vmatrix}a&b&c\\d&e&f\\g&h&i\end{vmatrix}}=aei+bfg+cdh-ceg-bdi-afh.\ }
28126:
28062:
26850:
26658:
25066:
18809:
Historically, determinants were used long before matrices: A determinant was originally defined as a property of a
3281:
3269:
382:
27609:
23583:
13966:
3733:
2607:
361:{\displaystyle {\begin{vmatrix}a&b&c\\d&e&f\\g&h&i\end{vmatrix}}=aei+bfg+cdh-ceg-bdi-afh.}
28024:
27960:
27625:
24119:
18926:
17921:{\displaystyle {\frac {d\det(A)}{d\alpha }}=\det(A)\operatorname {tr} \left(A^{-1}{\frac {dA}{d\alpha }}\right).}
15909:
13927:
13698:
486:
7962:
7919:
7876:
7790:
7653:
27478:
26560:
26424:
25164:
25113:
24872:
24687:
can be quickly calculated, since they are the products of the respective diagonal entries. The determinant of
24429:
20651:
19699:
3021:), or it can be denoted directly in terms of the matrix entries by writing enclosing bars instead of brackets:
27309:
23512:
of vector spaces. Minors of a matrix can also be cast in this setting, by considering lower alternating forms
19924:
17565:
16422:
5715:{\displaystyle |a_{1},\dots ,a_{j},\dots a_{i},\dots ,a_{n}|=-|a_{1},\dots ,a_{i},\dots ,a_{j},\dots ,a_{n}|.}
27578:
27162:
25359:
25305:
24154:
22599:
18423:
17739:
9382:
7330:
24468:
22656:. Adding yet another layer of abstraction, this is captured by saying that the determinant is a morphism of
14618:
27802:
27674:
27643:
26085:
20450:
19744:
18983:
18890:
4851:
25677:
23484:(as opposed to the determinant associated to an endomorphism) is sometimes also called the determinant of
18482:
1555:
under the mapping. The parallelogram defined by the rows of the above matrix is the one with vertices at
27907:
27757:
27573:
27129:
27023:
21421:
18999:
17558:
The Leibniz formula shows that the determinant of real (or analogously for complex) square matrices is a
17366:
12292:
10430:
1616:
is the area of the parallelogram, and thus represents the scale factor by which areas are transformed by
935:
The determinant has several key properties that can be proved by direct evaluation of the definition for
24138:
For matrices over non-commutative rings, multilinearity and alternating properties are incompatible for
1967:
Thus the determinant gives the scaling factor and the orientation induced by the mapping represented by
27812:
27706:
27271:"Division-free algorithms for the determinant and the Pfaffian: algebraic and combinatorial approaches"
27187:
26789:
25622:
Lin, Minghua; Sra, Suvrit (2014). "Completely strong superadditivity of generalized matrix functions".
20500:
20474:
19032:
18810:
18477:
8293:
8183:
7409:, both sides of the equation are alternating and multilinear as a function depending on the columns of
2672:.) In particular, if the determinant is zero, then this parallelotope has volume zero and is not fully
2669:
2598:
561:
509:
26882:
Garibaldi, Skip (2004), "The characteristic polynomial and determinant are not ad hoc constructions",
24192:
18626:
are column vectors of length 3, then the gradient over one of the three vectors may be written as the
17345:{\displaystyle \operatorname {tr} \left(I-A^{-1}\right)\leq \log \det(A)\leq \operatorname {tr} (A-I)}
12043:
of two square matrices of the same size is not in general expressible in terms of the determinants of
9308:
6826:
4950:
4630:
3570:
28052:
27701:
25730:
Kondratyuk, L. A.; Krivoruchenko, M. I. (1992). "Superconducting quark matter in SU(2) color group".
23515:
23457:
23164:
23134:
22877:
21873:
is defined to be the determinant of the matrix describing it, with respect to an arbitrary choice of
19007:
18991:
15931:
14033:
13766:
11911:{\displaystyle \det(X+cr)=\det(X)\det \left(1+rX^{-1}c\right)=\det(X)+r\,\operatorname {adj} (X)\,c.}
8349:
7913:
5423:
in defining the determinant, since without it the existence of an appropriate function is not clear.
4410:
517:
24960:
23926:
22082:
18975:
13420:{\displaystyle (A_{11}+A_{22})(B_{11}+B_{22})-(A_{11}B_{11}+A_{12}B_{21}+A_{21}B_{12}+A_{22}B_{22})}
4449:
if two of the integers are equal, and otherwise as the signature of the permutation defined by the
28044:
27927:
27619:
27495:
25843:
25320:
25236:
24423:
24235:
23711:
22626:
22384:{\displaystyle \operatorname {GL} _{n}(f):\operatorname {GL} _{n}(R)\to \operatorname {GL} _{n}(S)}
21618:
20419:) in a specified interval if and only if the given functions and all their derivatives up to order
18916:
16833:{\displaystyle (AB)_{J}^{I}=\sum _{K}A_{K}^{I}B_{J}^{K},\operatorname {tr} (A)=\sum _{I}A_{I}^{I}.}
14669:
14665:
14521:
12642:
12055:
8098:
7765:
2685:
27655:
Determinants explained in an easy fashion in the 4th chapter as a part of a Linear Algebra course.
25377:
25310:
24908:
reached by decomposition methods has been improved by different methods. If two matrices of order
24710:
24170:
22727:
21923:
17600:
28121:
28090:
28019:
27797:
27667:
27568:
26413:
a contradiction. There is no useful notion of multi-linear functions over a non-commutative ring.
25797:
25363:
24115:
23577:
21225:
20012:
19735:
time, which is comparable to more common methods of solving systems of linear equations, such as
19003:
18402:{\displaystyle \det(I+\epsilon X)=1+\operatorname {tr} (X)\epsilon +O\left(\epsilon ^{2}\right).}
14288:{\displaystyle \det(A)=\prod _{i=1}^{n}\lambda _{i}=\lambda _{1}\lambda _{2}\cdots \lambda _{n}.}
8070:
7829:
5983:
3564:
432:
with the same determinant, equal to the product of the diagonal entries of the row echelon form.
25149:
is based on the following idea: one replaces permutations (as in the Leibniz rule) by so-called
23242:
21641:
The above identities concerning the determinant of products and inverses of matrices imply that
6383:{\displaystyle A={\begin{bmatrix}-2&-1&2\\2&1&4\\-3&3&-1\end{bmatrix}}.}
504:
Determinants occur throughout mathematics. For example, a matrix is often used to represent the
28116:
27854:
27787:
27777:
26634:
25921:
Institute de France in Paris on November 30, 1812, and which was subsequently published in the
25411:
24419:
24373:
So, the determinant can be computed for almost free from the result of a Gaussian elimination.
24294:
24239:
24103:
23983:
22593:
22185:
21949:
21881:. By the similarity invariance, this determinant is independent of the choice of the basis for
21874:
20831:
20722:
20555:
19894:
18940:
18519:
15216:
14994:
14172:
14059:
13511:
12487:
12344:
12310:
11165:
10510:
9814:
9788:
9762:
9736:
9668:
9357:
8740:
5499:
5388:
4683:
3536:
938:
97:
54:
27649:
25200:, but the bit length of intermediate values can become exponentially long. By comparison, the
22735:
22273:
21834:
18450:
14595:
6961:
6792:{\displaystyle E={\begin{bmatrix}18&-3&2\\0&3&4\\0&0&-1\end{bmatrix}}}
6697:{\displaystyle D={\begin{bmatrix}5&-3&2\\13&3&4\\0&0&-1\end{bmatrix}}}
6602:{\displaystyle C={\begin{bmatrix}-3&5&2\\3&13&4\\0&0&-1\end{bmatrix}}}
6508:{\displaystyle B={\begin{bmatrix}-3&-1&2\\3&1&4\\0&3&-1\end{bmatrix}}}
6035:
28111:
27869:
27864:
27859:
27792:
27737:
27179:
26601:"Dodgson condensation: The historical and mathematical development of an experimental method"
25816:
Campbell, H: "Linear Algebra With Applications", pages 111–112. Appleton Century Crofts, 1971
25028:
24384:
24234:
Determinants are mainly used as a theoretical tool. They are rarely calculated explicitly in
22564:
of the determinant of such a matrix is equal to the determinant of the matrix reduced modulo
21826:
21822:
21406:{\displaystyle D(f)=\left({\frac {\partial f_{i}}{\partial x_{j}}}\right)_{1\leq i,j\leq n}.}
21041:
20539:
19918:
19917:-matrix consisting of the three vectors is zero. The same idea is also used in the theory of
19087:
18861:
17252:, the trace operator gives the following tight lower and upper bounds on the log determinant
13060:
9698:
8150:
7777:
7013:
6912:
6864:
6235:
3765:
3617:
2976:
2661:
2401:
2073:
26929:
25565:
Sothanaphan, Nat (January 2017). "Determinants of block matrices with noncommuting blocks".
25002:
24529:
23545:
19782:
is zero if and only if the column vectors (or, equivalently, the row vectors) of the matrix
19762:
7432:
6097:
6071:
3268:, i.e. one with the same number of rows and columns: the determinant can be defined via the
27879:
27844:
27831:
27722:
27530:
27466:
27400:
27305:
27169:
27010:
26921:
25739:
25610:
25271:
25158:
24351:
24007:
24003:
23626:
23613:
23430:
22661:
22229:
22179:
21417:
19860:
19575:
19229:
18979:
18871:
18597:
14589:
12473:
11041:
9346:
7694:
7648:
6262:
5994:
5436:
4811:
3706:
3514:
written beside it as in the illustration. This scheme for calculating the determinant of a
565:
490:
425:
50:
26930:"A condensation-based application of Cramer's rule for solving large-scale linear systems"
25897:
24931:
22259:
20901:
20693:
19038:
13737:
11518:{\displaystyle \det \left(I_{\mathit {m}}+AB\right)=\det \left(I_{\mathit {n}}+BA\right),}
8:
28057:
27937:
27912:
27762:
27630:
27605:
25519:
24355:
24178:
24127:
23996:
23659:
21943:
20422:
19988:
19756:
18931:
18893:(1811, 1812), who formally stated the theorem relating to the product of two matrices of
18864:(1773) treated determinants of the second and third order and applied it to questions of
18291:
17624:
17370:
12020:
11431:
have dimensions allowing them to be multiplied in either order forming a square matrix):
11220:
11194:
8502:
7496:
3811:
3785:
2693:
25743:
24753:
24730:
24323:
24248:
21628:, the determinant can be used to measure the rate of expansion of a map near the poles.
14028:
The determinant is closely related to two other central concepts in linear algebra, the
13914:{\displaystyle \det(aI+b\mathbf {i} )=a^{2}\det(I)+b^{2}\det(\mathbf {i} )=a^{2}+b^{2}.}
11987:
11964:
388:
27767:
27519:
27408:
27356:
26909:
26891:
26859:
26746:
26700:
26476:
25755:
25623:
25592:
25574:
25547:
25539:
25404:
25315:
25300:
25291:
25207:
24911:
24776:
24690:
24670:
24650:
24630:
24610:
24590:
24570:
24565:
24506:
24271:
23688:
23664:
23487:
23410:
23222:
23194:
22927:
22907:
22854:
22834:
22814:
22787:
22767:
22585:
22567:
22547:
22414:
22394:
22253:
22235:
22211:
22115:
22062:
22056:
22038:
21903:
21306:
20877:
20857:
20761:
20631:
20507:. In the case of an orthogonal basis, the magnitude of the determinant is equal to the
20462:
20402:
19785:
19296:
19276:
19256:
19067:
18865:
17934:
17795:
17775:
15175:
14968:
14948:
14928:
14571:
14547:
14527:
14503:
14408:
14388:
14368:
14348:
14328:
14308:
14299:
14089:
14039:
13678:
13658:
13089:
13040:
12625:{\displaystyle \det(A+B)=\det(A)+\det(B)+{\text{tr}}(A){\text{tr}}(B)-{\text{tr}}(AB).}
12100:
12080:
12060:
11944:
11924:
11377:{\displaystyle \det {\begin{pmatrix}A&B\\B&A\end{pmatrix}}=\det(A-B)\det(A+B).}
11266:
11246:
11018:
10998:
10539:
10490:
9978:
9591:
8908:
8768:
8355:
8273:
8253:
8233:
8123:
7856:
7784:
7630:
7610:
7478:
7455:
7412:
7392:
7303:
7283:
7167:
6397:
6269:
6011:
5525:
4945:
4927:
4888:
4713:
4344:
2697:
2372:
1327:
421:
109:
27121:
26803:
26784:
21811:{\displaystyle \det(A)=\det(X)^{-1}\det(B)\det(X)=\det(B)\det(X)^{-1}\det(X)=\det(B).}
21165:
11188:
blocks, again under appropriate commutativity conditions among the individual blocks.
5722:
This formula can be applied iteratively when several columns are swapped. For example
27965:
27922:
27849:
27742:
27586:
27546:
27534:
27523:
27482:
27452:
27386:
27368:
27331:
27291:
27246:
27228:
27210:
27191:
27146:
27111:
27091:
27077:
27064:
27046:
27028:
26996:
26970:
26837:
26827:
26770:
26662:
26550:
26539:
Proceedings of the 1997 international symposium on Symbolic and algebraic computation
26430:
25948:
http://www-history.mcs.st-and.ac.uk/history/HistTopics/Matrices_and_determinants.html
25759:
25596:
25285:
25201:
24367:
22541:
22268:
22108:
21642:
21036:
20520:
20512:
20446:
19686:{\displaystyle A\,\operatorname {adj} (A)=\operatorname {adj} (A)\,A=\det(A)\,I_{n}.}
18875:
18874:(1801) made the next advance. Like Lagrange, he made much use of determinants in the
18831:
9996:
8117:
7500:
6029:
5982:
Yet more generally, any permutation of the columns multiplies the determinant by the
1705:
this already is the signed area, yet it may be expressed more conveniently using the
1519:
573:
498:
478:
417:
116:
105:
26750:
25551:
25274:. Unfortunately this interesting method does not always work in its original form.
13241:{\displaystyle \det(A)+\det(B)+A_{11}B_{22}+B_{11}A_{22}-A_{12}B_{21}-B_{12}A_{21}.}
1971:. When the determinant is equal to one, the linear mapping defined by the matrix is
27970:
27874:
27727:
27511:
27436:
27360:
27281:
27018:
26988:
26944:
26901:
26869:
26808:
26798:
26738:
26692:
26612:
26542:
25893:
25747:
25656:
25584:
25531:
24415:
24411:
24359:
24166:
23617:
22872:
22809:
21898:
20684:
19885:
19740:
19736:
19114:
18820:
17547:
15199:
14661:
12304:
12300:
9838:
8109:
whose entries are the determinants of all quadratic submatrices of a given matrix.
5287:
3007:
1739:
becomes the signed area in question, which can be determined by the pattern of the
513:
429:
27654:
27640:
Compute determinants of matrices up to order 6 using Laplace expansion you choose.
28029:
27822:
27782:
27772:
27637:
27589:
27475:
Companion Encyclopedia of the History and Philosophy of the Mathematical Sciences
27462:
27396:
27301:
27166:
27006:
26917:
26874:
26719:
23971:
23621:
22657:
22589:
22035:
is invertible (in the sense that there is an inverse matrix whose entries are in
21229:
20524:
18856:
gave the general method of expanding a determinant in terms of its complementary
17543:
16851:
expansion of the logarithm when the expansion converges. If every eigenvalue of
16848:
16483:
9377:
8106:
7956:
7773:
6229:
4916:
4906:
3700:
1636:
1632:
962:
450:
11152:{\displaystyle \det {\begin{pmatrix}A&B\\C&D\end{pmatrix}}=\det(AD-BC).}
28034:
27955:
27690:
27158:
25154:
24318:
24223:
23748:
20575:
20571:
20504:
19011:
18967:
18883:
18857:
18285:
17620:
17539:
16629:{\displaystyle \det(A)={\frac {(-1)^{n}}{n!}}B_{n}(s_{1},s_{2},\ldots ,s_{n}).}
14083:
8144:
7999:
7321:
5419:
3967:
3510:
3503:
3264:
There are various equivalent ways to define the determinant of a square matrix
1960:
1740:
1536:
541:
101:
27364:
26992:
26949:
26617:
25588:
25353:
24773:
for an odd number of permutations). Once such a LU decomposition is known for
21981:
still holds, as do all the properties that result from that characterization.
12820:{\displaystyle (A_{11}+B_{11})(A_{22}+B_{22})-(A_{12}+B_{12})(A_{21}+B_{21}).}
9691:-matrix above continues to hold, under appropriate further assumptions, for a
7783:
Because the determinant respects multiplication and inverses, it is in fact a
28105:
28067:
27990:
27950:
27917:
27897:
27515:
27280:, Lecture Notes in Comput. Sci., vol. 2122, Springer, pp. 119–135,
26841:
26650:
25469:
25261:
24186:
23509:
23505:
21159:
18944:
18827:
18627:
17535:
14018:
11772:{\displaystyle \det(X+AB)=\det(X)\det \left(I_{\mathit {n}}+BX^{-1}A\right),}
11035:
4993:
4806:
4453:
tuple of integers. With the Levi-Civita symbol, the Leibniz formula becomes
2713:
1972:
1955:
1548:
549:
58:
27286:
26678:"Sylvester's Identity and Multistep Integer-Preserving Gaussian Elimination"
25661:
19802:
are linearly dependent. For example, given two linearly independent vectors
19216:{\displaystyle x_{i}={\frac {\det(A_{i})}{\det(A)}}\qquad i=1,2,3,\ldots ,n}
9580:{\displaystyle (\det A)I=A\operatorname {adj} A=(\operatorname {adj} A)\,A.}
7753:{\displaystyle \operatorname {SL} _{n}(K)\subset \operatorname {GL} _{n}(K)}
1682:
representing the parallelogram's sides. The signed area can be expressed as
556:-dimensional volume are transformed under the endomorphism. This is used in
470:
Adding a multiple of one row to another row does not change the determinant.
28000:
27889:
27839:
27732:
24317:-matrix. Thus, the number of required operations grows very quickly: it is
23979:
21867:
18963:
18879:
17205:
14110:
9692:
7761:
2465:{\displaystyle \mathbf {a} _{1},\mathbf {a} _{2},\ldots ,\mathbf {a} _{n},}
545:
482:
26742:
26723:
26546:
25983:
8105:
matrices. This formula can also be recast as a multiplicative formula for
27980:
27902:
27747:
24362:. One can restrict the computation to elementary matrices of determinant
24219:
23801:, but also includes several further cases including the determinant of a
22805:
21625:
21429:
21028:
19888:
18849:
18843:
18413:
16704:. The product and trace of such matrices are defined in a natural way as
12461:{\displaystyle {\sqrt{\det(A+B)}}\geq {\sqrt{\det(A)}}+{\sqrt{\det(B)}},}
9999:, then it follows with results from the section on multiplicativity that
8763:-matrices gives back the Leibniz formula mentioned above. Similarly, the
6221:{\displaystyle \det(A)=a_{11}a_{22}\cdots a_{nn}=\prod _{i=1}^{n}a_{ii}.}
4207:
3699:
exhausting the entire set. The set of all such permutations, called the
3560:
2063:
1628:
1627:
The absolute value of the determinant together with the sign becomes the
1552:
505:
42:
31:
30:
This article is about mathematics. For determinants in epidemiology, see
27440:
19313:. This follows immediately by column expansion of the determinant, i.e.
19010:; compound determinants by Sylvester, Reiss, and Picquet; Jacobians and
18951:
introduced the modern notation for the determinant using vertical bars.
5291:: whenever two columns of a matrix are identical, its determinant is 0:
467:
Multiplying a row by a number multiplies the determinant by this number.
381:
matrix can be defined in several equivalent ways, the most common being
28009:
27752:
27175:
26913:
26704:
25751:
25543:
25436:
24182:
23921:
23802:
22107:, this means that the determinant is +1 or −1. Such a matrix is called
20458:
18959:
18860:: Vandermonde had already given a special case. Immediately following,
17559:
15005:
and also equals the sum of the eigenvalues. Thus, for complex matrices
14672:
asserts that this is equivalent to the determinants of the submatrices
14565:
14029:
7760:. More generally, the word "special" indicates the subgroup of another
1532:
1090:{\displaystyle {\begin{vmatrix}a&b\\a&b\end{vmatrix}}=ab-ba=0.}
521:
93:
26785:"Triangular Factorization and Inversion by Fast Matrix Multiplication"
26541:. ISSAC '97. Kihei, Maui, Hawaii, United States: ACM. pp. 28–31.
20503:
of the basis is consistent with or opposite to the orientation of the
12007:
have the same characteristic polynomials (hence the same eigenvalues).
27807:
27594:
26896:
26813:
24288:
24242:, however, does frequently use calculations related to determinants.
24145:, so there is no good definition of the determinant in this setting.
23682:
22252:. Since it respects the multiplication in both groups, this map is a
21032:
20895:
20454:
20016:
18995:
18971:
18417:
12213:{\displaystyle \det(A+B+C)+\det(C)\geq \det(A+C)+\det(B+C){\text{,}}}
9590:
Thus the adjugate matrix can be used for expressing the inverse of a
8140:
8060:{\displaystyle \operatorname {GL} _{n}(K)/\operatorname {SL} _{n}(K)}
7161:
3612:
2390:
1652:
is the signed area, one may consider a matrix containing two vectors
409:
196:{\displaystyle {\begin{vmatrix}a&b\\c&d\end{vmatrix}}=ad-bc,}
26905:
26696:
26600:
25535:
14564:. By means of this polynomial, determinants can be used to find the
11162:
This formula has been generalized to matrices composed of more than
27975:
27418:
Cayley, Arthur (1841), "On a theorem in the geometry of position",
26481:
25935:
25579:
23975:
23580:
of a matrix is defined as the determinant, except that the factors
22537:
holds. In other words, the displayed commutative diagram commutes.
20499:. In that case, the sign of the determinant determines whether the
20480:
18987:
18955:
17769:
7690:
7596:{\displaystyle \det \left(A^{-1}\right)={\frac {1}{\det(A)}}=^{-1}}
6120:, then its determinant equals the product of the diagonal entries:
3533:
Generalizing the above to higher dimensions, the determinant of an
2668:. (The sign shows whether the transformation preserves or reverses
1980:
557:
525:
497:
of a linear endomorphism, which does not depend on the choice of a
26864:
25645:"Inequalities of Generalized Matrix Functions via Tensor Products"
25628:
22711:{\displaystyle \det :\operatorname {GL} _{n}\to \mathbb {G} _{m}.}
22168:{\displaystyle \operatorname {GL} _{n}(R)\rightarrow R^{\times },}
16639:
This formula can also be used to find the determinant of a matrix
16482:
The formula can be expressed in terms of the complete exponential
7389:
This key fact can be proven by observing that, for a fixed matrix
27659:
27136:, Revised and enlarged by William H. Metzler, New York, NY: Dover
25890:
The Theory of Determinants in the historical Order of Development
24370:, its determinant is the product of the entries of its diagonal.
22851:
can be formulated in a coordinate-free manner by considering the
20574:
of the determinant of real vectors is equal to the volume of the
20479:
The determinant can be thought of as assigning a number to every
18925:
used the functional determinant which Sylvester later called the
14305:
From this, one immediately sees that the determinant of a matrix
9502:{\displaystyle (\operatorname {adj} (A))_{i,j}=(-1)^{i+j}M_{ji}.}
2676:-dimensional, which indicates that the dimension of the image of
35:
25106:. This exponent has been further lowered, as of 2016, to 2.373.
24607:
in each column, and otherwise zeros), a lower triangular matrix
9650:{\displaystyle A^{-1}={\frac {1}{\det A}}\operatorname {adj} A.}
8896:{\displaystyle \det(A)=\sum _{i=1}^{n}(-1)^{i+j}a_{i,j}M_{i,j}.}
8481:{\displaystyle \det(A)=\sum _{j=1}^{n}(-1)^{i+j}a_{i,j}M_{i,j},}
6252:
which gives a non-zero contribution is the identity permutation.
1100:
This holds similarly if the two columns are the same. Moreover,
1017:
is 1. Second, the determinant is zero if two rows are the same:
92:. Its value characterizes some properties of the matrix and the
27985:
25025:, then there is an algorithm computing the determinant in time
24118:, one may define a positive real-valued determinant called the
24092:{\displaystyle \det(I+A)=\exp(\operatorname {tr} (\log(I+A))).}
23653:
21285:{\displaystyle f:\mathbf {R} ^{n}\rightarrow \mathbf {R} ^{n},}
20687:
20449:, this implies the given functions are linearly dependent. See
18912:
18852:(1771) first recognized determinants as independent functions.
15904:. Such expressions are deducible from combinatorial arguments,
5989:
If some column can be expressed as a linear combination of the
3499:
1706:
1631:
of the parallelogram. The signed area is the same as the usual
544:
is expressed by a determinant, and the determinant of a linear
533:
416:) signed products of matrix entries. It can be computed by the
27535:"Recherches sur le calcul intégral et sur le systéme du monde"
26532:"On the worst-case complexity of integer Gaussian elimination"
24226:
form the class closest to matrices with commutative elements.
23191:
is given by multiplying with some scalar, i.e., an element in
14164:{\displaystyle \lambda _{1},\lambda _{2},\ldots ,\lambda _{n}}
1984:
is related to these ideas. In 2D, it can be interpreted as an
1010:{\displaystyle {\begin{pmatrix}1&0\\0&1\end{pmatrix}}}
634:{\displaystyle {\begin{pmatrix}a&b\\c&d\end{pmatrix}}}
493:
does not depend on the chosen basis. This allows defining the
27227:, Undergraduate Texts in Mathematics (3 ed.), Springer,
27209:, Undergraduate Texts in Mathematics (2 ed.), Springer,
19015:
18587:{\displaystyle A={\begin{bmatrix}a&b&c\end{bmatrix}}}
13494:{\displaystyle {\text{tr}}(A){\text{tr}}(B)-{\text{tr}}(AB).}
4404:
25102:
algorithm for computing the determinant exists based on the
24133:
21212:{\displaystyle f\colon \mathbf {R} ^{2}\to \mathbf {R} ^{2}}
21158:, or any other combination of pairs of vertices that form a
19891:
by the former two vectors exactly if the determinant of the
14023:
24122:
using the canonical trace. In fact, corresponding to every
18838:
stated, without proof, Cramer's rule. Both Cramer and also
18826:
Determinants proper originated separately from the work of
17365:. This relationship can be derived via the formula for the
15160:{\displaystyle \operatorname {tr} (A)=\log(\det(\exp(A))).}
14182:
times in this list.) Then, it turns out the determinant of
8143:
in terms of determinants of smaller matrices, known as its
1709:
of the complementary angle to a perpendicular vector, e.g.
1702:
645:" or by vertical bars around the matrix, and is defined as
100:, by the matrix. In particular, the determinant is nonzero
25270:
fame) invented a method for computing determinants called
24102:
Another infinite-dimensional notion of determinant is the
23576:
Determinants as treated above admit several variants: the
22431:. The determinant respects these maps, i.e., the identity
21670:. Indeed, repeatedly applying the above identities yields
15076:{\displaystyle \det(\exp(A))=\exp(\operatorname {tr} (A))}
14298:
The product of all non-zero eigenvalues is referred to as
14019:
Properties of the determinant in relation to other notions
11650:{\displaystyle \det \left(I_{\mathit {m}}+cr\right)=1+rc.}
9411:
is the transpose of the matrix of the cofactors, that is,
8499:. For example, the Laplace expansion along the first row (
1547:. In either case, the images of the basis vectors form a
27245:. Graduate Texts in Mathematics. New York, NY: Springer.
27110:, Society for Industrial and Applied Mathematics (SIAM),
24854:{\displaystyle \det(A)=\varepsilon \det(L)\cdot \det(U).}
21892:
12639:
This can be shown by writing out each term in components
7239:{\displaystyle \det \left(A^{\textsf {T}}\right)=\det(A)}
5372:{\displaystyle |a_{1},\dots ,v,\dots ,v,\dots ,a_{n}|=0.}
512:, and determinants can be used to solve these equations (
27646:
Calculator for matrix determinants, up to the 8th order.
27161:(1947) "Some identities in the theory of determinants",
25837:
A Brief History of Linear Algebra and Matrix Theory at:
25611:
http://www.ee.ic.ac.uk/hp/staff/dmb/matrix/proof003.html
23909:{\displaystyle \det(a+ib+jc+kd)=a^{2}+b^{2}+c^{2}+d^{2}}
22028:{\displaystyle A\in \operatorname {Mat} _{n\times n}(R)}
20628:
is the linear map given by multiplication with a matrix
19031:
Determinants can be used to describe the solutions of a
3692:{\displaystyle \sigma (1),\sigma (2),\ldots ,\sigma (n)}
489:, the determinant of the matrix that represents it on a
27433:
Introduction à l'analyse des lignes courbes algébriques
25729:
23454:. For this reason, the highest non-zero exterior power
21151:{\displaystyle {\frac {1}{6}}\cdot |\det(a-b,b-c,c-d)|}
20511:
of the lengths of the basis vectors. For instance, an
12283:{\displaystyle \det(A+B)\geq \det(A)+\det(B){\text{.}}}
460:
The exchange of two rows multiplies the determinant by
27650:
Matrices and Linear Algebra on the Earliest Uses Pages
26356:
26311:
26269:
26223:
26181:
23571:
20821:{\displaystyle f:\mathbf {R} ^{n}\to \mathbf {R} ^{m}}
20621:{\displaystyle f:\mathbf {R} ^{n}\to \mathbf {R} ^{n}}
20084:
19443:
19353:
18868:; he proved many special cases of general identities.
18560:
18476:. The above formula shows that its Lie algebra is the
16147:
15001:) is by definition the sum of the diagonal entries of
14700:
14086:
entries. Then, by the Fundamental Theorem of Algebra,
13611:
13569:
11569:
From this general result several consequences follow.
11301:
11088:
10835:
10679:
10636:
10578:
10278:
10122:
10079:
10021:
9928:
9859:
9313:
8922:
8693:
8651:
8609:
8540:
6725:
6630:
6535:
6438:
6307:
4675:
4000:
3854:
3314:
3036:
2745:
2303:
2234:
2172:
1900:
1868:
1467:
1359:
1267:
1228:
1115:
1032:
976:
827:
785:
702:
663:
600:
225:
141:
26847:
26583:
26506:
26462:
26152:
25840:"A Brief History of Linear Algebra and Matrix Theory"
25468:, §III.8, Proposition 1 proves this result using the
25210:
25167:
25116:
25069:
25031:
25005:
24963:
24934:
24914:
24875:
24802:
24779:
24756:
24733:
24713:
24693:
24673:
24653:
24633:
24613:
24593:
24573:
24532:
24509:
24471:
24432:
24387:
24326:
24297:
24274:
24251:
24195:
24019:
23929:
23814:
23794:{\displaystyle A=\operatorname {Mat} _{n\times n}(F)}
23756:
23714:
23691:
23667:
23629:
23586:
23548:
23518:
23490:
23460:
23433:
23413:
23281:
23245:
23225:
23197:
23167:
23137:
22953:
22930:
22910:
22880:
22857:
22837:
22817:
22790:
22770:
22738:
22673:
22629:
22602:
22570:
22550:
22527:{\displaystyle f(\det((a_{i,j})))=\det((f(a_{i,j})))}
22440:
22417:
22397:
22308:
22276:
22238:
22214:
22188:
22126:
22085:
22065:
22041:
21990:
21952:
21926:
21906:
21837:
21679:
21474:
21317:
21245:
21175:
21082:
21044:
21031:
bounded by four points, they can be used to identify
20936:
20904:
20880:
20860:
20834:
20784:
20764:
20725:
20696:
20654:
20634:
20584:
20425:
20405:
20028:
19991:
19927:
19897:
19863:
19808:
19788:
19765:
19702:
19612:
19578:
19322:
19299:
19279:
19259:
19232:
19126:
19090:
19070:
19041:
18639:
18600:
18548:
18522:
18485:
18453:
18426:
18323:
18294:
18099:
17960:
17937:
17821:
17798:
17778:
17742:
17636:
17603:
17568:
17385:
17261:
17047:
16864:
16713:
16516:
16425:
16126:
15943:
15356:
15316:{\displaystyle \det(A)=\exp(\operatorname {tr} (L)).}
15267:
15219:
15099:
15018:
14971:
14951:
14931:
14681:
14621:
14598:
14574:
14550:
14530:
14506:
14437:
14411:
14391:
14371:
14351:
14331:
14311:
14199:
14118:
14092:
14062:
14042:
13969:
13930:
13803:
13769:
13740:
13701:
13681:
13661:
13540:
13514:
13439:
13260:
13115:
13092:
13063:
13043:
12839:
12694:
12645:
12519:
12490:
12373:
12347:
12313:
12226:
12123:
12103:
12083:
12063:
12023:
11990:
11967:
11947:
11927:
11789:
11682:
11594:
11440:
11292:
11269:
11249:
11223:
11197:
11168:
11079:
11044:
11021:
11001:
10565:
10542:
10513:
10493:
10433:
10008:
9981:
9850:
9817:
9791:
9765:
9739:
9701:
9671:
9603:
9521:
9420:
9385:
9311:
9293:-term Laplace expansion along a row or column can be
8916:
8796:
8771:
8743:
8534:
8505:
8381:
8358:
8296:
8276:
8256:
8236:
8186:
8153:
8126:
8073:
8007:
7965:
7922:
7879:
7873:. This homomorphism is surjective and its kernel is
7859:
7832:
7793:
7764:
of matrices of determinant one. Examples include the
7702:
7656:
7633:
7613:
7513:
7481:
7458:
7435:
7415:
7395:
7333:
7306:
7286:
7271:
7197:
7170:
7068:
7016:
6964:
6915:
6867:
6829:
6713:
6618:
6523:
6426:
6400:
6295:
6272:
6238:
6126:
6100:
6074:
6038:
6014:
5728:
5551:
5528:
5502:
5445:
5391:
5299:
5019:
4953:
4930:
4891:
4854:
4814:
4795:{\displaystyle A={\big (}a_{1},\dots ,a_{n}{\big )},}
4739:
4716:
4686:
4633:
4462:
4413:
4352:
4219:
3979:
3842:
3814:
3788:
3768:
3736:
3709:
3640:
3620:
3573:
3539:
3308:
3030:
2979:
2733:
2610:
2478:
2410:
2375:
2163:
2076:
1762:
1353:
1330:
1109:
1026:
970:
941:
776:
654:
594:
391:
219:
135:
27584:
25366:
from the original on 2021-12-11 – via YouTube.
25281:
24109:
23986:, also arise as special cases of this construction.
21228:, much of the above carries over by considering the
16406:
where the sum is taken over the set of all integers
16117:
In the general case, this may also be obtained from
1324:
Finally, if any column is multiplied by some number
22114:The determinant being multiplicative, it defines a
19113:is nonzero. In this case, the solution is given by
19014:by Sylvester; and symmetric gauche determinants by
7912:(the matrices with determinant one). Hence, by the
7146:{\displaystyle |A|=-|E|=-(18\cdot 3\cdot (-1))=54.}
3518:matrix does not carry over into higher dimensions.
1527:If the matrix entries are real numbers, the matrix
26405:
25642:
25403:
25216:
25192:
25141:
25094:
25055:
25017:
24991:
24949:
24920:
24900:
24853:
24785:
24765:
24742:
24719:
24699:
24679:
24659:
24647:. The determinants of the two triangular matrices
24639:
24619:
24599:
24579:
24553:
24515:
24492:
24457:
24402:
24335:
24309:
24280:
24260:
24210:
24130:there is a notion of Fuglede−Kadison determinant.
24091:
23962:
23908:
23793:
23735:
23697:
23673:
23642:
23604:
23560:
23534:
23496:
23476:
23446:
23419:
23396:
23264:
23231:
23203:
23183:
23153:
23120:
22936:
22916:
22896:
22863:
22843:
22823:
22796:
22776:
22756:
22710:
22648:
22615:
22576:
22556:
22526:
22423:
22403:
22383:
22294:
22244:
22220:
22200:
22167:
22099:
22071:
22047:
22027:
21973:
21934:
21912:
21855:
21810:
21653:are similar, if there exists an invertible matrix
21606:
21405:
21284:
21211:
21150:
21068:
21016:
20919:
20886:
20866:
20846:
20820:
20770:
20750:
20711:
20675:
20640:
20620:
20565:
20437:
20411:
20388:
20003:
19977:
19909:
19876:
19849:
19794:
19774:
19727:
19685:
19591:
19561:
19305:
19285:
19265:
19245:
19215:
19105:
19076:
19056:
18793:
18618:
18586:
18534:
18505:
18468:
18439:
18401:
18306:
18272:
18079:
17943:
17920:
17804:
17784:
17760:
17725:
17611:
17589:
17523:
17344:
17193:
17021:
16832:
16628:
16471:
16395:
16106:
15889:
15315:
15243:
15159:
15075:
14977:
14957:
14937:
14914:
14649:
14604:
14580:
14556:
14536:
14512:
14489:
14417:
14397:
14377:
14357:
14337:
14317:
14287:
14163:
14098:
14074:
14048:
14007:
13955:
13913:
13786:
13755:
13726:
13687:
13667:
13647:
13526:
13493:
13419:
13240:
13098:
13078:
13049:
13026:
12819:
12677:
12624:
12502:
12460:
12359:
12325:
12282:
12212:
12109:
12089:
12069:
12035:
11999:
11976:
11953:
11933:
11910:
11771:
11649:
11517:
11387:
11376:
11275:
11255:
11235:
11209:
11180:
11151:
11062:
11027:
11007:
10973:
10548:
10525:
10499:
10479:
10416:
9987:
9964:
9829:
9803:
9777:
9751:
9725:
9683:
9649:
9579:
9501:
9403:
9338:
9281:
8895:
8777:
8755:
8726:
8517:
8480:
8364:
8340:
8282:
8262:
8242:
8222:
8172:
8132:
8086:
8059:
7990:
7947:
7904:
7865:
7845:
7818:
7752:
7681:
7639:
7619:
7595:
7487:
7464:
7444:
7421:
7401:
7378:
7312:
7292:
7238:
7176:
7145:
7048:
6999:
6947:
6899:
6845:
6791:
6696:
6601:
6507:
6406:
6382:
6278:
6244:
6220:
6112:
6086:
6060:
6020:
5974:
5714:
5534:
5514:
5488:
5403:
5371:
5271:
4984:
4936:
4897:
4877:
4839:) is composed of the entries of the matrix in the
4827:
4794:
4722:
4698:
4660:
4612:
4437:
4392:{\displaystyle \varepsilon _{i_{1},\ldots ,i_{n}}}
4391:
4331:
4195:
3966:the Leibniz formula for its determinant is, using
3955:
3823:
3800:
3774:
3754:
3722:
3691:
3626:
3603:
3551:
3452:
3253:
2998:
2962:
2692:produces a linear transformation which is neither
2648:
2587:
2464:
2381:
2358:
2146:
1944:
1503:
1336:
1313:
1089:
1009:
953:
919:
756:
633:
445:matrices that has the four following properties:
400:
360:
195:
27603:
26426:Supersymmetry for mathematicians: An introduction
25443:, 2nd Edition, Addison-Wesley, 1971, pp 173, 191.
21636:
14171:. (Here it is understood that an eigenvalue with
4580:
1988:formed by imagining two vectors each with origin
28103:
27472:
27325:
26967:Mathematics for Physical Science and Engineering
26643:Elementary Linear Algebra (Applications Version)
26450:
25784:
25358:(video lecture). WildLinAlg. Sydney, Australia:
24836:
24821:
24803:
24388:
24161:as value on some pair of arguments implies that
24020:
24010:by an appropriate generalization of the formula
23815:
23715:
23504:and similarly for more involved objects such as
23347:
22674:
22484:
22447:
21953:
21793:
21778:
21756:
21744:
21729:
21717:
21695:
21680:
21558:
21101:
20966:
20731:
19766:
19657:
19547:
19435:
19345:
19323:
19164:
19143:
19091:
18752:
18704:
18656:
18513:consisting of those matrices having trace zero.
18454:
18324:
18199:
18124:
18100:
18031:
17967:
17854:
17828:
17643:
17425:
17306:
16865:
16517:
16127:
15622:
15466:
15361:
15268:
15127:
15019:
14592:of this polynomial, i.e., those complex numbers
14460:
14200:
13985:
13970:
13868:
13843:
13804:
13131:
13116:
13064:
12556:
12541:
12520:
12479:
12435:
12409:
12377:
12263:
12248:
12227:
12187:
12166:
12151:
12124:
11867:
11826:
11814:
11790:
11719:
11707:
11683:
11595:
11478:
11441:
11353:
11335:
11293:
11122:
11080:
10930:
10918:
10827:
10815:
10784:
10754:
10671:
10628:
10616:
10570:
10434:
10373:
10361:
10270:
10258:
10227:
10197:
10114:
10071:
10059:
10013:
9920:
9905:
9893:
9851:
9626:
9525:
8797:
8382:
8101:is a generalization of that product formula for
7568:
7547:
7514:
7436:
7364:
7352:
7334:
7224:
7198:
6127:
5474:
5446:
5011:is expressible as a similar linear combination:
4855:
4670:
4463:
4220:
3980:
2611:
777:
655:
27504:Journal für die reine und angewandte Mathematik
26983:Kleiner, Israel (2007), Kleiner, Israel (ed.),
26044:Linear Algebra and Its Applications 6th Edition
25239:allows rapid calculation of the determinant of
24006:defines the determinant for operators known as
21885:and therefore only depends on the endomorphism
21420:, appears in the higher-dimensional version of
20578:spanned by those vectors. As a consequence, if
19850:{\displaystyle v_{1},v_{2}\in \mathbf {R} ^{3}}
5993:columns (i.e. the columns of the matrix form a
5431:These rules have several further consequences:
3480:negative for an odd number. For the example of
27058:
26110:
24189:(i.e., matrices whose entries are elements of
14490:{\displaystyle \chi _{A}(t)=\det(t\cdot I-A).}
9695:, i.e., a matrix composed of four submatrices
1620:. (The parallelogram formed by the columns of
385:, which expresses the determinant as a sum of
27675:
27449:An introduction to the history of mathematics
27435:, Genève: Frères Cramer & Cl. Philibert,
26782:
26495:
25095:{\displaystyle \operatorname {O} (n^{2.376})}
23612:occurring in Leibniz's rule are omitted. The
22584:(the latter determinant being computed using
21631:
21220:parallelogram to that of the original square.
20453:. Another such use of the determinant is the
9329:
9316:
5046:
4784:
4748:
27626:Determinant Interactive Program and Tutorial
27551:: CS1 maint: multiple names: authors list (
26822:Dummit, David S.; Foote, Richard M. (2004),
23654:Determinants for finite-dimensional algebras
23605:{\displaystyle \operatorname {sgn}(\sigma )}
22263:The determinant is a natural transformation.
19603:. The rule is also implied by the identity
18842:were led to determinants by the question of
17038:is the identity matrix. More generally, if
14428:The characteristic polynomial is defined as
14008:{\displaystyle \det(I)=\det(\mathbf {i} )=1}
10995:size further formulas hold. For example, if
4652:
4634:
4432:
4414:
4210:for the product, this can be shortened into
3755:{\displaystyle \operatorname {sgn}(\sigma )}
3598:
3574:
2649:{\displaystyle \det(A)=\pm {\text{vol}}(P),}
27:In mathematics, invariant of square matrices
27451:(6 ed.), Saunders College Publishing,
27174:
27016:
26927:
26821:
26422:
26068:
26056:
25994:
25812:
25810:
25808:
25772:
25716:
25704:
25564:
25505:
25493:
25481:
24727:of the corresponding permutation (which is
23239:satisfying the following identity (for all
22732:The determinant of a linear transformation
21821:The determinant is therefore also called a
21035:. The volume of any tetrahedron, given its
18939:. About the time of Jacobi's last memoirs,
14544:is the identity matrix of the same size as
13956:{\displaystyle {\text{tr}}(\mathbf {i} )=0}
13727:{\displaystyle {\text{tr}}(\mathbf {i} )=0}
9349:, each the product of the determinant of a
6813:add 3 times the third column to the second
2604:-dimensional volume of this parallelotope,
1543:, and one that maps them to the columns of
27682:
27668:
27566:
27539:Histoire de l'Académie Royale des Sciences
27410:Théorie générale des equations algébriques
27107:Matrix Analysis and Applied Linear Algebra
25925:, Cahier 17, Tome 10, pages 29–112 (1815).
25728:A proof can be found in the Appendix B of
25351:
24750:for an even number of permutations and is
24465:, which is a significant improvement over
9360:and the determinant of the complementary (
7991:{\displaystyle \operatorname {GL} _{n}(K)}
7948:{\displaystyle \operatorname {SL} _{n}(K)}
7905:{\displaystyle \operatorname {SL} _{n}(K)}
7819:{\displaystyle \operatorname {GL} _{n}(K)}
7682:{\displaystyle \operatorname {GL} _{n}(K)}
7324:equals the product of their determinants:
6000:Adding a scalar multiple of one column to
2660:-dimensional volume scaling factor of the
2070:is written in terms of its column vectors
27326:Trefethen, Lloyd; Bau III, David (1997),
27285:
27061:Commutative Algebra: Constructive Methods
26948:
26895:
26881:
26873:
26863:
26812:
26802:
26616:
26529:
26480:
26126:
26086:Natural transformation § Determinant
25660:
25627:
25578:
25517:
25401:
25193:{\displaystyle \operatorname {O} (n^{3})}
25161:(or LU decomposition) method is of order
25142:{\displaystyle \operatorname {O} (n^{4})}
24901:{\displaystyle \operatorname {O} (n^{3})}
24793:, its determinant is readily computed as
24458:{\displaystyle \operatorname {O} (n^{3})}
24354:consists of left multiplying a matrix by
24198:
24134:Related notions for non-commutative rings
22695:
21592:
21508:
20980:
20676:{\displaystyle S\subset \mathbf {R} ^{n}}
20468:
19755:Determinants can be used to characterize
19728:{\displaystyle \operatorname {O} (n^{3})}
19669:
19650:
19616:
18816:The Nine Chapters on the Mathematical Art
17241:
17226:are zero and the remaining polynomial is
17187:
17015:
14024:Eigenvalues and characteristic polynomial
11901:
11885:
10991:If the blocks are square matrices of the
10780:
10776:
10753:
10223:
10219:
10196:
9570:
7211:
6394:Computation of the determinant of matrix
5426:
1851:
1847:
1833:
1806:
1802:
1786:
27350:
27344:
27134:A treatise on the theory of determinants
27083:Categories for the Working Mathematician
27076:
27059:Lombardi, Henri; Quitté, Claude (2015),
26764:
26474:
26114:
26098:
26080:
25876:
25805:
25465:
24503:For example, LU decomposition expresses
24376:
23212:
22258:
21645:have the same determinant: two matrices
21164:
19978:{\displaystyle f_{1}(x),\dots ,f_{n}(x)}
18915:also presented one on the subject. (See
18420:. For example, the special linear group
17623:. Its derivative can be expressed using
17590:{\displaystyle \mathbf {R} ^{n\times n}}
16472:{\displaystyle \sum _{l=1}^{n}lk_{l}=n.}
8180:is defined to be the determinant of the
3498:
1954:
1603:, as shown in the accompanying diagram.
1518:
520:of a square matrix, whose roots are the
424:of determinants of submatrices, or with
119:is the product of its diagonal entries.
27615:MacTutor History of Mathematics Archive
27529:
26982:
26718:
26675:
26587:
25984:http://jeff560.tripod.com/matrices.html
25908:
25864:
24346:
22660:, from the general linear group to the
22616:{\displaystyle \operatorname {GL} _{n}}
22055:) if and only if its determinant is an
20562:switches the orientation of the basis.
18994:, by Cayley; continuants by Sylvester;
18853:
18846:passing through a given set of points.
18440:{\displaystyle \operatorname {SL} _{n}}
17761:{\displaystyle \operatorname {adj} (A)}
11243:, the following formula holds (even if
10984:Both results can be combined to derive
9404:{\displaystyle \operatorname {adj} (A)}
7379:{\displaystyle \det(AB)=\det(A)\det(B)}
2656:and hence describes more generally the
1839:
1819:
1793:
1777:
420:, which expresses the determinant as a
108:and the corresponding linear map is an
14:
28104:
28073:Comparison of linear algebra libraries
27494:
27430:
27417:
27406:
27380:
27353:Elements of the history of mathematics
26964:
26783:Bunch, J. R.; Hopcroft, J. E. (1974).
26598:
25970:
25946:History of matrices and determinants:
25621:
25424:
24493:{\displaystyle \operatorname {O} (n!)}
22728:Exterior algebra § Linear algebra
21893:Square matrices over commutative rings
21305:matrix whose entries are given by the
20530:More generally, if the determinant of
19750:
19253:is the matrix formed by replacing the
19064:. This equation has a unique solution
18948:
18922:
18889:The next contributor of importance is
18839:
18835:
18823:in 1545 by a determinant-like entity.
18090:Yet another equivalent formulation is
14650:{\displaystyle \chi _{A}(\lambda )=0.}
11782:For a column and row vector as above:
8120:expresses the determinant of a matrix
3475:from the second row first column, and
2724:columns, so that it can be written as
1753:according to the following equations:
579:
34:. For determinants in immunology, see
27663:
27585:
27264:(7th ed.), Pearson Prentice Hall
27143:Linear Algebra: A Modern Introduction
27140:
27103:
27017:Kung, Joseph P.S.; Rota, Gian-Carlo;
26649:
26640:
26635:Linear algebra § Further reading
26530:Fang, Xin Gui; Havas, George (1997).
25675:
23219:the determinant to be the element in
20538:represents an orientation-preserving
20451:the Wronskian and linear independence
17931:Expressed in terms of the entries of
14668:if all its eigenvalues are positive.
13251:We can then write the cross-terms as
9665:The formula for the determinant of a
7429:. Moreover, they both take the value
6853:times the second column to the first
5489:{\displaystyle \det(cA)=c^{n}\det(A)}
4878:{\displaystyle \det \left(I\right)=1}
3634:from this set to itself, with values
2016:. The bivector magnitude (denoted by
27446:
27330:(1st ed.), Philadelphia: SIAM,
27268:
27259:
27240:
27222:
27204:
27128:
27104:Meyer, Carl D. (February 15, 2001),
26826:(3rd ed.), Hoboken, NJ: Wiley,
26518:
26029:
26017:
26005:
25958:
25825:
25700:
25649:Electronic Journal of Linear Algebra
25452:
25339:
24177:-determinant on quantum groups, the
23989:
22540:For example, the determinant of the
20778:. More generally, if the linear map
19696:Cramer's rule can be implemented in
18901:rows, which for the special case of
18506:{\displaystyle {\mathfrak {sl}}_{n}}
18412:This identity is used in describing
8737:Unwinding the determinants of these
8112:
7320:of equal size, the determinant of a
1514:
481:. This implies that, given a linear
27500:"De Determinantibus functionalibus"
27043:Linear Algebra and Its Applications
27040:
26928:Habgood, Ken; Arel, Itamar (2012).
26645:(9th ed.), Wiley International
26605:Linear Algebra and Its Applications
26041:
25567:Linear Algebra and Its Applications
25063:. This means, for example, that an
23978:of a skew-symmetric matrix and the
23572:Generalizations and related notions
22721:
20457:, which gives a criterion when two
18990:, in connection with the theory of
18986:, and Scott; skew determinants and
18830:in 1683 in Japan and parallelly of
18492:
18489:
17546:, which is, in turn, less than the
13106:, so the expression can be written
11661:More generally, for any invertible
10480:{\displaystyle \det(A)(D-CA^{-1}B)}
6808:add the second column to the first
4676:Characterization of the determinant
930:
477:The determinant is invariant under
24:
27689:
27473:Grattan-Guinness, I., ed. (2003),
27278:Computational discrete mathematics
25719:, Observation 7.1.2, Theorem 7.2.5
25168:
25117:
25070:
24876:
24864:
24472:
24433:
23616:generalizes both by introducing a
22391:given by replacing all entries in
21564:
21356:
21341:
19703:
18741:
18693:
18645:
17981:
17964:
17619:. In particular, it is everywhere
17109:
17064:
16947:
16902:
16855:is less than 1 in absolute value,
16847:identity can be obtained from the
15190:corresponds to the eigenvalue exp(
11733:
11609:
11492:
11455:
9371:
9320:
7272:Multiplicativity and matrix groups
3559:matrix is an expression involving
3471:from the first row second column,
3275:
2571:
25:
28138:
27560:
27090:(2nd ed.), Springer-Verlag,
27041:Lay, David C. (August 22, 2005),
26804:10.1090/S0025-5718-1974-0331751-8
26584:Fisikopoulos & Peñaranda 2016
26507:Fisikopoulos & Peñaranda 2016
26463:Fisikopoulos & Peñaranda 2016
24110:Operators in von Neumann algebras
21027:By calculating the volume of the
16843:An important arbitrary dimension
13037:The terms which are quadratic in
11566:identity matrices, respectively.
9660:
8341:{\displaystyle (-1)^{i+j}M_{i,j}}
8223:{\displaystyle (n-1)\times (n-1)}
7062:Combining these equalities gives
3521:
3286:
1551:that represents the image of the
28086:
28085:
28063:Basic Linear Algebra Subprograms
27821:
27541:(seconde partie), Paris: 267–376
27262:Linear Algebra With Applications
27086:, Graduate Texts in Mathematics
27045:(3rd ed.), Addison Wesley,
26960:from the original on 2019-05-05.
25936:http://jeff560.tripod.com/d.html
25923:Journal de l'Ecole Polytechnique
25520:"Determinants of Block Matrices"
25284:
25267:Alice's Adventures in Wonderland
25235:have already been computed, the
24211:{\displaystyle \mathbb {Z} _{2}}
22093:
21928:
21617:The Jacobian also occurs in the
21597:
21580:
21543:
21513:
21501:
21269:
21254:
21199:
21184:
20808:
20793:
20663:
20608:
20593:
19837:
19026:
18780:
18772:
18746:
18732:
18724:
18698:
18684:
18676:
18650:
17605:
17571:
15930:the signed constant term of the
13992:
13940:
13875:
13823:
13780:
13711:
13554:
12472:th root of the determinant is a
12341:are Hermitian positive-definite
9339:{\displaystyle {\tbinom {n}{k}}}
6846:{\displaystyle -{\frac {13}{3}}}
4985:{\displaystyle a_{j}=r\cdot v+w}
4661:{\displaystyle \{1,\ldots ,n\}.}
4623:where the sum is taken over all
3604:{\displaystyle \{1,2,\dots ,n\}}
3282:Leibniz formula for determinants
2533:
2502:
2449:
2428:
2413:
2343:
2274:
2212:
2126:
2107:
2093:
2040:, which is also the determinant
27961:Seven-dimensional cross product
26760:from the original on 2006-09-01
26714:from the original on 2012-10-25
26592:
26577:
26523:
26512:
26500:
26489:
26468:
26456:
26443:
26416:
26131:
26120:
26104:
26092:
26074:
26062:
26050:
26035:
26023:
26011:
25999:
25988:
25976:
25964:
25952:
25940:
25934:Origins of mathematical terms:
25928:
25914:
25902:
25882:
25870:
25858:
25831:
25819:
25790:
25778:
25766:
25722:
25710:
25694:
25669:
25643:Paksoy; Turkmen; Zhang (2014).
25636:
25615:
25603:
25558:
25511:
25499:
25487:
25410:. Dover Publications. pp.
25406:Applications of Tensor Analysis
24627:and an upper triangular matrix
23685:as a vector space over a field
23535:{\displaystyle \bigwedge ^{k}V}
23477:{\displaystyle \bigwedge ^{n}V}
23184:{\displaystyle \bigwedge ^{n}T}
23154:{\displaystyle \bigwedge ^{n}V}
22897:{\displaystyle \bigwedge ^{n}V}
20566:Volume and Jacobian determinant
19179:
19021:
17246:For a positive definite matrix
13924:This result followed just from
13787:{\displaystyle B=b\mathbf {i} }
11961:of the same size, the matrices
11393:Sylvester's determinant theorem
11388:Sylvester's determinant theorem
10986:Sylvester's determinant theorem
8525:) gives the following formula:
4438:{\displaystyle \{1,\ldots ,n\}}
4045:
3899:
2294:
2287:
2225:
487:finite-dimensional vector space
27479:Johns Hopkins University Press
27420:Cambridge Mathematical Journal
27207:Introduction to Linear Algebra
27027:, Cambridge University Press,
26937:Journal of Discrete Algorithms
26429:, American Mathematical Soc.,
25475:
25458:
25446:
25430:
25418:
25395:
25370:
25352:Wildberger, Norman J. (2010).
25345:
25333:
25187:
25174:
25136:
25123:
25104:Coppersmith–Winograd algorithm
25089:
25076:
25050:
25047:
25041:
25035:
24992:{\displaystyle M(n)\geq n^{a}}
24973:
24967:
24944:
24938:
24895:
24882:
24845:
24839:
24830:
24824:
24812:
24806:
24487:
24478:
24452:
24439:
24426:). These methods are of order
24397:
24391:
24229:
24083:
24080:
24077:
24065:
24056:
24047:
24035:
24023:
23963:{\displaystyle N_{L/F}:L\to F}
23954:
23851:
23818:
23788:
23782:
23724:
23599:
23593:
23356:
23350:
23057:
22991:
22748:
22690:
22637:
22630:
22521:
22518:
22515:
22496:
22490:
22487:
22478:
22475:
22472:
22453:
22450:
22444:
22378:
22372:
22356:
22353:
22347:
22328:
22322:
22286:
22149:
22146:
22140:
22100:{\displaystyle R=\mathbf {Z} }
22022:
22016:
21962:
21956:
21847:
21802:
21796:
21787:
21781:
21766:
21759:
21753:
21747:
21738:
21732:
21726:
21720:
21705:
21698:
21689:
21683:
21637:Determinant of an endomorphism
21584:
21576:
21573:
21561:
21550:
21547:
21539:
21533:
21505:
21497:
21489:
21483:
21327:
21321:
21264:
21194:
21144:
21140:
21104:
21097:
21008:
21002:
20958:
20955:
20949:
20943:
20914:
20908:
20803:
20744:
20740:
20734:
20727:
20706:
20700:
20603:
20372:
20366:
20361:
20349:
20329:
20323:
20318:
20306:
20291:
20285:
20280:
20268:
20229:
20223:
20200:
20194:
20176:
20170:
20150:
20144:
20124:
20118:
20103:
20097:
20073:
20067:
20064:
20032:
19972:
19966:
19944:
19938:
19722:
19709:
19666:
19660:
19647:
19641:
19629:
19623:
19556:
19550:
19339:
19326:
19173:
19167:
19159:
19146:
19100:
19094:
18966:determinants by Sylvester and
18761:
18755:
18713:
18707:
18665:
18659:
18366:
18360:
18342:
18327:
18314:, the identity matrix, yields
18208:
18202:
18166:
18160:
18154:
18145:
18133:
18127:
18118:
18103:
18040:
18034:
18016:
18009:
17976:
17970:
17863:
17857:
17837:
17831:
17755:
17749:
17692:
17686:
17652:
17646:
17474:
17468:
17435:
17428:
17339:
17327:
17315:
17309:
17127:
17117:
16965:
16955:
16880:
16868:
16796:
16790:
16724:
16714:
16648:with multidimensional indices
16620:
16575:
16545:
16535:
16526:
16520:
16295:
16285:
16136:
16130:
16095:
16077:
15934:, determined recursively from
15842:
15836:
15736:
15730:
15674:
15668:
15631:
15625:
15550:
15544:
15518:
15512:
15475:
15469:
15413:
15407:
15370:
15364:
15307:
15304:
15298:
15289:
15277:
15271:
15232:
15226:
15151:
15148:
15145:
15139:
15130:
15124:
15112:
15106:
15070:
15067:
15061:
15052:
15040:
15037:
15031:
15022:
14638:
14632:
14481:
14463:
14454:
14448:
14209:
14203:
13996:
13988:
13979:
13973:
13944:
13936:
13879:
13871:
13852:
13846:
13827:
13807:
13715:
13707:
13485:
13476:
13465:
13459:
13451:
13445:
13414:
13322:
13316:
13290:
13287:
13261:
13140:
13134:
13125:
13119:
13073:
13067:
12811:
12785:
12782:
12756:
12750:
12724:
12721:
12695:
12616:
12607:
12596:
12590:
12582:
12576:
12565:
12559:
12550:
12544:
12535:
12523:
12444:
12438:
12418:
12412:
12392:
12380:
12272:
12266:
12257:
12251:
12242:
12230:
12202:
12190:
12181:
12169:
12160:
12154:
12145:
12127:
12056:positive semidefinite matrices
11898:
11892:
11876:
11870:
11823:
11817:
11808:
11793:
11716:
11710:
11701:
11686:
11574:For the case of column vector
11368:
11356:
11350:
11338:
11143:
11125:
10988:, which is also stated below.
10961:
10933:
10927:
10921:
10824:
10818:
10791:
10781:
10773:
10757:
10625:
10619:
10474:
10446:
10443:
10437:
10404:
10376:
10370:
10364:
10267:
10261:
10234:
10224:
10216:
10200:
10068:
10062:
9914:
9908:
9902:
9896:
9567:
9555:
9531:
9522:
9468:
9458:
9440:
9436:
9430:
9421:
9398:
9392:
8843:
8833:
8806:
8800:
8428:
8418:
8391:
8385:
8307:
8297:
8217:
8205:
8199:
8187:
8054:
8048:
8027:
8021:
7985:
7979:
7942:
7936:
7899:
7893:
7826:into the multiplicative group
7813:
7807:
7747:
7741:
7722:
7716:
7676:
7670:
7581:
7577:
7571:
7565:
7556:
7550:
7373:
7367:
7361:
7355:
7346:
7337:
7233:
7227:
7134:
7131:
7122:
7107:
7097:
7089:
7078:
7070:
7042:
7034:
7026:
7018:
6993:
6985:
6974:
6966:
6941:
6933:
6925:
6917:
6893:
6885:
6877:
6869:
6136:
6130:
5965:
5892:
5884:
5811:
5800:
5730:
5705:
5633:
5622:
5553:
5483:
5477:
5458:
5449:
5359:
5301:
5261:
5215:
5207:
5164:
5143:
5033:
5025:
4472:
4466:
4319:
4313:
4275:
4269:
4229:
4223:
4185:
4179:
4157:
4151:
4134:
4128:
3989:
3983:
3749:
3743:
3686:
3680:
3665:
3659:
3650:
3644:
3484:, the single transposition of
2640:
2634:
2620:
2614:
1798:
1788:
1782:
1772:
1450:
1432:
1214:
1197:
1191:
1174:
905:
893:
887:
884:
875:
866:
61:. The determinant of a matrix
13:
1:
27631:Linear algebra: determinants.
27163:American Mathematical Monthly
27145:(2nd ed.), Brooks/Cole,
26985:A history of abstract algebra
26884:American Mathematical Monthly
26628:
25676:Serre, Denis (Oct 18, 2010).
25360:University of New South Wales
24181:on Capelli matrices, and the
23736:{\displaystyle \det :A\to F.}
23705:, there is a determinant map
22649:{\displaystyle (-)^{\times }}
21624:When applied to the field of
18416:associated to certain matrix
17553:
17355:with equality if and only if
14385:is invertible if and only if
12678:{\displaystyle A_{ij},B_{ij}}
12480:Sum identity for 2×2 matrices
7647:) forms a group known as the
4671:Properties of the determinant
2703:
1531:can be used to represent two
27803:Eigenvalues and eigenvectors
26875:10.1016/j.comgeo.2015.12.001
26599:Abeles, Francine F. (2008).
26451:Trefethen & Bau III 1997
25982:History of matrix notation:
24720:{\displaystyle \varepsilon }
24587:(which has exactly a single
23161:is one-dimensional, the map
21935:{\displaystyle \mathbf {Z} }
21866:for some finite-dimensional
20558:), while if it is negative,
19745:singular value decomposition
19035:, written in matrix form as
17612:{\displaystyle \mathbf {R} }
15198:). In particular, given any
13794:in the above identity gives
12299:th root of determinant is a
10536:A similar result holds when
8765:Laplace expansion along the
8493:Laplace expansion along the
7280:, i.e., for square matrices
7264:rows, the determinant is an
7260:matrix as being composed of
7155:
6094:or, alternatively, whenever
2700:, and so is not invertible.
1975:and orientation-preserving.
7:
27610:"Matrices and determinants"
27574:Encyclopedia of Mathematics
27024:Combinatorics: The Rota Way
25277:
24120:Fuglede−Kadison determinant
21632:Abstract algebraic aspects
21422:integration by substitution
21295:the Jacobian matrix is the
18447:is defined by the equation
17367:Kullback-Leibler divergence
15910:Faddeev–LeVerrier algorithm
15182:, because every eigenvalue
13508:This has an application to
13504:which completes the proof.
13430:which can be recognized as
12017:The determinant of the sum
8290:-th column. The expression
8087:{\displaystyle K^{\times }}
7846:{\displaystyle K^{\times }}
6818:swap the first two columns
3567:. A permutation of the set
562:exterior differential forms
428:, which allows computing a
10:
28143:
27567:Suprunenko, D.A. (2001) ,
27351:Bourbaki, Nicolas (1994),
27188:Cambridge University Press
26790:Mathematics of Computation
26765:Bourbaki, Nicolas (1998),
26685:Mathematics of Computation
26632:
26423:Varadarajan, V. S (2004),
26111:Lombardi & Quitté 2015
25678:"Concavity of det over HPD
25378:"Determinants and Volumes"
24928:can be multiplied in time
24424:positive definite matrices
24358:for getting a matrix in a
24114:For operators in a finite
23265:{\displaystyle v_{i}\in V}
22784:-dimensional vector space
22725:
22208:-matrices with entries in
20570:As pointed out above, the
20475:Orientation (vector space)
20472:
19033:linear system of equations
18811:system of linear equations
18804:
18478:special linear Lie algebra
9512:For every matrix, one has
8230:-matrix that results from
7772:is 2 or 3 consists of all
7184:equals the determinant of
6256:
5007:, then the determinant of
3279:
2597:The determinant gives the
510:system of linear equations
29:
28081:
28043:
27999:
27936:
27888:
27830:
27819:
27715:
27697:
27531:Laplace, Pierre-Simon, de
27496:Jacobi, Carl Gustav Jakob
27365:10.1007/978-3-642-61693-8
26993:10.1007/978-0-8176-4685-1
26965:Harris, Frank E. (2014),
26950:10.1016/j.jda.2011.06.007
26655:Linear Algebra Done Right
26618:10.1016/j.laa.2007.11.022
26496:Bunch & Hopcroft 1974
25973:, Vol. II, p. 92, no. 462
25589:10.1016/j.laa.2016.10.004
25518:Silvester, J. R. (2000).
25494:Kung, Rota & Yan 2009
25382:textbooks.math.gatech.edu
25306:Cayley–Menger determinant
24310:{\displaystyle n\times n}
22596:between the two functors
22201:{\displaystyle n\times n}
22182:(the group of invertible
21974:{\displaystyle \det(I)=1}
21451:) of some other function
21424:: for suitable functions
20847:{\displaystyle m\times n}
20751:{\displaystyle |\det(A)|}
20399:It is non-zero (for some
19910:{\displaystyle 3\times 3}
18992:orthogonal transformation
18535:{\displaystyle 3\times 3}
18288:. The special case where
17542:, which is less than the
17212:then all coefficients of
15932:characteristic polynomial
15244:{\displaystyle \exp(L)=A}
14588:: they are precisely the
14075:{\displaystyle n\times n}
14034:characteristic polynomial
13527:{\displaystyle 2\times 2}
12503:{\displaystyle 2\times 2}
12360:{\displaystyle n\times n}
12326:{\displaystyle n\times n}
11181:{\displaystyle 2\times 2}
10526:{\displaystyle 1\times 1}
9830:{\displaystyle n\times n}
9804:{\displaystyle n\times m}
9778:{\displaystyle m\times n}
9752:{\displaystyle m\times m}
9684:{\displaystyle 2\times 2}
8756:{\displaystyle 2\times 2}
7914:first isomorphism theorem
5515:{\displaystyle n\times n}
5404:{\displaystyle n\times n}
4710:as being composed of its
4699:{\displaystyle n\times n}
3552:{\displaystyle n\times n}
3299:matrix is the following:
3295:for the determinant of a
954:{\displaystyle 2\times 2}
518:characteristic polynomial
206:and the determinant of a
27620:University of St Andrews
27516:10.1515/crll.1841.22.319
27431:Cramer, Gabriel (1750),
27407:Bézout, Étienne (1779),
27381:Cajori, Florian (1993),
27328:Numerical Linear Algebra
27260:Leon, Steven J. (2006),
25799:A History of Mathematics
25732:Zeitschrift für Physik A
25327:
25321:Determinantal conjecture
25237:matrix determinant lemma
24236:numerical linear algebra
23211:. Some authors such as (
22831:over a commutative ring
22757:{\displaystyle T:V\to V}
22295:{\displaystyle f:R\to S}
21856:{\displaystyle T:V\to V}
21619:inverse function theorem
18469:{\displaystyle \det A=1}
17204:is expanded as a formal
16416:satisfying the equation
14988:
14925:being positive, for all
14605:{\displaystyle \lambda }
14405:is not an eigenvalue of
12685:. The left-hand side is
12484:For the special case of
12333:matrices. Therefore, if
9305:determinant as a sum of
7766:special orthogonal group
7000:{\displaystyle |D|=-|C|}
6061:{\displaystyle a_{ij}=0}
1978:The object known as the
96:represented, on a given
28127:Homogeneous polynomials
27287:10.1007/3-540-45506-X_9
26767:Algebra I, Chapters 1-3
26676:Bareiss, Erwin (1968),
26069:Dummit & Foote 2004
26057:Dummit & Foote 2004
26046:. Pearson. p. 172.
25995:Habgood & Arel 2012
25773:Horn & Johnson 2018
25717:Horn & Johnson 2018
25705:Horn & Johnson 2018
25662:10.13001/1081-3810.1622
25609:Proofs can be found in
25506:Horn & Johnson 2018
25482:Horn & Johnson 2018
25056:{\displaystyle O(M(n))}
24403:{\displaystyle \det(A)}
22592:, the determinant is a
21920:, such as the integers
21825:. The determinant of a
21226:differentiable function
21069:{\displaystyle a,b,c,d}
19106:{\displaystyle \det(A)}
17812:is invertible, we have
15912:. That is, for generic
15902:Cayley-Hamilton theorem
14325:is zero if and only if
13079:{\displaystyle \det(A)}
12293:Brunn–Minkowski theorem
9726:{\displaystyle A,B,C,D}
8372:, one has the equality
8173:{\displaystyle M_{i,j}}
7853:of nonzero elements of
7160:The determinant of the
7049:{\displaystyle |E|=|D|}
6948:{\displaystyle |B|=|C|}
6900:{\displaystyle |A|=|B|}
6245:{\displaystyle \sigma }
5284:
4913:
4848:
4730:columns, so denoted as
4627:-tuples of integers in
3775:{\displaystyle \sigma }
3627:{\displaystyle \sigma }
2999:{\displaystyle a_{1,1}}
2404:defined by the vectors
2147:{\displaystyle A=\left}
1539:vectors to the rows of
449:The determinant of the
27788:Row and column vectors
27644:Determinant Calculator
27180:Johnson, Charles Royal
26851:Computational Geometry
26641:Anton, Howard (2005),
26407:
26032:, §VII.6, Theorem 6.10
25472:of the exterior power.
25260:Charles Dodgson (i.e.
25227:If the determinant of
25218:
25194:
25143:
25096:
25057:
25019:
25018:{\displaystyle a>2}
24993:
24951:
24922:
24902:
24855:
24787:
24767:
24744:
24721:
24701:
24681:
24661:
24641:
24621:
24601:
24581:
24555:
24554:{\displaystyle A=PLU.}
24517:
24494:
24459:
24420:Cholesky decomposition
24404:
24337:
24311:
24282:
24262:
24240:Computational geometry
24212:
24104:functional determinant
24093:
23984:central simple algebra
23964:
23910:
23795:
23737:
23699:
23675:
23644:
23606:
23562:
23561:{\displaystyle k<n}
23536:
23498:
23478:
23448:
23427:-tuples of vectors in
23421:
23398:
23266:
23233:
23205:
23185:
23155:
23122:
22938:
22918:
22898:
22865:
22845:
22825:
22798:
22778:
22758:
22712:
22650:
22617:
22594:natural transformation
22588:). In the language of
22578:
22558:
22528:
22425:
22411:by their images under
22405:
22385:
22296:
22264:
22246:
22222:
22202:
22169:
22101:
22073:
22049:
22029:
21975:
21936:
21914:
21857:
21812:
21608:
21407:
21286:
21221:
21213:
21152:
21070:
21018:
20921:
20888:
20868:
20848:
20828:is represented by the
20822:
20772:
20752:
20713:
20677:
20642:
20622:
20469:Orientation of a basis
20439:
20413:
20390:
20005:
19979:
19919:differential equations
19911:
19878:
19851:
19796:
19776:
19775:{\displaystyle \det A}
19729:
19687:
19593:
19563:
19424:
19307:
19287:
19267:
19247:
19217:
19107:
19078:
19058:
18795:
18620:
18588:
18536:
18507:
18470:
18441:
18403:
18308:
18274:
18081:
17945:
17922:
17806:
17786:
17762:
17727:
17613:
17591:
17525:
17346:
17242:Upper and lower bounds
17195:
17113:
17068:
17023:
16951:
16906:
16834:
16630:
16473:
16446:
16397:
16281:
16108:
16024:
15891:
15317:
15245:
15206:, that is, any matrix
15161:
15086:or, for real matrices
15077:
14979:
14959:
14939:
14916:
14651:
14606:
14582:
14558:
14538:
14524:of the polynomial and
14514:
14491:
14419:
14399:
14379:
14359:
14339:
14319:
14289:
14235:
14190:of these eigenvalues,
14173:algebraic multiplicity
14165:
14100:
14076:
14050:
14009:
13957:
13915:
13788:
13757:
13728:
13689:
13669:
13649:
13528:
13495:
13421:
13242:
13100:
13080:
13051:
13028:
12821:
12679:
12626:
12504:
12462:
12361:
12327:
12284:
12214:
12111:
12091:
12071:
12037:
12012:
12001:
11978:
11955:
11935:
11912:
11773:
11651:
11519:
11378:
11277:
11257:
11237:
11211:
11182:
11153:
11064:
11029:
11009:
10975:
10556:is invertible, namely
10550:
10527:
10501:
10481:
10418:
9989:
9966:
9831:
9805:
9779:
9753:
9727:
9685:
9651:
9581:
9503:
9405:
9340:
9283:
8897:
8832:
8779:
8757:
8728:
8519:
8482:
8417:
8366:
8342:
8284:
8264:
8244:
8224:
8174:
8134:
8088:
8061:
7992:
7949:
7906:
7867:
7847:
7820:
7754:
7683:
7641:
7621:
7597:
7489:
7466:
7446:
7445:{\displaystyle \det B}
7423:
7403:
7380:
7314:
7294:
7240:
7178:
7147:
7050:
7001:
6949:
6901:
6847:
6793:
6698:
6603:
6509:
6408:
6384:
6280:
6246:
6222:
6201:
6114:
6113:{\displaystyle i<j}
6088:
6087:{\displaystyle i>j}
6062:
6022:
5976:
5716:
5536:
5516:
5490:
5427:Immediate consequences
5405:
5373:
5273:
4986:
4938:
4924:th column of a matrix
4899:
4879:
4829:
4796:
4724:
4700:
4662:
4614:
4439:
4393:
4333:
4298:
4197:
3957:
3825:
3802:
3776:
3756:
3724:
3703:, is commonly denoted
3693:
3628:
3605:
3553:
3506:
3454:
3255:
3000:
2964:
2650:
2589:
2466:
2383:
2360:
2148:
1986:oriented plane segment
1964:
1946:
1606:The absolute value of
1524:
1505:
1338:
1315:
1091:
1011:
955:
921:
758:
641:is denoted either by "
635:
402:
371:The determinant of an
362:
197:
27793:Row and column spaces
27738:Scalar multiplication
27447:Eves, Howard (1990),
27345:Historical references
27269:Rote, Günter (2001),
27141:Poole, David (2006),
26743:10.1145/122272.122273
26731:ACM SIGNUM Newsletter
26547:10.1145/258726.258740
26408:
25785:Grattan-Guinness 2003
25311:Dieudonné determinant
25219:
25195:
25144:
25097:
25058:
25020:
24994:
24952:
24923:
24903:
24856:
24788:
24768:
24745:
24722:
24702:
24682:
24662:
24642:
24622:
24602:
24582:
24556:
24518:
24495:
24460:
24405:
24381:Some methods compute
24377:Decomposition methods
24338:
24312:
24283:
24263:
24213:
24171:Dieudonné determinant
24094:
24008:trace class operators
23965:
23911:
23796:
23738:
23700:
23676:
23645:
23643:{\displaystyle S_{n}}
23607:
23563:
23537:
23499:
23479:
23449:
23447:{\displaystyle R^{n}}
23422:
23399:
23267:
23234:
23206:
23186:
23156:
23123:
22944:induces a linear map
22939:
22919:
22899:
22866:
22846:
22826:
22804:or, more generally a
22799:
22779:
22759:
22713:
22651:
22618:
22579:
22559:
22529:
22426:
22406:
22386:
22297:
22262:
22247:
22223:
22203:
22170:
22102:
22074:
22050:
22030:
21976:
21937:
21915:
21858:
21827:linear transformation
21813:
21609:
21443:), the integral over
21416:Its determinant, the
21408:
21287:
21214:
21168:
21153:
21071:
21019:
20922:
20889:
20869:
20849:
20823:
20773:
20753:
20714:
20690:, then the volume of
20678:
20643:
20623:
20540:linear transformation
20440:
20414:
20391:
20006:
19980:
19912:
19879:
19877:{\displaystyle v_{3}}
19852:
19797:
19777:
19730:
19688:
19594:
19592:{\displaystyle a_{j}}
19564:
19404:
19308:
19293:by the column vector
19288:
19268:
19248:
19246:{\displaystyle A_{i}}
19218:
19108:
19079:
19059:
18796:
18621:
18619:{\displaystyle a,b,c}
18589:
18537:
18508:
18471:
18442:
18404:
18309:
18275:
18082:
17946:
17923:
17807:
17787:
17763:
17728:
17614:
17592:
17526:
17347:
17196:
17093:
17048:
17024:
16931:
16886:
16835:
16631:
16474:
16426:
16398:
16261:
16109:
16004:
15892:
15318:
15246:
15162:
15078:
14980:
14960:
14940:
14917:
14670:Sylvester's criterion
14652:
14607:
14583:
14559:
14539:
14515:
14492:
14420:
14400:
14380:
14360:
14340:
14320:
14290:
14215:
14166:
14101:
14077:
14051:
14010:
13958:
13916:
13789:
13758:
13729:
13690:
13670:
13650:
13529:
13496:
13422:
13243:
13101:
13081:
13052:
13029:
12822:
12680:
12627:
12505:
12463:
12362:
12328:
12303:, when restricted to
12285:
12215:
12112:
12092:
12072:
12038:
12002:
11979:
11956:
11936:
11913:
11774:
11652:
11520:
11379:
11278:
11258:
11238:
11212:
11183:
11154:
11065:
11063:{\displaystyle CD=DC}
11030:
11010:
10976:
10551:
10528:
10502:
10482:
10419:
9990:
9967:
9832:
9806:
9780:
9754:
9728:
9686:
9652:
9582:
9504:
9406:
9341:
9284:
8898:
8812:
8780:
8758:
8729:
8520:
8483:
8397:
8367:
8343:
8285:
8265:
8245:
8225:
8175:
8135:
8089:
8062:
7993:
7950:
7907:
7868:
7848:
7821:
7778:special unitary group
7755:
7684:
7642:
7622:
7598:
7490:
7467:
7447:
7424:
7404:
7381:
7315:
7295:
7276:The determinant is a
7241:
7179:
7148:
7051:
7002:
6950:
6902:
6848:
6794:
6699:
6604:
6510:
6409:
6385:
6281:
6247:
6223:
6181:
6115:
6089:
6063:
6023:
5977:
5717:
5537:
5517:
5491:
5435:The determinant is a
5406:
5374:
5274:
4987:
4939:
4900:
4880:
4830:
4828:{\displaystyle a_{i}}
4797:
4725:
4701:
4663:
4615:
4440:
4394:
4334:
4278:
4198:
3958:
3826:
3803:
3777:
3757:
3725:
3723:{\displaystyle S_{n}}
3694:
3629:
3606:
3554:
3502:
3455:
3256:
3001:
2965:
2662:linear transformation
2651:
2590:
2467:
2384:
2361:
2149:
1958:
1947:
1522:
1506:
1339:
1316:
1092:
1012:
956:
922:
759:
636:
584:The determinant of a
403:
363:
198:
122:The determinant of a
27928:Gram–Schmidt process
27880:Gaussian elimination
27606:Robertson, Edmund F.
27241:Lang, Serge (2002).
27223:Lang, Serge (1987),
27205:Lang, Serge (1985),
26150:
25846:on 10 September 2012
25455:, §VI.7, Theorem 7.5
25272:Dodgson condensation
25257:are column vectors.
25208:
25165:
25159:Gaussian elimination
25151:closed ordered walks
25114:
25067:
25029:
25003:
24961:
24950:{\displaystyle M(n)}
24932:
24912:
24873:
24800:
24777:
24754:
24731:
24711:
24691:
24671:
24651:
24631:
24611:
24591:
24571:
24530:
24507:
24469:
24430:
24385:
24352:Gaussian elimination
24347:Gaussian elimination
24324:
24295:
24272:
24249:
24193:
24017:
24004:Fredholm determinant
23927:
23812:
23754:
23712:
23689:
23665:
23627:
23584:
23546:
23516:
23488:
23458:
23431:
23411:
23279:
23243:
23223:
23195:
23165:
23135:
22951:
22928:
22908:
22878:
22855:
22835:
22815:
22788:
22768:
22736:
22671:
22662:multiplicative group
22627:
22600:
22568:
22548:
22438:
22415:
22395:
22306:
22274:
22236:
22230:multiplicative group
22212:
22186:
22180:general linear group
22124:
22083:
22063:
22039:
21988:
21950:
21942:, as opposed to the
21924:
21904:
21835:
21823:similarity invariant
21677:
21472:
21418:Jacobian determinant
21315:
21243:
21173:
21080:
21042:
20934:
20920:{\displaystyle f(S)}
20902:
20878:
20858:
20832:
20782:
20762:
20758:times the volume of
20723:
20712:{\displaystyle f(S)}
20694:
20652:
20632:
20582:
20423:
20403:
20026:
19989:
19925:
19895:
19861:
19806:
19786:
19763:
19700:
19610:
19576:
19320:
19297:
19277:
19257:
19230:
19124:
19088:
19068:
19057:{\displaystyle Ax=b}
19039:
18929:. In his memoirs in
18917:Cauchy–Binet formula
18637:
18598:
18546:
18520:
18483:
18451:
18424:
18321:
18292:
18097:
17958:
17935:
17819:
17796:
17792:. In particular, if
17776:
17740:
17634:
17601:
17566:
17383:
17259:
17045:
16862:
16711:
16514:
16423:
16124:
15941:
15354:
15265:
15217:
15097:
15016:
14969:
14949:
14929:
14679:
14619:
14596:
14572:
14548:
14528:
14504:
14435:
14409:
14389:
14369:
14349:
14345:is an eigenvalue of
14329:
14309:
14197:
14116:
14090:
14060:
14040:
13967:
13928:
13801:
13767:
13756:{\displaystyle A=aI}
13738:
13699:
13679:
13659:
13538:
13512:
13437:
13258:
13113:
13090:
13086:, and similarly for
13061:
13041:
12837:
12692:
12643:
12517:
12488:
12474:homogeneous function
12371:
12345:
12311:
12224:
12121:
12101:
12081:
12061:
12021:
11988:
11965:
11945:
11925:
11921:For square matrices
11787:
11680:
11592:
11438:
11290:
11267:
11247:
11221:
11195:
11166:
11077:
11042:
11019:
10999:
10563:
10540:
10511:
10491:
10431:
10427:which simplifies to
10006:
9979:
9848:
9815:
9789:
9763:
9737:
9699:
9669:
9601:
9519:
9418:
9383:
9309:
8914:
8794:
8769:
8741:
8532:
8503:
8491:which is called the
8379:
8356:
8294:
8274:
8254:
8234:
8184:
8151:
8124:
8099:Cauchy–Binet formula
8071:
8005:
7963:
7920:
7877:
7857:
7830:
7791:
7700:
7695:special linear group
7654:
7649:general linear group
7631:
7611:
7511:
7479:
7456:
7433:
7413:
7393:
7331:
7304:
7284:
7195:
7168:
7066:
7014:
6962:
6913:
6865:
6827:
6711:
6616:
6521:
6424:
6398:
6293:
6270:
6263:Gaussian elimination
6236:
6124:
6098:
6072:
6036:
6012:
5726:
5549:
5526:
5500:
5443:
5437:homogeneous function
5389:
5297:
5017:
4951:
4928:
4889:
4852:
4812:
4737:
4714:
4684:
4631:
4460:
4411:
4350:
4217:
3977:
3840:
3812:
3786:
3766:
3734:
3707:
3638:
3618:
3571:
3537:
3306:
3028:
2977:
2731:
2608:
2476:
2408:
2373:
2161:
2074:
1760:
1535:: one that maps the
1351:
1328:
1107:
1024:
968:
939:
774:
652:
592:
570:changes of variables
568:, in particular for
566:Jacobian determinant
426:Gaussian elimination
389:
217:
133:
67:is commonly denoted
57:of the entries of a
28058:Numerical stability
27938:Multilinear algebra
27913:Inner product space
27763:Linear independence
27604:O'Connor, John J.;
27441:10.3931/e-rara-4048
26724:"An empty exercise"
26071:, §11.4, Theorem 30
26042:Lay, David (2021).
25744:1992ZPhyA.344...99K
25231:and the inverse of
24356:elementary matrices
24179:Capelli determinant
24128:von Neumann algebra
23997:Functional analysis
23660:associative algebra
23650:in Leibniz's rule.
23215:) use this fact to
21307:partial derivatives
21162:over the vertices.
20438:{\displaystyle n-1}
20365:
20322:
20284:
20222:
20193:
20169:
20004:{\displaystyle n-1}
19751:Linear independence
19599:are the columns of
18307:{\displaystyle A=I}
17371:multivariate normal
16826:
16780:
16765:
16737:
15906:Newton's identities
15254:the determinant of
12220:with the corollary
12036:{\displaystyle A+B}
11236:{\displaystyle B=C}
11210:{\displaystyle A=D}
9203:
9175:
9152:
9129:
9077:
9055:
9038:
9021:
8518:{\displaystyle i=1}
6414:
6286:using that method:
5986:of the permutation.
5285:The determinant is
4914:The determinant is
3824:{\displaystyle -1.}
3801:{\displaystyle +1,}
3013:The determinant of
1959:The volume of this
580:Two by two matrices
548:determines how the
27768:Linear combination
27636:2008-12-04 at the
27587:Weisstein, Eric W.
27078:Mac Lane, Saunders
26651:Axler, Sheldon Jay
26403:
26401:
26381:
26336:
26294:
26248:
26206:
25888:Muir, Sir Thomas,
25752:10.1007/BF01291027
25402:McConnell (1957).
25316:Slater determinant
25301:Cauchy determinant
25292:Mathematics portal
25214:
25190:
25139:
25110:having complexity
25092:
25053:
25015:
24989:
24947:
24918:
24898:
24851:
24783:
24766:{\displaystyle -1}
24763:
24743:{\displaystyle +1}
24740:
24717:
24697:
24677:
24657:
24637:
24617:
24597:
24577:
24566:permutation matrix
24551:
24513:
24490:
24455:
24400:
24336:{\displaystyle n!}
24333:
24307:
24291:) products for an
24278:
24261:{\displaystyle n!}
24258:
24208:
24089:
23960:
23906:
23791:
23733:
23695:
23683:finite-dimensional
23671:
23640:
23602:
23558:
23532:
23494:
23474:
23444:
23417:
23394:
23262:
23229:
23201:
23181:
23151:
23118:
23116:
22934:
22914:
22894:
22861:
22841:
22821:
22794:
22774:
22754:
22708:
22646:
22613:
22586:modular arithmetic
22574:
22554:
22524:
22421:
22401:
22381:
22292:
22265:
22254:group homomorphism
22242:
22218:
22198:
22165:
22116:group homomorphism
22097:
22069:
22057:invertible element
22045:
22025:
21971:
21932:
21910:
21853:
21808:
21604:
21403:
21282:
21222:
21209:
21148:
21066:
21014:
20917:
20884:
20864:
20844:
20818:
20768:
20748:
20709:
20673:
20638:
20618:
20554:matrix, this is a
20447:analytic functions
20435:
20409:
20386:
20377:
20339:
20296:
20258:
20210:
20181:
20157:
20001:
19975:
19921:: given functions
19907:
19874:
19847:
19792:
19772:
19757:linearly dependent
19725:
19683:
19589:
19572:where the vectors
19559:
19528:
19395:
19303:
19283:
19263:
19243:
19213:
19103:
19074:
19054:
18947:began their work.
18866:elimination theory
18791:
18789:
18630:of the other two:
18616:
18584:
18578:
18532:
18503:
18466:
18437:
18399:
18304:
18270:
18077:
17941:
17918:
17802:
17782:
17758:
17723:
17609:
17587:
17521:
17342:
17191:
17019:
16830:
16812:
16811:
16766:
16751:
16750:
16723:
16626:
16469:
16393:
16260:
16258:
16104:
15887:
15885:
15313:
15241:
15176:matrix exponential
15157:
15073:
14975:
14955:
14935:
14912:
14906:
14647:
14602:
14578:
14554:
14534:
14510:
14487:
14415:
14395:
14375:
14365:. In other words,
14355:
14335:
14315:
14300:pseudo-determinant
14285:
14161:
14106:must have exactly
14096:
14072:
14046:
14005:
13953:
13911:
13784:
13753:
13724:
13685:
13665:
13645:
13639:
13594:
13524:
13491:
13417:
13238:
13096:
13076:
13047:
13024:
12817:
12675:
12637:
12622:
12500:
12458:
12367:matrices, one has
12357:
12323:
12307:positive-definite
12280:
12210:
12107:
12087:
12067:
12033:
12000:{\displaystyle BA}
11997:
11977:{\displaystyle AB}
11974:
11951:
11931:
11908:
11769:
11647:
11515:
11374:
11326:
11273:
11253:
11233:
11207:
11178:
11149:
11113:
11060:
11025:
11005:
10971:
10969:
10902:
10804:
10747:
10737:
10661:
10603:
10546:
10523:
10497:
10477:
10414:
10412:
10345:
10247:
10190:
10180:
10104:
10046:
9985:
9962:
9953:
9884:
9827:
9801:
9775:
9749:
9723:
9681:
9647:
9592:nonsingular matrix
9577:
9499:
9401:
9336:
9334:
9279:
9242:
9206:
9183:
9155:
9132:
9109:
9063:
9041:
9024:
9007:
8909:Vandermonde matrix
8893:
8775:
8753:
8724:
8718:
8676:
8634:
8592:
8515:
8478:
8362:
8338:
8280:
8260:
8240:
8220:
8170:
8130:
8084:
8057:
7988:
7945:
7916:, this shows that
7902:
7863:
7843:
7816:
7785:group homomorphism
7750:
7679:
7637:
7617:
7593:
7495:with entries in a
7485:
7462:
7442:
7419:
7399:
7376:
7310:
7290:
7278:multiplicative map
7268:-linear function.
7236:
7174:
7143:
7046:
6997:
6945:
6897:
6843:
6789:
6783:
6694:
6688:
6599:
6593:
6505:
6499:
6404:
6393:
6380:
6371:
6276:
6242:
6218:
6110:
6084:
6058:
6018:
5995:linearly dependent
5972:
5712:
5532:
5512:
5486:
5401:
5369:
5269:
5267:
4982:
4946:linear combination
4934:
4895:
4875:
4825:
4792:
4720:
4696:
4658:
4610:
4526:
4435:
4399:is defined on the
4389:
4345:Levi-Civita symbol
4329:
4257:
4193:
4121:
4090:
3953:
3944:
3821:
3798:
3772:
3752:
3720:
3689:
3624:
3613:bijective function
3601:
3549:
3507:
3450:
3366:
3251:
3242:
3017:is denoted by det(
2996:
2960:
2951:
2646:
2585:
2462:
2379:
2356:
2332:
2263:
2201:
2144:
2138:
1992:, and coordinates
1965:
1942:
1915:
1886:
1525:
1501:
1492:
1396:
1334:
1311:
1302:
1253:
1162:
1087:
1057:
1007:
1001:
951:
917:
857:
813:
754:
727:
688:
631:
625:
574:multiple integrals
422:linear combination
401:{\displaystyle n!}
398:
358:
277:
193:
166:
18:Matrix determinant
28099:
28098:
27966:Geometric algebra
27923:Kronecker product
27758:Linear projection
27743:Vector projection
27337:978-0-89871-361-9
27297:978-3-540-42775-9
27252:978-0-387-95385-4
27197:978-0-521-54823-6
27117:978-0-89871-454-8
27052:978-0-321-28713-7
27002:978-0-8176-4684-4
26668:978-3-319-11079-0
26436:978-0-8218-3574-6
26083:, §I.4. See also
25217:{\displaystyle n}
25202:Bareiss Algorithm
24921:{\displaystyle n}
24786:{\displaystyle A}
24707:is just the sign
24700:{\displaystyle P}
24680:{\displaystyle U}
24660:{\displaystyle L}
24640:{\displaystyle U}
24620:{\displaystyle L}
24600:{\displaystyle 1}
24580:{\displaystyle P}
24516:{\displaystyle A}
24368:triangular matrix
24281:{\displaystyle n}
24167:quasideterminants
23990:Infinite matrices
23974:, as well as the
23698:{\displaystyle F}
23674:{\displaystyle A}
23528:
23497:{\displaystyle V}
23470:
23420:{\displaystyle n}
23296:
23232:{\displaystyle R}
23204:{\displaystyle R}
23177:
23147:
23003:
22983:
22967:
22937:{\displaystyle T}
22917:{\displaystyle V}
22890:
22864:{\displaystyle n}
22844:{\displaystyle R}
22824:{\displaystyle n}
22797:{\displaystyle V}
22777:{\displaystyle n}
22577:{\displaystyle m}
22557:{\displaystyle m}
22542:complex conjugate
22424:{\displaystyle f}
22404:{\displaystyle R}
22302:, there is a map
22269:ring homomorphism
22245:{\displaystyle R}
22221:{\displaystyle R}
22072:{\displaystyle R}
22048:{\displaystyle R}
21913:{\displaystyle R}
21370:
21091:
20994:
20982:
20887:{\displaystyle n}
20867:{\displaystyle A}
20771:{\displaystyle S}
20641:{\displaystyle A}
20546:is an orthogonal
20521:orthonormal basis
20513:orthogonal matrix
20412:{\displaystyle x}
20019:is defined to be
19857:, a third vector
19795:{\displaystyle A}
19306:{\displaystyle b}
19286:{\displaystyle A}
19266:{\displaystyle i}
19177:
19077:{\displaystyle x}
18962:, and Sylvester;
18876:theory of numbers
17998:
17944:{\displaystyle A}
17908:
17849:
17805:{\displaystyle A}
17785:{\displaystyle A}
17713:
17664:
17538:is less than the
17516:
17490:
17460:
17446:
17420:
17150:
17082:
16978:
16920:
16802:
16741:
16563:
16350:
16142:
16100:
16076:
16073:
16002:
15971:
15968:
15965:
15829:
15649:
15555:
15493:
15388:
15326:For example, for
14978:{\displaystyle n}
14958:{\displaystyle 1}
14938:{\displaystyle k}
14666:positive definite
14581:{\displaystyle A}
14557:{\displaystyle A}
14537:{\displaystyle I}
14513:{\displaystyle t}
14418:{\displaystyle A}
14398:{\displaystyle 0}
14378:{\displaystyle A}
14358:{\displaystyle A}
14338:{\displaystyle 0}
14318:{\displaystyle A}
14099:{\displaystyle A}
14049:{\displaystyle A}
14036:of a matrix. Let
13934:
13705:
13688:{\displaystyle b}
13668:{\displaystyle a}
13474:
13457:
13443:
13099:{\displaystyle B}
13050:{\displaystyle A}
12636:Proof of identity
12635:
12605:
12588:
12574:
12453:
12427:
12401:
12295:implies that the
12278:
12208:
12110:{\displaystyle C}
12090:{\displaystyle B}
12070:{\displaystyle A}
11954:{\displaystyle B}
11934:{\displaystyle A}
11276:{\displaystyle B}
11256:{\displaystyle A}
11028:{\displaystyle D}
11008:{\displaystyle C}
10669:
10667:
10549:{\displaystyle D}
10500:{\displaystyle D}
10112:
10110:
9988:{\displaystyle A}
9633:
9327:
9215:
8778:{\displaystyle j}
8365:{\displaystyle i}
8283:{\displaystyle j}
8263:{\displaystyle i}
8243:{\displaystyle A}
8133:{\displaystyle A}
8118:Laplace expansion
8113:Laplace expansion
8107:compound matrices
8067:is isomorphic to
7866:{\displaystyle K}
7774:rotation matrices
7689:(respectively, a
7640:{\displaystyle K}
7620:{\displaystyle n}
7560:
7488:{\displaystyle A}
7465:{\displaystyle A}
7422:{\displaystyle A}
7402:{\displaystyle B}
7313:{\displaystyle B}
7293:{\displaystyle A}
7213:
7177:{\displaystyle A}
7060:
7059:
6841:
6407:{\displaystyle A}
6279:{\displaystyle A}
6030:triangular matrix
6021:{\displaystyle A}
5535:{\displaystyle A}
4937:{\displaystyle A}
4898:{\displaystyle I}
4723:{\displaystyle n}
4478:
4235:
4099:
3762:of a permutation
3449:
2632:
2570:
2382:{\displaystyle A}
1766:
1515:Geometric meaning
1337:{\displaystyle r}
499:coordinate system
479:matrix similarity
418:Laplace expansion
117:triangular matrix
16:(Redirected from
28134:
28089:
28088:
27971:Exterior algebra
27908:Hadamard product
27825:
27813:Linear equations
27684:
27677:
27670:
27661:
27660:
27622:
27600:
27599:
27581:
27556:
27550:
27542:
27526:
27491:
27469:
27443:
27427:
27414:
27403:
27377:
27355:, translated by
27340:
27322:
27321:
27320:
27314:
27308:, archived from
27289:
27275:
27265:
27256:
27237:
27219:
27201:
27186:(2nd ed.).
27176:Horn, Roger Alan
27155:
27137:
27125:
27120:, archived from
27100:
27073:
27055:
27037:
27013:
26979:
26961:
26959:
26952:
26934:
26924:
26899:
26878:
26877:
26867:
26844:
26824:Abstract algebra
26818:
26816:
26806:
26797:(125): 231–236.
26779:
26761:
26759:
26728:
26715:
26713:
26691:(102): 565–578,
26682:
26672:
26657:(3rd ed.).
26646:
26623:
26622:
26620:
26611:(2–3): 429–438.
26596:
26590:
26581:
26575:
26574:
26572:
26571:
26565:
26559:. Archived from
26536:
26527:
26521:
26516:
26510:
26504:
26498:
26493:
26487:
26486:
26484:
26472:
26466:
26460:
26454:
26447:
26441:
26440:
26420:
26414:
26412:
26410:
26409:
26404:
26402:
26386:
26385:
26341:
26340:
26299:
26298:
26257:
26253:
26252:
26211:
26210:
26135:
26129:
26124:
26118:
26108:
26102:
26096:
26090:
26078:
26072:
26066:
26060:
26054:
26048:
26047:
26039:
26033:
26027:
26021:
26015:
26009:
26003:
25997:
25992:
25986:
25980:
25974:
25968:
25962:
25956:
25950:
25944:
25938:
25932:
25926:
25918:
25912:
25906:
25900:
25886:
25880:
25874:
25868:
25862:
25856:
25855:
25853:
25851:
25842:. Archived from
25835:
25829:
25823:
25817:
25814:
25803:
25794:
25788:
25782:
25776:
25770:
25764:
25763:
25726:
25720:
25714:
25708:
25698:
25692:
25691:
25673:
25667:
25666:
25664:
25640:
25634:
25633:
25631:
25619:
25613:
25607:
25601:
25600:
25582:
25562:
25556:
25555:
25530:(501): 460–467.
25515:
25509:
25503:
25497:
25491:
25485:
25479:
25473:
25462:
25456:
25450:
25444:
25434:
25428:
25422:
25416:
25415:
25409:
25399:
25393:
25392:
25390:
25388:
25374:
25368:
25367:
25349:
25343:
25337:
25294:
25289:
25288:
25248:
25223:
25221:
25220:
25215:
25199:
25197:
25196:
25191:
25186:
25185:
25148:
25146:
25145:
25140:
25135:
25134:
25101:
25099:
25098:
25093:
25088:
25087:
25062:
25060:
25059:
25054:
25024:
25022:
25021:
25016:
24998:
24996:
24995:
24990:
24988:
24987:
24956:
24954:
24953:
24948:
24927:
24925:
24924:
24919:
24907:
24905:
24904:
24899:
24894:
24893:
24860:
24858:
24857:
24852:
24792:
24790:
24789:
24784:
24772:
24770:
24769:
24764:
24749:
24747:
24746:
24741:
24726:
24724:
24723:
24718:
24706:
24704:
24703:
24698:
24686:
24684:
24683:
24678:
24666:
24664:
24663:
24658:
24646:
24644:
24643:
24638:
24626:
24624:
24623:
24618:
24606:
24604:
24603:
24598:
24586:
24584:
24583:
24578:
24560:
24558:
24557:
24552:
24522:
24520:
24519:
24514:
24499:
24497:
24496:
24491:
24464:
24462:
24461:
24456:
24451:
24450:
24416:QR decomposition
24412:LU decomposition
24409:
24407:
24406:
24401:
24365:
24360:row echelon form
24342:
24340:
24339:
24334:
24316:
24314:
24313:
24308:
24287:
24285:
24284:
24279:
24267:
24265:
24264:
24259:
24217:
24215:
24214:
24209:
24207:
24206:
24201:
24152:
24144:
24098:
24096:
24095:
24090:
23969:
23967:
23966:
23961:
23947:
23946:
23942:
23915:
23913:
23912:
23907:
23905:
23904:
23892:
23891:
23879:
23878:
23866:
23865:
23800:
23798:
23797:
23792:
23778:
23777:
23742:
23740:
23739:
23734:
23704:
23702:
23701:
23696:
23680:
23678:
23677:
23672:
23649:
23647:
23646:
23641:
23639:
23638:
23611:
23609:
23608:
23603:
23567:
23565:
23564:
23559:
23541:
23539:
23538:
23533:
23527:
23519:
23503:
23501:
23500:
23495:
23483:
23481:
23480:
23475:
23469:
23461:
23453:
23451:
23450:
23445:
23443:
23442:
23426:
23424:
23423:
23418:
23403:
23401:
23400:
23395:
23390:
23389:
23371:
23370:
23343:
23339:
23338:
23337:
23319:
23318:
23304:
23300:
23295:
23287:
23271:
23269:
23268:
23263:
23255:
23254:
23238:
23236:
23235:
23230:
23210:
23208:
23207:
23202:
23190:
23188:
23187:
23182:
23176:
23168:
23160:
23158:
23157:
23152:
23146:
23138:
23127:
23125:
23124:
23119:
23117:
23110:
23109:
23088:
23087:
23072:
23071:
23052:
23051:
23033:
23032:
23020:
23019:
23002:
22994:
22982:
22974:
22966:
22958:
22943:
22941:
22940:
22935:
22923:
22921:
22920:
22915:
22903:
22901:
22900:
22895:
22889:
22881:
22870:
22868:
22867:
22862:
22850:
22848:
22847:
22842:
22830:
22828:
22827:
22822:
22803:
22801:
22800:
22795:
22783:
22781:
22780:
22775:
22763:
22761:
22760:
22755:
22722:Exterior algebra
22717:
22715:
22714:
22709:
22704:
22703:
22698:
22689:
22688:
22658:algebraic groups
22655:
22653:
22652:
22647:
22645:
22644:
22622:
22620:
22619:
22614:
22612:
22611:
22583:
22581:
22580:
22575:
22563:
22561:
22560:
22555:
22533:
22531:
22530:
22525:
22514:
22513:
22471:
22470:
22430:
22428:
22427:
22422:
22410:
22408:
22407:
22402:
22390:
22388:
22387:
22382:
22368:
22367:
22343:
22342:
22318:
22317:
22301:
22299:
22298:
22293:
22251:
22249:
22248:
22243:
22227:
22225:
22224:
22219:
22207:
22205:
22204:
22199:
22174:
22172:
22171:
22166:
22161:
22160:
22136:
22135:
22106:
22104:
22103:
22098:
22096:
22078:
22076:
22075:
22070:
22054:
22052:
22051:
22046:
22034:
22032:
22031:
22026:
22012:
22011:
21980:
21978:
21977:
21972:
21941:
21939:
21938:
21933:
21931:
21919:
21917:
21916:
21911:
21899:commutative ring
21862:
21860:
21859:
21854:
21817:
21815:
21814:
21809:
21777:
21776:
21716:
21715:
21669:
21643:similar matrices
21613:
21611:
21610:
21605:
21600:
21591:
21587:
21583:
21546:
21529:
21528:
21516:
21504:
21493:
21492:
21464:
21412:
21410:
21409:
21404:
21399:
21398:
21375:
21371:
21369:
21368:
21367:
21354:
21353:
21352:
21339:
21304:
21291:
21289:
21288:
21283:
21278:
21277:
21272:
21263:
21262:
21257:
21218:
21216:
21215:
21210:
21208:
21207:
21202:
21193:
21192:
21187:
21169:A nonlinear map
21157:
21155:
21154:
21149:
21147:
21100:
21092:
21084:
21075:
21073:
21072:
21067:
21023:
21021:
21020:
21015:
20995:
20993:
20989:
20985:
20984:
20983:
20965:
20926:
20924:
20923:
20918:
20893:
20891:
20890:
20885:
20873:
20871:
20870:
20865:
20853:
20851:
20850:
20845:
20827:
20825:
20824:
20819:
20817:
20816:
20811:
20802:
20801:
20796:
20777:
20775:
20774:
20769:
20757:
20755:
20754:
20749:
20747:
20730:
20718:
20716:
20715:
20710:
20682:
20680:
20679:
20674:
20672:
20671:
20666:
20647:
20645:
20644:
20639:
20627:
20625:
20624:
20619:
20617:
20616:
20611:
20602:
20601:
20596:
20553:
20549:
20515:with entries in
20444:
20442:
20441:
20436:
20418:
20416:
20415:
20410:
20395:
20393:
20392:
20387:
20382:
20381:
20364:
20347:
20321:
20304:
20283:
20266:
20218:
20189:
20165:
20143:
20142:
20117:
20116:
20096:
20095:
20063:
20062:
20044:
20043:
20010:
20008:
20007:
20002:
19985:(supposed to be
19984:
19982:
19981:
19976:
19965:
19964:
19937:
19936:
19916:
19914:
19913:
19908:
19883:
19881:
19880:
19875:
19873:
19872:
19856:
19854:
19853:
19848:
19846:
19845:
19840:
19831:
19830:
19818:
19817:
19801:
19799:
19798:
19793:
19781:
19779:
19778:
19773:
19734:
19732:
19731:
19726:
19721:
19720:
19692:
19690:
19689:
19684:
19679:
19678:
19598:
19596:
19595:
19590:
19588:
19587:
19568:
19566:
19565:
19560:
19546:
19545:
19533:
19532:
19525:
19524:
19508:
19507:
19490:
19489:
19478:
19477:
19455:
19454:
19434:
19433:
19423:
19418:
19400:
19399:
19392:
19391:
19365:
19364:
19338:
19337:
19312:
19310:
19309:
19304:
19292:
19290:
19289:
19284:
19272:
19270:
19269:
19264:
19252:
19250:
19249:
19244:
19242:
19241:
19222:
19220:
19219:
19214:
19178:
19176:
19162:
19158:
19157:
19141:
19136:
19135:
19112:
19110:
19109:
19104:
19083:
19081:
19080:
19075:
19063:
19061:
19060:
19055:
18932:Crelle's Journal
18910:
18800:
18798:
18797:
18792:
18790:
18783:
18775:
18751:
18750:
18749:
18735:
18727:
18703:
18702:
18701:
18687:
18679:
18655:
18654:
18653:
18625:
18623:
18622:
18617:
18593:
18591:
18590:
18585:
18583:
18582:
18541:
18539:
18538:
18533:
18512:
18510:
18509:
18504:
18502:
18501:
18496:
18495:
18475:
18473:
18472:
18467:
18446:
18444:
18443:
18438:
18436:
18435:
18408:
18406:
18405:
18400:
18395:
18391:
18390:
18313:
18311:
18310:
18305:
18279:
18277:
18276:
18271:
18269:
18265:
18264:
18242:
18238:
18234:
18233:
18195:
18191:
18190:
18086:
18084:
18083:
18078:
18073:
18072:
18064:
18060:
18059:
18027:
18026:
17999:
17997:
17996:
17995:
17979:
17962:
17950:
17948:
17947:
17942:
17927:
17925:
17924:
17919:
17914:
17910:
17909:
17907:
17899:
17891:
17889:
17888:
17850:
17848:
17840:
17823:
17811:
17809:
17808:
17803:
17791:
17789:
17788:
17783:
17767:
17765:
17764:
17759:
17732:
17730:
17729:
17724:
17719:
17715:
17714:
17712:
17704:
17696:
17665:
17663:
17655:
17638:
17625:Jacobi's formula
17618:
17616:
17615:
17610:
17608:
17596:
17594:
17593:
17588:
17586:
17585:
17574:
17548:root mean square
17530:
17528:
17527:
17522:
17517:
17515:
17511:
17510:
17491:
17483:
17481:
17461:
17453:
17448:
17447:
17439:
17421:
17419:
17418:
17414:
17413:
17387:
17364:
17351:
17349:
17348:
17343:
17296:
17292:
17291:
17290:
17251:
17237:
17225:
17215:
17211:
17200:
17198:
17197:
17192:
17186:
17185:
17180:
17176:
17175:
17171:
17170:
17151:
17146:
17145:
17144:
17135:
17134:
17115:
17112:
17107:
17083:
17081:
17070:
17067:
17062:
17037:
17028:
17026:
17025:
17020:
17014:
17013:
17008:
17004:
17003:
16999:
16998:
16979:
16974:
16973:
16972:
16953:
16950:
16945:
16921:
16919:
16908:
16905:
16900:
16846:
16839:
16837:
16836:
16831:
16825:
16820:
16810:
16779:
16774:
16764:
16759:
16749:
16736:
16731:
16703:
16675:
16647:
16635:
16633:
16632:
16627:
16619:
16618:
16600:
16599:
16587:
16586:
16574:
16573:
16564:
16562:
16554:
16553:
16552:
16533:
16478:
16476:
16475:
16470:
16459:
16458:
16445:
16440:
16415:
16402:
16400:
16399:
16394:
16389:
16388:
16387:
16386:
16376:
16372:
16371:
16351:
16349:
16345:
16344:
16335:
16334:
16333:
16332:
16317:
16316:
16315:
16308:
16307:
16283:
16280:
16275:
16259:
16249:
16248:
16227:
16226:
16211:
16210:
16191:
16190:
16172:
16171:
16159:
16158:
16113:
16111:
16110:
16105:
16098:
16074:
16071:
16070:
16066:
16065:
16046:
16045:
16023:
16018:
16003:
15995:
15987:
15986:
15969:
15966:
15963:
15953:
15952:
15929:
15915:
15896:
15894:
15893:
15888:
15886:
15879:
15875:
15874:
15870:
15869:
15827:
15826:
15822:
15821:
15796:
15795:
15790:
15786:
15785:
15781:
15780:
15749:
15748:
15743:
15739:
15717:
15713:
15712:
15687:
15686:
15681:
15677:
15650:
15642:
15614:
15610:
15609:
15605:
15604:
15579:
15575:
15574:
15553:
15531:
15530:
15525:
15521:
15494:
15486:
15458:
15454:
15453:
15449:
15448:
15426:
15425:
15420:
15416:
15389:
15381:
15347:, respectively,
15346:
15339:
15332:
15322:
15320:
15319:
15314:
15257:
15250:
15248:
15247:
15242:
15209:
15205:
15197:
15193:
15189:
15185:
15181:
15173:
15166:
15164:
15163:
15158:
15089:
15082:
15080:
15079:
15074:
15008:
15004:
14984:
14982:
14981:
14976:
14964:
14962:
14961:
14956:
14944:
14942:
14941:
14936:
14921:
14919:
14918:
14913:
14911:
14910:
14903:
14902:
14880:
14879:
14862:
14861:
14820:
14819:
14797:
14796:
14779:
14778:
14759:
14758:
14736:
14735:
14718:
14717:
14691:
14690:
14662:Hermitian matrix
14656:
14654:
14653:
14648:
14631:
14630:
14611:
14609:
14608:
14603:
14587:
14585:
14584:
14579:
14563:
14561:
14560:
14555:
14543:
14541:
14540:
14535:
14519:
14517:
14516:
14511:
14496:
14494:
14493:
14488:
14447:
14446:
14424:
14422:
14421:
14416:
14404:
14402:
14401:
14396:
14384:
14382:
14381:
14376:
14364:
14362:
14361:
14356:
14344:
14342:
14341:
14336:
14324:
14322:
14321:
14316:
14294:
14292:
14291:
14286:
14281:
14280:
14268:
14267:
14258:
14257:
14245:
14244:
14234:
14229:
14186:is equal to the
14185:
14181:
14177:
14170:
14168:
14167:
14162:
14160:
14159:
14141:
14140:
14128:
14127:
14105:
14103:
14102:
14097:
14081:
14079:
14078:
14073:
14055:
14053:
14052:
14047:
14014:
14012:
14011:
14006:
13995:
13962:
13960:
13959:
13954:
13943:
13935:
13932:
13920:
13918:
13917:
13912:
13907:
13906:
13894:
13893:
13878:
13867:
13866:
13842:
13841:
13826:
13793:
13791:
13790:
13785:
13783:
13762:
13760:
13759:
13754:
13733:
13731:
13730:
13725:
13714:
13706:
13703:
13694:
13692:
13691:
13686:
13674:
13672:
13671:
13666:
13654:
13652:
13651:
13646:
13644:
13643:
13599:
13598:
13557:
13533:
13531:
13530:
13525:
13500:
13498:
13497:
13492:
13475:
13472:
13458:
13455:
13444:
13441:
13426:
13424:
13423:
13418:
13413:
13412:
13403:
13402:
13390:
13389:
13380:
13379:
13367:
13366:
13357:
13356:
13344:
13343:
13334:
13333:
13315:
13314:
13302:
13301:
13286:
13285:
13273:
13272:
13247:
13245:
13244:
13239:
13234:
13233:
13224:
13223:
13211:
13210:
13201:
13200:
13188:
13187:
13178:
13177:
13165:
13164:
13155:
13154:
13105:
13103:
13102:
13097:
13085:
13083:
13082:
13077:
13056:
13054:
13053:
13048:
13033:
13031:
13030:
13025:
13020:
13019:
13010:
13009:
12997:
12996:
12987:
12986:
12974:
12973:
12964:
12963:
12951:
12950:
12941:
12940:
12928:
12927:
12918:
12917:
12905:
12904:
12895:
12894:
12882:
12881:
12872:
12871:
12859:
12858:
12849:
12848:
12830:Expanding gives
12826:
12824:
12823:
12818:
12810:
12809:
12797:
12796:
12781:
12780:
12768:
12767:
12749:
12748:
12736:
12735:
12720:
12719:
12707:
12706:
12684:
12682:
12681:
12676:
12674:
12673:
12658:
12657:
12631:
12629:
12628:
12623:
12606:
12603:
12589:
12586:
12575:
12572:
12509:
12507:
12506:
12501:
12471:
12467:
12465:
12464:
12459:
12454:
12452:
12447:
12433:
12428:
12426:
12421:
12407:
12402:
12400:
12395:
12375:
12366:
12364:
12363:
12358:
12340:
12336:
12332:
12330:
12329:
12324:
12301:concave function
12298:
12289:
12287:
12286:
12281:
12279:
12276:
12219:
12217:
12216:
12211:
12209:
12206:
12116:
12114:
12113:
12108:
12096:
12094:
12093:
12088:
12076:
12074:
12073:
12068:
12042:
12040:
12039:
12034:
12006:
12004:
12003:
11998:
11983:
11981:
11980:
11975:
11960:
11958:
11957:
11952:
11940:
11938:
11937:
11932:
11917:
11915:
11914:
11909:
11863:
11859:
11855:
11854:
11778:
11776:
11775:
11770:
11765:
11761:
11757:
11756:
11738:
11737:
11736:
11670:
11656:
11654:
11653:
11648:
11628:
11624:
11614:
11613:
11612:
11565:
11555:
11524:
11522:
11521:
11516:
11511:
11507:
11497:
11496:
11495:
11474:
11470:
11460:
11459:
11458:
11423:matrix (so that
11422:
11408:
11395:states that for
11383:
11381:
11380:
11375:
11331:
11330:
11283:do not commute)
11282:
11280:
11279:
11274:
11262:
11260:
11259:
11254:
11242:
11240:
11239:
11234:
11216:
11214:
11213:
11208:
11187:
11185:
11184:
11179:
11158:
11156:
11155:
11150:
11118:
11117:
11069:
11067:
11066:
11061:
11034:
11032:
11031:
11026:
11014:
11012:
11011:
11006:
10980:
10978:
10977:
10972:
10970:
10957:
10956:
10911:
10907:
10906:
10899:
10898:
10880:
10879:
10859:
10858:
10808:
10803:
10802:
10801:
10772:
10771:
10748:
10743:
10742:
10741:
10734:
10733:
10716:
10715:
10691:
10690:
10666:
10665:
10608:
10607:
10555:
10553:
10552:
10547:
10532:
10530:
10529:
10524:
10506:
10504:
10503:
10498:
10486:
10484:
10483:
10478:
10470:
10469:
10423:
10421:
10420:
10415:
10413:
10400:
10399:
10354:
10350:
10349:
10339:
10338:
10315:
10314:
10290:
10289:
10251:
10246:
10245:
10244:
10215:
10214:
10191:
10186:
10185:
10184:
10177:
10176:
10155:
10154:
10137:
10136:
10109:
10108:
10051:
10050:
9994:
9992:
9991:
9986:
9971:
9969:
9968:
9963:
9958:
9957:
9889:
9888:
9839:Schur complement
9836:
9834:
9833:
9828:
9810:
9808:
9807:
9802:
9784:
9782:
9781:
9776:
9758:
9756:
9755:
9750:
9732:
9730:
9729:
9724:
9690:
9688:
9687:
9682:
9656:
9654:
9653:
9648:
9634:
9632:
9621:
9616:
9615:
9586:
9584:
9583:
9578:
9508:
9506:
9505:
9500:
9495:
9494:
9482:
9481:
9454:
9453:
9410:
9408:
9407:
9402:
9345:
9343:
9342:
9337:
9335:
9333:
9332:
9319:
9288:
9286:
9285:
9280:
9275:
9271:
9270:
9269:
9257:
9256:
9241:
9211:
9210:
9202:
9191:
9174:
9163:
9151:
9140:
9128:
9117:
9076:
9071:
9054:
9049:
9037:
9032:
9020:
9015:
9002:
9001:
8985:
8984:
8973:
8972:
8961:
8960:
8902:
8900:
8899:
8894:
8889:
8888:
8873:
8872:
8857:
8856:
8831:
8826:
8787:is the equality
8784:
8782:
8781:
8776:
8762:
8760:
8759:
8754:
8733:
8731:
8730:
8725:
8723:
8722:
8681:
8680:
8639:
8638:
8597:
8596:
8524:
8522:
8521:
8516:
8496:
8487:
8485:
8484:
8479:
8474:
8473:
8458:
8457:
8442:
8441:
8416:
8411:
8371:
8369:
8368:
8363:
8347:
8345:
8344:
8339:
8337:
8336:
8321:
8320:
8289:
8287:
8286:
8281:
8270:-th row and the
8269:
8267:
8266:
8261:
8250:by removing the
8249:
8247:
8246:
8241:
8229:
8227:
8226:
8221:
8179:
8177:
8176:
8171:
8169:
8168:
8139:
8137:
8136:
8131:
8093:
8091:
8090:
8085:
8083:
8082:
8066:
8064:
8063:
8058:
8044:
8043:
8034:
8017:
8016:
7997:
7995:
7994:
7989:
7975:
7974:
7954:
7952:
7951:
7946:
7932:
7931:
7911:
7909:
7908:
7903:
7889:
7888:
7872:
7870:
7869:
7864:
7852:
7850:
7849:
7844:
7842:
7841:
7825:
7823:
7822:
7817:
7803:
7802:
7759:
7757:
7756:
7751:
7737:
7736:
7712:
7711:
7688:
7686:
7685:
7680:
7666:
7665:
7646:
7644:
7643:
7638:
7626:
7624:
7623:
7618:
7602:
7600:
7599:
7594:
7592:
7591:
7561:
7559:
7542:
7537:
7533:
7532:
7494:
7492:
7491:
7486:
7471:
7469:
7468:
7463:
7451:
7449:
7448:
7443:
7428:
7426:
7425:
7420:
7408:
7406:
7405:
7400:
7385:
7383:
7382:
7377:
7319:
7317:
7316:
7311:
7299:
7297:
7296:
7291:
7259:
7245:
7243:
7242:
7237:
7220:
7216:
7215:
7214:
7183:
7181:
7180:
7175:
7152:
7150:
7149:
7144:
7100:
7092:
7081:
7073:
7055:
7053:
7052:
7047:
7045:
7037:
7029:
7021:
7006:
7004:
7003:
6998:
6996:
6988:
6977:
6969:
6954:
6952:
6951:
6946:
6944:
6936:
6928:
6920:
6906:
6904:
6903:
6898:
6896:
6888:
6880:
6872:
6852:
6850:
6849:
6844:
6842:
6834:
6798:
6796:
6795:
6790:
6788:
6787:
6703:
6701:
6700:
6695:
6693:
6692:
6608:
6606:
6605:
6600:
6598:
6597:
6514:
6512:
6511:
6506:
6504:
6503:
6415:
6413:
6411:
6410:
6405:
6392:
6389:
6387:
6386:
6381:
6376:
6375:
6285:
6283:
6282:
6277:
6251:
6249:
6248:
6243:
6227:
6225:
6224:
6219:
6214:
6213:
6200:
6195:
6177:
6176:
6161:
6160:
6151:
6150:
6119:
6117:
6116:
6111:
6093:
6091:
6090:
6085:
6067:
6065:
6064:
6059:
6051:
6050:
6027:
6025:
6024:
6019:
5981:
5979:
5978:
5973:
5968:
5963:
5962:
5944:
5943:
5931:
5930:
5918:
5917:
5905:
5904:
5895:
5887:
5882:
5881:
5863:
5862:
5850:
5849:
5837:
5836:
5824:
5823:
5814:
5803:
5798:
5797:
5782:
5781:
5769:
5768:
5756:
5755:
5743:
5742:
5733:
5721:
5719:
5718:
5713:
5708:
5703:
5702:
5684:
5683:
5665:
5664:
5646:
5645:
5636:
5625:
5620:
5619:
5601:
5600:
5585:
5584:
5566:
5565:
5556:
5541:
5539:
5538:
5533:
5521:
5519:
5518:
5513:
5495:
5493:
5492:
5487:
5473:
5472:
5410:
5408:
5407:
5402:
5378:
5376:
5375:
5370:
5362:
5357:
5356:
5314:
5313:
5304:
5278:
5276:
5275:
5270:
5268:
5264:
5259:
5258:
5228:
5227:
5218:
5210:
5205:
5204:
5177:
5176:
5167:
5150:
5146:
5141:
5140:
5122:
5121:
5085:
5084:
5060:
5059:
5050:
5049:
5036:
5028:
4991:
4989:
4988:
4983:
4963:
4962:
4944:is written as a
4943:
4941:
4940:
4935:
4904:
4902:
4901:
4896:
4884:
4882:
4881:
4876:
4868:
4834:
4832:
4831:
4826:
4824:
4823:
4801:
4799:
4798:
4793:
4788:
4787:
4781:
4780:
4762:
4761:
4752:
4751:
4729:
4727:
4726:
4721:
4705:
4703:
4702:
4697:
4667:
4665:
4664:
4659:
4626:
4619:
4617:
4616:
4611:
4606:
4605:
4604:
4603:
4579:
4578:
4577:
4576:
4556:
4555:
4554:
4553:
4541:
4540:
4525:
4524:
4523:
4505:
4504:
4492:
4491:
4448:
4444:
4442:
4441:
4436:
4402:
4398:
4396:
4395:
4390:
4388:
4387:
4386:
4385:
4367:
4366:
4338:
4336:
4335:
4330:
4328:
4324:
4323:
4322:
4297:
4292:
4256:
4255:
4254:
4202:
4200:
4199:
4194:
4189:
4188:
4161:
4160:
4120:
4119:
4118:
4095:
4094:
4087:
4086:
4068:
4067:
4037:
4036:
4018:
4017:
3962:
3960:
3959:
3954:
3949:
3948:
3941:
3940:
3922:
3921:
3891:
3890:
3872:
3871:
3830:
3828:
3827:
3822:
3807:
3805:
3804:
3799:
3781:
3779:
3778:
3773:
3761:
3759:
3758:
3753:
3730:. The signature
3729:
3727:
3726:
3721:
3719:
3718:
3698:
3696:
3695:
3690:
3633:
3631:
3630:
3625:
3610:
3608:
3607:
3602:
3558:
3556:
3555:
3550:
3517:
3459:
3457:
3456:
3451:
3447:
3371:
3370:
3298:
3260:
3258:
3257:
3252:
3247:
3246:
3239:
3238:
3216:
3215:
3198:
3197:
3156:
3155:
3133:
3132:
3115:
3114:
3095:
3094:
3072:
3071:
3054:
3053:
3008:commutative ring
3005:
3003:
3002:
2997:
2995:
2994:
2969:
2967:
2966:
2961:
2956:
2955:
2948:
2947:
2925:
2924:
2907:
2906:
2865:
2864:
2842:
2841:
2824:
2823:
2804:
2803:
2781:
2780:
2763:
2762:
2655:
2653:
2652:
2647:
2633:
2630:
2594:
2592:
2591:
2586:
2581:
2577:
2568:
2561:
2560:
2542:
2541:
2536:
2530:
2529:
2511:
2510:
2505:
2499:
2498:
2471:
2469:
2468:
2463:
2458:
2457:
2452:
2437:
2436:
2431:
2422:
2421:
2416:
2388:
2386:
2385:
2380:
2369:This means that
2365:
2363:
2362:
2357:
2352:
2351:
2346:
2337:
2336:
2283:
2282:
2277:
2268:
2267:
2221:
2220:
2215:
2206:
2205:
2153:
2151:
2150:
2145:
2143:
2139:
2135:
2134:
2129:
2116:
2115:
2110:
2102:
2101:
2096:
2062:
2049:
2035:
2015:
2003:
1991:
1951:
1949:
1948:
1943:
1920:
1919:
1891:
1890:
1859:
1846:
1842:
1832:
1828:
1827:
1822:
1801:
1796:
1791:
1785:
1780:
1775:
1767:
1764:
1752:
1738:
1723:
1696:
1681:
1666:
1651:
1623:
1619:
1615:
1602:
1590:
1570:
1558:
1546:
1542:
1530:
1510:
1508:
1507:
1502:
1497:
1496:
1401:
1400:
1343:
1341:
1340:
1335:
1320:
1318:
1317:
1312:
1307:
1306:
1299:
1282:
1258:
1257:
1213:
1190:
1167:
1166:
1159:
1136:
1096:
1094:
1093:
1088:
1062:
1061:
1016:
1014:
1013:
1008:
1006:
1005:
960:
958:
957:
952:
931:First properties
926:
924:
923:
918:
862:
861:
854:
818:
817:
763:
761:
760:
755:
732:
731:
693:
692:
644:
640:
638:
637:
632:
630:
629:
587:
555:
539:
531:
463:
456:
444:
430:row echelon form
415:
407:
405:
404:
399:
380:
367:
365:
364:
359:
282:
281:
209:
202:
200:
199:
194:
171:
170:
125:
91:
89:
81:
74:
66:
21:
28142:
28141:
28137:
28136:
28135:
28133:
28132:
28131:
28102:
28101:
28100:
28095:
28077:
28039:
27995:
27932:
27884:
27826:
27817:
27783:Change of basis
27773:Multilinear map
27711:
27693:
27688:
27638:Wayback Machine
27563:
27544:
27543:
27510:(22): 320–359,
27489:
27477:, vol. 1,
27459:
27393:
27375:
27347:
27338:
27318:
27316:
27312:
27298:
27273:
27253:
27235:
27217:
27198:
27184:Matrix Analysis
27153:
27118:
27098:
27071:
27053:
27035:
27003:
26977:
26957:
26932:
26906:10.2307/4145188
26834:
26777:
26757:
26726:
26711:
26697:10.2307/2004533
26680:
26669:
26637:
26631:
26626:
26597:
26593:
26582:
26578:
26569:
26567:
26563:
26557:
26534:
26528:
26524:
26517:
26513:
26505:
26501:
26494:
26490:
26473:
26469:
26461:
26457:
26448:
26444:
26437:
26421:
26417:
26400:
26399:
26380:
26379:
26374:
26368:
26367:
26362:
26352:
26351:
26335:
26334:
26329:
26323:
26322:
26317:
26307:
26306:
26293:
26292:
26287:
26281:
26280:
26275:
26265:
26264:
26255:
26254:
26247:
26246:
26241:
26235:
26234:
26229:
26219:
26218:
26205:
26204:
26199:
26193:
26192:
26187:
26177:
26176:
26163:
26153:
26151:
26148:
26147:
26136:
26132:
26125:
26121:
26109:
26105:
26097:
26093:
26079:
26075:
26067:
26063:
26055:
26051:
26040:
26036:
26028:
26024:
26016:
26012:
26004:
26000:
25993:
25989:
25981:
25977:
25969:
25965:
25957:
25953:
25945:
25941:
25933:
25929:
25919:
25915:
25907:
25903:
25887:
25883:
25875:
25871:
25863:
25859:
25849:
25847:
25838:
25836:
25832:
25824:
25820:
25815:
25806:
25795:
25791:
25783:
25779:
25775:, § 0.8.10
25771:
25767:
25727:
25723:
25715:
25711:
25699:
25695:
25683:
25674:
25670:
25641:
25637:
25620:
25616:
25608:
25604:
25563:
25559:
25536:10.2307/3620776
25516:
25512:
25504:
25500:
25492:
25488:
25480:
25476:
25464:Alternatively,
25463:
25459:
25451:
25447:
25435:
25431:
25423:
25419:
25400:
25396:
25386:
25384:
25376:
25375:
25371:
25350:
25346:
25338:
25334:
25330:
25325:
25290:
25283:
25280:
25240:
25209:
25206:
25205:
25181:
25177:
25166:
25163:
25162:
25130:
25126:
25115:
25112:
25111:
25083:
25079:
25068:
25065:
25064:
25030:
25027:
25026:
25004:
25001:
25000:
24983:
24979:
24962:
24959:
24958:
24933:
24930:
24929:
24913:
24910:
24909:
24889:
24885:
24874:
24871:
24870:
24867:
24865:Further methods
24801:
24798:
24797:
24778:
24775:
24774:
24755:
24752:
24751:
24732:
24729:
24728:
24712:
24709:
24708:
24692:
24689:
24688:
24672:
24669:
24668:
24652:
24649:
24648:
24632:
24629:
24628:
24612:
24609:
24608:
24592:
24589:
24588:
24572:
24569:
24568:
24531:
24528:
24527:
24508:
24505:
24504:
24470:
24467:
24466:
24446:
24442:
24431:
24428:
24427:
24386:
24383:
24382:
24379:
24363:
24349:
24325:
24322:
24321:
24296:
24293:
24292:
24273:
24270:
24269:
24250:
24247:
24246:
24232:
24202:
24197:
24196:
24194:
24191:
24190:
24155:regular element
24150:
24139:
24136:
24112:
24018:
24015:
24014:
23992:
23972:field extension
23938:
23934:
23930:
23928:
23925:
23924:
23900:
23896:
23887:
23883:
23874:
23870:
23861:
23857:
23813:
23810:
23809:
23767:
23763:
23755:
23752:
23751:
23713:
23710:
23709:
23690:
23687:
23686:
23666:
23663:
23662:
23656:
23634:
23630:
23628:
23625:
23624:
23622:symmetric group
23585:
23582:
23581:
23574:
23547:
23544:
23543:
23523:
23517:
23514:
23513:
23510:chain complexes
23489:
23486:
23485:
23465:
23459:
23456:
23455:
23438:
23434:
23432:
23429:
23428:
23412:
23409:
23408:
23385:
23381:
23366:
23362:
23333:
23329:
23314:
23310:
23309:
23305:
23291:
23286:
23282:
23280:
23277:
23276:
23250:
23246:
23244:
23241:
23240:
23224:
23221:
23220:
23196:
23193:
23192:
23172:
23166:
23163:
23162:
23142:
23136:
23133:
23132:
23115:
23114:
23105:
23101:
23083:
23079:
23067:
23063:
23053:
23047:
23043:
23028:
23024:
23015:
23011:
23008:
23007:
22998:
22987:
22978:
22962:
22954:
22952:
22949:
22948:
22929:
22926:
22925:
22909:
22906:
22905:
22885:
22879:
22876:
22875:
22856:
22853:
22852:
22836:
22833:
22832:
22816:
22813:
22812:
22789:
22786:
22785:
22769:
22766:
22765:
22737:
22734:
22733:
22730:
22724:
22699:
22694:
22693:
22684:
22680:
22672:
22669:
22668:
22640:
22636:
22628:
22625:
22624:
22607:
22603:
22601:
22598:
22597:
22590:category theory
22569:
22566:
22565:
22549:
22546:
22545:
22503:
22499:
22460:
22456:
22439:
22436:
22435:
22416:
22413:
22412:
22396:
22393:
22392:
22363:
22359:
22338:
22334:
22313:
22309:
22307:
22304:
22303:
22275:
22272:
22271:
22237:
22234:
22233:
22213:
22210:
22209:
22187:
22184:
22183:
22156:
22152:
22131:
22127:
22125:
22122:
22121:
22092:
22084:
22081:
22080:
22064:
22061:
22060:
22040:
22037:
22036:
22001:
21997:
21989:
21986:
21985:
21951:
21948:
21947:
21927:
21925:
21922:
21921:
21905:
21902:
21901:
21895:
21836:
21833:
21832:
21769:
21765:
21708:
21704:
21678:
21675:
21674:
21658:
21639:
21634:
21596:
21579:
21557:
21553:
21542:
21524:
21520:
21512:
21500:
21479:
21475:
21473:
21470:
21469:
21452:
21439:(the domain of
21376:
21363:
21359:
21355:
21348:
21344:
21340:
21338:
21334:
21333:
21316:
21313:
21312:
21296:
21273:
21268:
21267:
21258:
21253:
21252:
21244:
21241:
21240:
21230:Jacobian matrix
21203:
21198:
21197:
21188:
21183:
21182:
21174:
21171:
21170:
21143:
21096:
21083:
21081:
21078:
21077:
21043:
21040:
21039:
20979:
20978:
20974:
20973:
20969:
20964:
20935:
20932:
20931:
20903:
20900:
20899:
20879:
20876:
20875:
20859:
20856:
20855:
20833:
20830:
20829:
20812:
20807:
20806:
20797:
20792:
20791:
20783:
20780:
20779:
20763:
20760:
20759:
20743:
20726:
20724:
20721:
20720:
20695:
20692:
20691:
20667:
20662:
20661:
20653:
20650:
20649:
20633:
20630:
20629:
20612:
20607:
20606:
20597:
20592:
20591:
20583:
20580:
20579:
20568:
20551:
20547:
20525:Euclidean space
20477:
20471:
20424:
20421:
20420:
20404:
20401:
20400:
20376:
20375:
20348:
20343:
20337:
20332:
20305:
20300:
20294:
20267:
20262:
20255:
20254:
20249:
20244:
20239:
20233:
20232:
20214:
20208:
20203:
20185:
20179:
20161:
20154:
20153:
20138:
20134:
20132:
20127:
20112:
20108:
20106:
20091:
20087:
20080:
20079:
20058:
20054:
20039:
20035:
20027:
20024:
20023:
19990:
19987:
19986:
19960:
19956:
19932:
19928:
19926:
19923:
19922:
19896:
19893:
19892:
19868:
19864:
19862:
19859:
19858:
19841:
19836:
19835:
19826:
19822:
19813:
19809:
19807:
19804:
19803:
19787:
19784:
19783:
19764:
19761:
19760:
19753:
19716:
19712:
19701:
19698:
19697:
19674:
19670:
19611:
19608:
19607:
19583:
19579:
19577:
19574:
19573:
19541:
19537:
19527:
19526:
19520:
19516:
19514:
19509:
19497:
19493:
19491:
19485:
19481:
19479:
19467:
19463:
19461:
19456:
19450:
19446:
19439:
19438:
19429:
19425:
19419:
19408:
19394:
19393:
19387:
19383:
19381:
19376:
19371:
19366:
19360:
19356:
19349:
19348:
19333:
19329:
19321:
19318:
19317:
19298:
19295:
19294:
19278:
19275:
19274:
19258:
19255:
19254:
19237:
19233:
19231:
19228:
19227:
19163:
19153:
19149:
19142:
19140:
19131:
19127:
19125:
19122:
19121:
19089:
19086:
19085:
19084:if and only if
19069:
19066:
19065:
19040:
19037:
19036:
19029:
19024:
18902:
18807:
18788:
18787:
18779:
18771:
18764:
18745:
18744:
18740:
18737:
18736:
18731:
18723:
18716:
18697:
18696:
18692:
18689:
18688:
18683:
18675:
18668:
18649:
18648:
18644:
18640:
18638:
18635:
18634:
18599:
18596:
18595:
18577:
18576:
18571:
18566:
18556:
18555:
18547:
18544:
18543:
18521:
18518:
18517:
18497:
18488:
18487:
18486:
18484:
18481:
18480:
18452:
18449:
18448:
18431:
18427:
18425:
18422:
18421:
18386:
18382:
18378:
18322:
18319:
18318:
18293:
18290:
18289:
18260:
18256:
18252:
18226:
18222:
18221:
18217:
18186:
18182:
18178:
18098:
18095:
18094:
18065:
18052:
18048:
18044:
18043:
18019:
18015:
17988:
17984:
17980:
17963:
17961:
17959:
17956:
17955:
17936:
17933:
17932:
17900:
17892:
17890:
17881:
17877:
17876:
17872:
17841:
17824:
17822:
17820:
17817:
17816:
17797:
17794:
17793:
17777:
17774:
17773:
17741:
17738:
17737:
17705:
17697:
17695:
17679:
17675:
17656:
17639:
17637:
17635:
17632:
17631:
17604:
17602:
17599:
17598:
17575:
17570:
17569:
17567:
17564:
17563:
17556:
17544:arithmetic mean
17506:
17502:
17498:
17482:
17480:
17452:
17438:
17434:
17406:
17402:
17398:
17391:
17386:
17384:
17381:
17380:
17373:distributions.
17356:
17283:
17279:
17272:
17268:
17260:
17257:
17256:
17247:
17244:
17227:
17217:
17213:
17209:
17181:
17166:
17162:
17158:
17140:
17136:
17130:
17126:
17116:
17114:
17108:
17097:
17089:
17085:
17084:
17074:
17069:
17063:
17052:
17046:
17043:
17042:
17033:
17009:
16994:
16990:
16986:
16968:
16964:
16954:
16952:
16946:
16935:
16927:
16923:
16922:
16912:
16907:
16901:
16890:
16863:
16860:
16859:
16849:Mercator series
16844:
16821:
16816:
16806:
16775:
16770:
16760:
16755:
16745:
16732:
16727:
16712:
16709:
16708:
16700:
16694:
16687:
16677:
16672:
16666:
16659:
16649:
16645:
16640:
16614:
16610:
16595:
16591:
16582:
16578:
16569:
16565:
16555:
16548:
16544:
16534:
16532:
16515:
16512:
16511:
16498:
16484:Bell polynomial
16454:
16450:
16441:
16430:
16424:
16421:
16420:
16412:
16407:
16382:
16378:
16377:
16367:
16363:
16359:
16358:
16340:
16336:
16328:
16324:
16323:
16319:
16318:
16303:
16299:
16298:
16294:
16284:
16282:
16276:
16265:
16257:
16256:
16244:
16240:
16222:
16218:
16206:
16202:
16199:
16198:
16186:
16182:
16167:
16163:
16154:
16150:
16146:
16125:
16122:
16121:
16061:
16057:
16053:
16029:
16025:
16019:
16008:
15994:
15976:
15972:
15948:
15944:
15942:
15939:
15938:
15928:
15917:
15913:
15884:
15883:
15865:
15861:
15857:
15817:
15813:
15809:
15791:
15776:
15772:
15768:
15761:
15757:
15756:
15744:
15723:
15719:
15718:
15708:
15704:
15700:
15682:
15661:
15657:
15656:
15655:
15651:
15641:
15634:
15619:
15618:
15600:
15596:
15592:
15570:
15566:
15562:
15526:
15505:
15501:
15500:
15499:
15495:
15485:
15478:
15463:
15462:
15444:
15440:
15436:
15421:
15400:
15396:
15395:
15394:
15390:
15380:
15373:
15357:
15355:
15352:
15351:
15341:
15334:
15327:
15266:
15263:
15262:
15255:
15218:
15215:
15214:
15207:
15203:
15195:
15191:
15187:
15183:
15179:
15171:
15098:
15095:
15094:
15087:
15017:
15014:
15013:
15006:
15002:
14991:
14970:
14967:
14966:
14950:
14947:
14946:
14930:
14927:
14926:
14905:
14904:
14892:
14888:
14886:
14881:
14869:
14865:
14863:
14851:
14847:
14844:
14843:
14838:
14833:
14828:
14822:
14821:
14809:
14805:
14803:
14798:
14786:
14782:
14780:
14768:
14764:
14761:
14760:
14748:
14744:
14742:
14737:
14725:
14721:
14719:
14707:
14703:
14696:
14695:
14686:
14682:
14680:
14677:
14676:
14626:
14622:
14620:
14617:
14616:
14597:
14594:
14593:
14573:
14570:
14569:
14549:
14546:
14545:
14529:
14526:
14525:
14505:
14502:
14501:
14442:
14438:
14436:
14433:
14432:
14410:
14407:
14406:
14390:
14387:
14386:
14370:
14367:
14366:
14350:
14347:
14346:
14330:
14327:
14326:
14310:
14307:
14306:
14276:
14272:
14263:
14259:
14253:
14249:
14240:
14236:
14230:
14219:
14198:
14195:
14194:
14183:
14179:
14175:
14155:
14151:
14136:
14132:
14123:
14119:
14117:
14114:
14113:
14091:
14088:
14087:
14061:
14058:
14057:
14041:
14038:
14037:
14026:
14021:
13991:
13968:
13965:
13964:
13939:
13931:
13929:
13926:
13925:
13902:
13898:
13889:
13885:
13874:
13862:
13858:
13837:
13833:
13822:
13802:
13799:
13798:
13779:
13768:
13765:
13764:
13739:
13736:
13735:
13710:
13702:
13700:
13697:
13696:
13680:
13677:
13676:
13660:
13657:
13656:
13638:
13637:
13632:
13626:
13625:
13617:
13607:
13606:
13593:
13592:
13587:
13581:
13580:
13575:
13565:
13564:
13553:
13539:
13536:
13535:
13513:
13510:
13509:
13506:
13471:
13454:
13440:
13438:
13435:
13434:
13408:
13404:
13398:
13394:
13385:
13381:
13375:
13371:
13362:
13358:
13352:
13348:
13339:
13335:
13329:
13325:
13310:
13306:
13297:
13293:
13281:
13277:
13268:
13264:
13259:
13256:
13255:
13229:
13225:
13219:
13215:
13206:
13202:
13196:
13192:
13183:
13179:
13173:
13169:
13160:
13156:
13150:
13146:
13114:
13111:
13110:
13091:
13088:
13087:
13062:
13059:
13058:
13057:are seen to be
13042:
13039:
13038:
13015:
13011:
13005:
13001:
12992:
12988:
12982:
12978:
12969:
12965:
12959:
12955:
12946:
12942:
12936:
12932:
12923:
12919:
12913:
12909:
12900:
12896:
12890:
12886:
12877:
12873:
12867:
12863:
12854:
12850:
12844:
12840:
12838:
12835:
12834:
12805:
12801:
12792:
12788:
12776:
12772:
12763:
12759:
12744:
12740:
12731:
12727:
12715:
12711:
12702:
12698:
12693:
12690:
12689:
12666:
12662:
12650:
12646:
12644:
12641:
12640:
12602:
12585:
12571:
12518:
12515:
12514:
12489:
12486:
12485:
12482:
12469:
12448:
12434:
12432:
12422:
12408:
12406:
12396:
12376:
12374:
12372:
12369:
12368:
12346:
12343:
12342:
12338:
12334:
12312:
12309:
12308:
12296:
12275:
12225:
12222:
12221:
12205:
12122:
12119:
12118:
12117:of equal size,
12102:
12099:
12098:
12082:
12079:
12078:
12062:
12059:
12058:
12022:
12019:
12018:
12015:
12010:
11989:
11986:
11985:
11966:
11963:
11962:
11946:
11943:
11942:
11926:
11923:
11922:
11847:
11843:
11833:
11829:
11788:
11785:
11784:
11749:
11745:
11732:
11731:
11727:
11726:
11722:
11681:
11678:
11677:
11662:
11608:
11607:
11603:
11602:
11598:
11593:
11590:
11589:
11578:and row vector
11557:
11547:
11545:
11536:
11491:
11490:
11486:
11485:
11481:
11454:
11453:
11449:
11448:
11444:
11439:
11436:
11435:
11414:
11400:
11390:
11325:
11324:
11319:
11313:
11312:
11307:
11297:
11296:
11291:
11288:
11287:
11268:
11265:
11264:
11248:
11245:
11244:
11222:
11219:
11218:
11196:
11193:
11192:
11167:
11164:
11163:
11112:
11111:
11106:
11100:
11099:
11094:
11084:
11083:
11078:
11075:
11074:
11043:
11040:
11039:
11020:
11017:
11016:
11000:
10997:
10996:
10968:
10967:
10949:
10945:
10909:
10908:
10901:
10900:
10894:
10890:
10888:
10882:
10881:
10872:
10868:
10863:
10851:
10847:
10831:
10830:
10806:
10805:
10794:
10790:
10764:
10760:
10749:
10736:
10735:
10726:
10722:
10720:
10708:
10704:
10698:
10697:
10692:
10686:
10682:
10675:
10674:
10670:
10668:
10660:
10659:
10654:
10648:
10647:
10642:
10632:
10631:
10609:
10602:
10601:
10596:
10590:
10589:
10584:
10574:
10573:
10566:
10564:
10561:
10560:
10541:
10538:
10537:
10512:
10509:
10508:
10492:
10489:
10488:
10462:
10458:
10432:
10429:
10428:
10411:
10410:
10392:
10388:
10352:
10351:
10344:
10343:
10331:
10327:
10316:
10307:
10303:
10297:
10296:
10291:
10285:
10281:
10274:
10273:
10249:
10248:
10237:
10233:
10207:
10203:
10192:
10179:
10178:
10172:
10168:
10166:
10160:
10159:
10147:
10143:
10138:
10129:
10125:
10118:
10117:
10113:
10111:
10103:
10102:
10097:
10091:
10090:
10085:
10075:
10074:
10052:
10045:
10044:
10039:
10033:
10032:
10027:
10017:
10016:
10009:
10007:
10004:
10003:
9980:
9977:
9976:
9952:
9951:
9946:
9940:
9939:
9934:
9924:
9923:
9883:
9882:
9877:
9871:
9870:
9865:
9855:
9854:
9849:
9846:
9845:
9816:
9813:
9812:
9790:
9787:
9786:
9764:
9761:
9760:
9738:
9735:
9734:
9700:
9697:
9696:
9670:
9667:
9666:
9663:
9625:
9620:
9608:
9604:
9602:
9599:
9598:
9520:
9517:
9516:
9487:
9483:
9471:
9467:
9443:
9439:
9419:
9416:
9415:
9384:
9381:
9380:
9378:adjugate matrix
9374:
9372:Adjugate matrix
9328:
9315:
9314:
9312:
9310:
9307:
9306:
9265:
9261:
9252:
9248:
9247:
9243:
9219:
9205:
9204:
9192:
9187:
9181:
9176:
9164:
9159:
9153:
9141:
9136:
9130:
9118:
9113:
9106:
9105:
9100:
9095:
9090:
9085:
9079:
9078:
9072:
9067:
9061:
9056:
9050:
9045:
9039:
9033:
9028:
9022:
9016:
9011:
9004:
9003:
8997:
8993:
8991:
8986:
8980:
8976:
8974:
8968:
8964:
8962:
8956:
8952:
8949:
8948:
8943:
8938:
8933:
8928:
8918:
8917:
8915:
8912:
8911:
8878:
8874:
8862:
8858:
8846:
8842:
8827:
8816:
8795:
8792:
8791:
8770:
8767:
8766:
8742:
8739:
8738:
8717:
8716:
8711:
8705:
8704:
8699:
8689:
8688:
8675:
8674:
8669:
8663:
8662:
8657:
8647:
8646:
8633:
8632:
8627:
8621:
8620:
8615:
8605:
8604:
8591:
8590:
8585:
8580:
8574:
8573:
8568:
8563:
8557:
8556:
8551:
8546:
8536:
8535:
8533:
8530:
8529:
8504:
8501:
8500:
8494:
8463:
8459:
8447:
8443:
8431:
8427:
8412:
8401:
8380:
8377:
8376:
8357:
8354:
8353:
8326:
8322:
8310:
8306:
8295:
8292:
8291:
8275:
8272:
8271:
8255:
8252:
8251:
8235:
8232:
8231:
8185:
8182:
8181:
8158:
8154:
8152:
8149:
8148:
8125:
8122:
8121:
8115:
8078:
8074:
8072:
8069:
8068:
8039:
8035:
8030:
8012:
8008:
8006:
8003:
8002:
7998:, and that the
7970:
7966:
7964:
7961:
7960:
7957:normal subgroup
7927:
7923:
7921:
7918:
7917:
7884:
7880:
7878:
7875:
7874:
7858:
7855:
7854:
7837:
7833:
7831:
7828:
7827:
7798:
7794:
7792:
7789:
7788:
7732:
7728:
7707:
7703:
7701:
7698:
7697:
7661:
7657:
7655:
7652:
7651:
7632:
7629:
7628:
7612:
7609:
7608:
7584:
7580:
7546:
7541:
7525:
7521:
7517:
7512:
7509:
7508:
7480:
7477:
7476:
7457:
7454:
7453:
7434:
7431:
7430:
7414:
7411:
7410:
7394:
7391:
7390:
7332:
7329:
7328:
7305:
7302:
7301:
7285:
7282:
7281:
7274:
7251:
7210:
7209:
7205:
7201:
7196:
7193:
7192:
7169:
7166:
7165:
7158:
7096:
7088:
7077:
7069:
7067:
7064:
7063:
7041:
7033:
7025:
7017:
7015:
7012:
7011:
6992:
6984:
6973:
6965:
6963:
6960:
6959:
6940:
6932:
6924:
6916:
6914:
6911:
6910:
6892:
6884:
6876:
6868:
6866:
6863:
6862:
6833:
6828:
6825:
6824:
6782:
6781:
6773:
6768:
6762:
6761:
6756:
6751:
6745:
6744:
6739:
6731:
6721:
6720:
6712:
6709:
6708:
6687:
6686:
6678:
6673:
6667:
6666:
6661:
6656:
6650:
6649:
6644:
6636:
6626:
6625:
6617:
6614:
6613:
6592:
6591:
6583:
6578:
6572:
6571:
6566:
6561:
6555:
6554:
6549:
6544:
6531:
6530:
6522:
6519:
6518:
6498:
6497:
6489:
6484:
6478:
6477:
6472:
6467:
6461:
6460:
6455:
6447:
6434:
6433:
6425:
6422:
6421:
6399:
6396:
6395:
6370:
6369:
6361:
6356:
6347:
6346:
6341:
6336:
6330:
6329:
6324:
6316:
6303:
6302:
6294:
6291:
6290:
6271:
6268:
6267:
6259:
6237:
6234:
6233:
6230:diagonal matrix
6206:
6202:
6196:
6185:
6169:
6165:
6156:
6152:
6146:
6142:
6125:
6122:
6121:
6099:
6096:
6095:
6073:
6070:
6069:
6043:
6039:
6037:
6034:
6033:
6013:
6010:
6009:
5964:
5958:
5954:
5939:
5935:
5926:
5922:
5913:
5909:
5900:
5896:
5891:
5883:
5877:
5873:
5858:
5854:
5845:
5841:
5832:
5828:
5819:
5815:
5810:
5799:
5793:
5789:
5777:
5773:
5764:
5760:
5751:
5747:
5738:
5734:
5729:
5727:
5724:
5723:
5704:
5698:
5694:
5679:
5675:
5660:
5656:
5641:
5637:
5632:
5621:
5615:
5611:
5596:
5592:
5580:
5576:
5561:
5557:
5552:
5550:
5547:
5546:
5527:
5524:
5523:
5501:
5498:
5497:
5468:
5464:
5444:
5441:
5440:
5429:
5390:
5387:
5386:
5358:
5352:
5348:
5309:
5305:
5300:
5298:
5295:
5294:
5266:
5265:
5260:
5254:
5250:
5223:
5219:
5214:
5206:
5200:
5196:
5172:
5168:
5163:
5148:
5147:
5142:
5136:
5132:
5111:
5107:
5074:
5070:
5055:
5051:
5045:
5044:
5037:
5032:
5024:
5020:
5018:
5015:
5014:
4958:
4954:
4952:
4949:
4948:
4929:
4926:
4925:
4907:identity matrix
4890:
4887:
4886:
4858:
4853:
4850:
4849:
4819:
4815:
4813:
4810:
4809:
4783:
4782:
4776:
4772:
4757:
4753:
4747:
4746:
4738:
4735:
4734:
4715:
4712:
4711:
4685:
4682:
4681:
4678:
4673:
4632:
4629:
4628:
4624:
4599:
4595:
4588:
4584:
4572:
4568:
4561:
4557:
4549:
4545:
4536:
4532:
4531:
4527:
4519:
4515:
4500:
4496:
4487:
4483:
4482:
4461:
4458:
4457:
4446:
4412:
4409:
4408:
4407:of integers in
4400:
4381:
4377:
4362:
4358:
4357:
4353:
4351:
4348:
4347:
4303:
4299:
4293:
4282:
4262:
4258:
4250:
4246:
4239:
4218:
4215:
4214:
4169:
4165:
4141:
4137:
4114:
4110:
4103:
4089:
4088:
4076:
4072:
4057:
4053:
4050:
4049:
4039:
4038:
4026:
4022:
4007:
4003:
3996:
3995:
3978:
3975:
3974:
3943:
3942:
3930:
3926:
3911:
3907:
3904:
3903:
3893:
3892:
3880:
3876:
3861:
3857:
3850:
3849:
3841:
3838:
3837:
3833:Given a matrix
3813:
3810:
3809:
3787:
3784:
3783:
3767:
3764:
3763:
3735:
3732:
3731:
3714:
3710:
3708:
3705:
3704:
3701:symmetric group
3639:
3636:
3635:
3619:
3616:
3615:
3572:
3569:
3568:
3538:
3535:
3534:
3531:
3515:
3365:
3364:
3359:
3354:
3348:
3347:
3342:
3337:
3331:
3330:
3325:
3320:
3310:
3309:
3307:
3304:
3303:
3296:
3293:Leibniz formula
3289:
3284:
3278:
3276:Leibniz formula
3270:Leibniz formula
3241:
3240:
3228:
3224:
3222:
3217:
3205:
3201:
3199:
3187:
3183:
3180:
3179:
3174:
3169:
3164:
3158:
3157:
3145:
3141:
3139:
3134:
3122:
3118:
3116:
3104:
3100:
3097:
3096:
3084:
3080:
3078:
3073:
3061:
3057:
3055:
3043:
3039:
3032:
3031:
3029:
3026:
3025:
2984:
2980:
2978:
2975:
2974:
2950:
2949:
2937:
2933:
2931:
2926:
2914:
2910:
2908:
2896:
2892:
2889:
2888:
2883:
2878:
2873:
2867:
2866:
2854:
2850:
2848:
2843:
2831:
2827:
2825:
2813:
2809:
2806:
2805:
2793:
2789:
2787:
2782:
2770:
2766:
2764:
2752:
2748:
2741:
2740:
2732:
2729:
2728:
2706:
2629:
2609:
2606:
2605:
2556:
2552:
2537:
2532:
2531:
2525:
2521:
2506:
2501:
2500:
2494:
2490:
2489:
2485:
2477:
2474:
2473:
2453:
2448:
2447:
2432:
2427:
2426:
2417:
2412:
2411:
2409:
2406:
2405:
2374:
2371:
2370:
2347:
2342:
2341:
2331:
2330:
2324:
2323:
2317:
2316:
2310:
2309:
2299:
2298:
2278:
2273:
2272:
2262:
2261:
2255:
2254:
2248:
2247:
2241:
2240:
2230:
2229:
2216:
2211:
2210:
2200:
2199:
2193:
2192:
2186:
2185:
2179:
2178:
2168:
2167:
2162:
2159:
2158:
2137:
2136:
2130:
2125:
2124:
2122:
2117:
2111:
2106:
2105:
2103:
2097:
2092:
2091:
2087:
2083:
2075:
2072:
2071:
2054:
2041:
2017:
2005:
1993:
1989:
1914:
1913:
1907:
1906:
1896:
1895:
1885:
1884:
1878:
1877:
1864:
1863:
1852:
1838:
1834:
1823:
1818:
1817:
1813:
1797:
1792:
1787:
1781:
1776:
1771:
1763:
1761:
1758:
1757:
1744:
1743:to be equal to
1725:
1710:
1683:
1668:
1653:
1643:
1637:identity matrix
1621:
1617:
1607:
1592:
1572:
1560:
1556:
1544:
1540:
1528:
1517:
1491:
1490:
1485:
1479:
1478:
1473:
1463:
1462:
1395:
1394:
1389:
1377:
1376:
1371:
1355:
1354:
1352:
1349:
1348:
1329:
1326:
1325:
1301:
1300:
1292:
1290:
1284:
1283:
1275:
1273:
1263:
1262:
1252:
1251:
1246:
1240:
1239:
1234:
1224:
1223:
1206:
1183:
1161:
1160:
1152:
1144:
1138:
1137:
1129:
1121:
1111:
1110:
1108:
1105:
1104:
1056:
1055:
1050:
1044:
1043:
1038:
1028:
1027:
1025:
1022:
1021:
1000:
999:
994:
988:
987:
982:
972:
971:
969:
966:
965:
963:identity matrix
940:
937:
936:
933:
856:
855:
847:
845:
839:
838:
833:
823:
822:
812:
811:
803:
797:
796:
791:
781:
780:
775:
772:
771:
726:
725:
720:
714:
713:
708:
698:
697:
687:
686:
681:
675:
674:
669:
659:
658:
653:
650:
649:
642:
624:
623:
618:
612:
611:
606:
596:
595:
593:
590:
589:
585:
582:
553:
537:
529:
461:
454:
451:identity matrix
436:
413:
390:
387:
386:
383:Leibniz formula
372:
276:
275:
270:
265:
259:
258:
253:
248:
242:
241:
236:
231:
221:
220:
218:
215:
214:
207:
165:
164:
159:
153:
152:
147:
137:
136:
134:
131:
130:
123:
85:
83:
76:
68:
62:
39:
28:
23:
22:
15:
12:
11:
5:
28140:
28130:
28129:
28124:
28122:Linear algebra
28119:
28114:
28097:
28096:
28094:
28093:
28082:
28079:
28078:
28076:
28075:
28070:
28065:
28060:
28055:
28053:Floating-point
28049:
28047:
28041:
28040:
28038:
28037:
28035:Tensor product
28032:
28027:
28022:
28020:Function space
28017:
28012:
28006:
28004:
27997:
27996:
27994:
27993:
27988:
27983:
27978:
27973:
27968:
27963:
27958:
27956:Triple product
27953:
27948:
27942:
27940:
27934:
27933:
27931:
27930:
27925:
27920:
27915:
27910:
27905:
27900:
27894:
27892:
27886:
27885:
27883:
27882:
27877:
27872:
27870:Transformation
27867:
27862:
27860:Multiplication
27857:
27852:
27847:
27842:
27836:
27834:
27828:
27827:
27820:
27818:
27816:
27815:
27810:
27805:
27800:
27795:
27790:
27785:
27780:
27775:
27770:
27765:
27760:
27755:
27750:
27745:
27740:
27735:
27730:
27725:
27719:
27717:
27716:Basic concepts
27713:
27712:
27710:
27709:
27704:
27698:
27695:
27694:
27691:Linear algebra
27687:
27686:
27679:
27672:
27664:
27658:
27657:
27652:
27647:
27641:
27628:
27623:
27601:
27582:
27562:
27561:External links
27559:
27558:
27557:
27527:
27492:
27487:
27470:
27457:
27444:
27428:
27415:
27404:
27391:
27378:
27373:
27346:
27343:
27342:
27341:
27336:
27323:
27296:
27266:
27257:
27251:
27238:
27233:
27225:Linear Algebra
27220:
27215:
27202:
27196:
27172:
27159:G. Baley Price
27156:
27151:
27138:
27126:
27116:
27101:
27096:
27074:
27069:
27056:
27051:
27038:
27033:
27019:Yan, Catherine
27014:
27001:
26987:, Birkhäuser,
26980:
26975:
26962:
26925:
26890:(9): 761–778,
26879:
26845:
26832:
26819:
26780:
26775:
26762:
26716:
26673:
26667:
26647:
26630:
26627:
26625:
26624:
26591:
26576:
26555:
26522:
26511:
26499:
26488:
26467:
26455:
26442:
26435:
26415:
26398:
26395:
26392:
26389:
26384:
26378:
26375:
26373:
26370:
26369:
26366:
26363:
26361:
26358:
26357:
26355:
26350:
26347:
26344:
26339:
26333:
26330:
26328:
26325:
26324:
26321:
26318:
26316:
26313:
26312:
26310:
26305:
26302:
26297:
26291:
26288:
26286:
26283:
26282:
26279:
26276:
26274:
26271:
26270:
26268:
26263:
26260:
26258:
26256:
26251:
26245:
26242:
26240:
26237:
26236:
26233:
26230:
26228:
26225:
26224:
26222:
26217:
26214:
26209:
26203:
26200:
26198:
26195:
26194:
26191:
26188:
26186:
26183:
26182:
26180:
26175:
26172:
26169:
26166:
26164:
26162:
26159:
26156:
26155:
26130:
26127:Garibaldi 2004
26119:
26103:
26091:
26073:
26061:
26049:
26034:
26022:
26010:
25998:
25987:
25975:
25963:
25951:
25939:
25927:
25913:
25901:
25881:
25877:Bourbaki (1994
25869:
25857:
25830:
25818:
25804:
25789:
25777:
25765:
25721:
25709:
25693:
25679:
25668:
25635:
25614:
25602:
25557:
25510:
25498:
25486:
25474:
25457:
25445:
25441:Linear Algebra
25429:
25417:
25394:
25369:
25355:Episode 4
25344:
25331:
25329:
25326:
25324:
25323:
25318:
25313:
25308:
25303:
25297:
25296:
25295:
25279:
25276:
25213:
25189:
25184:
25180:
25176:
25173:
25170:
25155:bit complexity
25138:
25133:
25129:
25125:
25122:
25119:
25091:
25086:
25082:
25078:
25075:
25072:
25052:
25049:
25046:
25043:
25040:
25037:
25034:
25014:
25011:
25008:
24986:
24982:
24978:
24975:
24972:
24969:
24966:
24946:
24943:
24940:
24937:
24917:
24897:
24892:
24888:
24884:
24881:
24878:
24866:
24863:
24862:
24861:
24850:
24847:
24844:
24841:
24838:
24835:
24832:
24829:
24826:
24823:
24820:
24817:
24814:
24811:
24808:
24805:
24782:
24762:
24759:
24739:
24736:
24716:
24696:
24676:
24656:
24636:
24616:
24596:
24576:
24562:
24561:
24550:
24547:
24544:
24541:
24538:
24535:
24512:
24489:
24486:
24483:
24480:
24477:
24474:
24454:
24449:
24445:
24441:
24438:
24435:
24399:
24396:
24393:
24390:
24378:
24375:
24348:
24345:
24332:
24329:
24306:
24303:
24300:
24277:
24257:
24254:
24231:
24228:
24224:Manin matrices
24205:
24200:
24135:
24132:
24111:
24108:
24100:
24099:
24088:
24085:
24082:
24079:
24076:
24073:
24070:
24067:
24064:
24061:
24058:
24055:
24052:
24049:
24046:
24043:
24040:
24037:
24034:
24031:
24028:
24025:
24022:
23991:
23988:
23959:
23956:
23953:
23950:
23945:
23941:
23937:
23933:
23918:
23917:
23903:
23899:
23895:
23890:
23886:
23882:
23877:
23873:
23869:
23864:
23860:
23856:
23853:
23850:
23847:
23844:
23841:
23838:
23835:
23832:
23829:
23826:
23823:
23820:
23817:
23790:
23787:
23784:
23781:
23776:
23773:
23770:
23766:
23762:
23759:
23749:matrix algebra
23744:
23743:
23732:
23729:
23726:
23723:
23720:
23717:
23694:
23670:
23655:
23652:
23637:
23633:
23601:
23598:
23595:
23592:
23589:
23573:
23570:
23557:
23554:
23551:
23531:
23526:
23522:
23506:vector bundles
23493:
23473:
23468:
23464:
23441:
23437:
23416:
23405:
23404:
23393:
23388:
23384:
23380:
23377:
23374:
23369:
23365:
23361:
23358:
23355:
23352:
23349:
23346:
23342:
23336:
23332:
23328:
23325:
23322:
23317:
23313:
23308:
23303:
23299:
23294:
23290:
23285:
23261:
23258:
23253:
23249:
23228:
23200:
23180:
23175:
23171:
23150:
23145:
23141:
23129:
23128:
23113:
23108:
23104:
23100:
23097:
23094:
23091:
23086:
23082:
23078:
23075:
23070:
23066:
23062:
23059:
23056:
23054:
23050:
23046:
23042:
23039:
23036:
23031:
23027:
23023:
23018:
23014:
23010:
23009:
23006:
23001:
22997:
22993:
22990:
22988:
22986:
22981:
22977:
22973:
22970:
22965:
22961:
22957:
22956:
22933:
22913:
22893:
22888:
22884:
22873:exterior power
22860:
22840:
22820:
22793:
22773:
22753:
22750:
22747:
22744:
22741:
22723:
22720:
22719:
22718:
22707:
22702:
22697:
22692:
22687:
22683:
22679:
22676:
22643:
22639:
22635:
22632:
22610:
22606:
22573:
22553:
22535:
22534:
22523:
22520:
22517:
22512:
22509:
22506:
22502:
22498:
22495:
22492:
22489:
22486:
22483:
22480:
22477:
22474:
22469:
22466:
22463:
22459:
22455:
22452:
22449:
22446:
22443:
22420:
22400:
22380:
22377:
22374:
22371:
22366:
22362:
22358:
22355:
22352:
22349:
22346:
22341:
22337:
22333:
22330:
22327:
22324:
22321:
22316:
22312:
22291:
22288:
22285:
22282:
22279:
22241:
22217:
22197:
22194:
22191:
22176:
22175:
22164:
22159:
22155:
22151:
22148:
22145:
22142:
22139:
22134:
22130:
22095:
22091:
22088:
22068:
22044:
22024:
22021:
22018:
22015:
22010:
22007:
22004:
22000:
21996:
21993:
21970:
21967:
21964:
21961:
21958:
21955:
21930:
21909:
21894:
21891:
21864:
21863:
21852:
21849:
21846:
21843:
21840:
21819:
21818:
21807:
21804:
21801:
21798:
21795:
21792:
21789:
21786:
21783:
21780:
21775:
21772:
21768:
21764:
21761:
21758:
21755:
21752:
21749:
21746:
21743:
21740:
21737:
21734:
21731:
21728:
21725:
21722:
21719:
21714:
21711:
21707:
21703:
21700:
21697:
21694:
21691:
21688:
21685:
21682:
21638:
21635:
21633:
21630:
21615:
21614:
21603:
21599:
21595:
21590:
21586:
21582:
21578:
21575:
21572:
21569:
21566:
21563:
21560:
21556:
21552:
21549:
21545:
21541:
21538:
21535:
21532:
21527:
21523:
21519:
21515:
21511:
21507:
21503:
21499:
21496:
21491:
21488:
21485:
21482:
21478:
21414:
21413:
21402:
21397:
21394:
21391:
21388:
21385:
21382:
21379:
21374:
21366:
21362:
21358:
21351:
21347:
21343:
21337:
21332:
21329:
21326:
21323:
21320:
21293:
21292:
21281:
21276:
21271:
21266:
21261:
21256:
21251:
21248:
21224:For a general
21206:
21201:
21196:
21191:
21186:
21181:
21178:
21146:
21142:
21139:
21136:
21133:
21130:
21127:
21124:
21121:
21118:
21115:
21112:
21109:
21106:
21103:
21099:
21095:
21090:
21087:
21065:
21062:
21059:
21056:
21053:
21050:
21047:
21025:
21024:
21013:
21010:
21007:
21004:
21001:
20998:
20992:
20988:
20977:
20972:
20968:
20963:
20960:
20957:
20954:
20951:
20948:
20945:
20942:
20939:
20916:
20913:
20910:
20907:
20883:
20863:
20843:
20840:
20837:
20815:
20810:
20805:
20800:
20795:
20790:
20787:
20767:
20746:
20742:
20739:
20736:
20733:
20729:
20708:
20705:
20702:
20699:
20670:
20665:
20660:
20657:
20637:
20615:
20610:
20605:
20600:
20595:
20590:
20587:
20576:parallelepiped
20572:absolute value
20567:
20564:
20519:represents an
20505:standard basis
20473:Main article:
20470:
20467:
20461:have a common
20434:
20431:
20428:
20408:
20397:
20396:
20385:
20380:
20374:
20371:
20368:
20363:
20360:
20357:
20354:
20351:
20346:
20342:
20338:
20336:
20333:
20331:
20328:
20325:
20320:
20317:
20314:
20311:
20308:
20303:
20299:
20295:
20293:
20290:
20287:
20282:
20279:
20276:
20273:
20270:
20265:
20261:
20257:
20256:
20253:
20250:
20248:
20245:
20243:
20240:
20238:
20235:
20234:
20231:
20228:
20225:
20221:
20217:
20213:
20209:
20207:
20204:
20202:
20199:
20196:
20192:
20188:
20184:
20180:
20178:
20175:
20172:
20168:
20164:
20160:
20156:
20155:
20152:
20149:
20146:
20141:
20137:
20133:
20131:
20128:
20126:
20123:
20120:
20115:
20111:
20107:
20105:
20102:
20099:
20094:
20090:
20086:
20085:
20083:
20078:
20075:
20072:
20069:
20066:
20061:
20057:
20053:
20050:
20047:
20042:
20038:
20034:
20031:
20013:differentiable
20000:
19997:
19994:
19974:
19971:
19968:
19963:
19959:
19955:
19952:
19949:
19946:
19943:
19940:
19935:
19931:
19906:
19903:
19900:
19871:
19867:
19844:
19839:
19834:
19829:
19825:
19821:
19816:
19812:
19791:
19771:
19768:
19752:
19749:
19724:
19719:
19715:
19711:
19708:
19705:
19694:
19693:
19682:
19677:
19673:
19668:
19665:
19662:
19659:
19656:
19653:
19649:
19646:
19643:
19640:
19637:
19634:
19631:
19628:
19625:
19622:
19619:
19615:
19586:
19582:
19570:
19569:
19558:
19555:
19552:
19549:
19544:
19540:
19536:
19531:
19523:
19519:
19515:
19513:
19510:
19506:
19503:
19500:
19496:
19492:
19488:
19484:
19480:
19476:
19473:
19470:
19466:
19462:
19460:
19457:
19453:
19449:
19445:
19444:
19442:
19437:
19432:
19428:
19422:
19417:
19414:
19411:
19407:
19403:
19398:
19390:
19386:
19382:
19380:
19377:
19375:
19372:
19370:
19367:
19363:
19359:
19355:
19354:
19352:
19347:
19344:
19341:
19336:
19332:
19328:
19325:
19302:
19282:
19273:-th column of
19262:
19240:
19236:
19224:
19223:
19212:
19209:
19206:
19203:
19200:
19197:
19194:
19191:
19188:
19185:
19182:
19175:
19172:
19169:
19166:
19161:
19156:
19152:
19148:
19145:
19139:
19134:
19130:
19102:
19099:
19096:
19093:
19073:
19053:
19050:
19047:
19044:
19028:
19025:
19023:
19020:
18998:(so called by
18854:Laplace (1772)
18806:
18803:
18802:
18801:
18786:
18782:
18778:
18774:
18770:
18767:
18765:
18763:
18760:
18757:
18754:
18748:
18743:
18739:
18738:
18734:
18730:
18726:
18722:
18719:
18717:
18715:
18712:
18709:
18706:
18700:
18695:
18691:
18690:
18686:
18682:
18678:
18674:
18671:
18669:
18667:
18664:
18661:
18658:
18652:
18647:
18643:
18642:
18615:
18612:
18609:
18606:
18603:
18581:
18575:
18572:
18570:
18567:
18565:
18562:
18561:
18559:
18554:
18551:
18531:
18528:
18525:
18500:
18494:
18491:
18465:
18462:
18459:
18456:
18434:
18430:
18410:
18409:
18398:
18394:
18389:
18385:
18381:
18377:
18374:
18371:
18368:
18365:
18362:
18359:
18356:
18353:
18350:
18347:
18344:
18341:
18338:
18335:
18332:
18329:
18326:
18303:
18300:
18297:
18286:big O notation
18282:
18281:
18268:
18263:
18259:
18255:
18251:
18248:
18245:
18241:
18237:
18232:
18229:
18225:
18220:
18216:
18213:
18210:
18207:
18204:
18201:
18198:
18194:
18189:
18185:
18181:
18177:
18174:
18171:
18168:
18165:
18162:
18159:
18156:
18153:
18150:
18147:
18144:
18141:
18138:
18135:
18132:
18129:
18126:
18123:
18120:
18117:
18114:
18111:
18108:
18105:
18102:
18088:
18087:
18076:
18071:
18068:
18063:
18058:
18055:
18051:
18047:
18042:
18039:
18036:
18033:
18030:
18025:
18022:
18018:
18014:
18011:
18008:
18005:
18002:
17994:
17991:
17987:
17983:
17978:
17975:
17972:
17969:
17966:
17940:
17929:
17928:
17917:
17913:
17906:
17903:
17898:
17895:
17887:
17884:
17880:
17875:
17871:
17868:
17865:
17862:
17859:
17856:
17853:
17847:
17844:
17839:
17836:
17833:
17830:
17827:
17801:
17781:
17757:
17754:
17751:
17748:
17745:
17734:
17733:
17722:
17718:
17711:
17708:
17703:
17700:
17694:
17691:
17688:
17685:
17682:
17678:
17674:
17671:
17668:
17662:
17659:
17654:
17651:
17648:
17645:
17642:
17621:differentiable
17607:
17584:
17581:
17578:
17573:
17562:function from
17555:
17552:
17540:geometric mean
17532:
17531:
17520:
17514:
17509:
17505:
17501:
17497:
17494:
17489:
17486:
17479:
17476:
17473:
17470:
17467:
17464:
17459:
17456:
17451:
17445:
17442:
17437:
17433:
17430:
17427:
17424:
17417:
17412:
17409:
17405:
17401:
17397:
17394:
17390:
17353:
17352:
17341:
17338:
17335:
17332:
17329:
17326:
17323:
17320:
17317:
17314:
17311:
17308:
17305:
17302:
17299:
17295:
17289:
17286:
17282:
17278:
17275:
17271:
17267:
17264:
17243:
17240:
17202:
17201:
17190:
17184:
17179:
17174:
17169:
17165:
17161:
17157:
17154:
17149:
17143:
17139:
17133:
17129:
17125:
17122:
17119:
17111:
17106:
17103:
17100:
17096:
17092:
17088:
17080:
17077:
17073:
17066:
17061:
17058:
17055:
17051:
17030:
17029:
17018:
17012:
17007:
17002:
16997:
16993:
16989:
16985:
16982:
16977:
16971:
16967:
16963:
16960:
16957:
16949:
16944:
16941:
16938:
16934:
16930:
16926:
16918:
16915:
16911:
16904:
16899:
16896:
16893:
16889:
16885:
16882:
16879:
16876:
16873:
16870:
16867:
16841:
16840:
16829:
16824:
16819:
16815:
16809:
16805:
16801:
16798:
16795:
16792:
16789:
16786:
16783:
16778:
16773:
16769:
16763:
16758:
16754:
16748:
16744:
16740:
16735:
16730:
16726:
16722:
16719:
16716:
16698:
16692:
16685:
16670:
16664:
16657:
16643:
16637:
16636:
16625:
16622:
16617:
16613:
16609:
16606:
16603:
16598:
16594:
16590:
16585:
16581:
16577:
16572:
16568:
16561:
16558:
16551:
16547:
16543:
16540:
16537:
16531:
16528:
16525:
16522:
16519:
16494:
16480:
16479:
16468:
16465:
16462:
16457:
16453:
16449:
16444:
16439:
16436:
16433:
16429:
16410:
16404:
16403:
16392:
16385:
16381:
16375:
16370:
16366:
16362:
16357:
16354:
16348:
16343:
16339:
16331:
16327:
16322:
16314:
16311:
16306:
16302:
16297:
16293:
16290:
16287:
16279:
16274:
16271:
16268:
16264:
16255:
16252:
16247:
16243:
16239:
16236:
16233:
16230:
16225:
16221:
16217:
16214:
16209:
16205:
16201:
16200:
16197:
16194:
16189:
16185:
16181:
16178:
16175:
16170:
16166:
16162:
16157:
16153:
16149:
16148:
16145:
16141:
16138:
16135:
16132:
16129:
16115:
16114:
16103:
16097:
16094:
16091:
16088:
16085:
16082:
16079:
16069:
16064:
16060:
16056:
16052:
16049:
16044:
16041:
16038:
16035:
16032:
16028:
16022:
16017:
16014:
16011:
16007:
16001:
15998:
15993:
15990:
15985:
15982:
15979:
15975:
15962:
15959:
15956:
15951:
15947:
15926:
15898:
15897:
15882:
15878:
15873:
15868:
15864:
15860:
15856:
15853:
15850:
15847:
15844:
15841:
15838:
15835:
15832:
15825:
15820:
15816:
15812:
15808:
15805:
15802:
15799:
15794:
15789:
15784:
15779:
15775:
15771:
15767:
15764:
15760:
15755:
15752:
15747:
15742:
15738:
15735:
15732:
15729:
15726:
15722:
15716:
15711:
15707:
15703:
15699:
15696:
15693:
15690:
15685:
15680:
15676:
15673:
15670:
15667:
15664:
15660:
15654:
15648:
15645:
15640:
15637:
15635:
15633:
15630:
15627:
15624:
15621:
15620:
15617:
15613:
15608:
15603:
15599:
15595:
15591:
15588:
15585:
15582:
15578:
15573:
15569:
15565:
15561:
15558:
15552:
15549:
15546:
15543:
15540:
15537:
15534:
15529:
15524:
15520:
15517:
15514:
15511:
15508:
15504:
15498:
15492:
15489:
15484:
15481:
15479:
15477:
15474:
15471:
15468:
15465:
15464:
15461:
15457:
15452:
15447:
15443:
15439:
15435:
15432:
15429:
15424:
15419:
15415:
15412:
15409:
15406:
15403:
15399:
15393:
15387:
15384:
15379:
15376:
15374:
15372:
15369:
15366:
15363:
15360:
15359:
15324:
15323:
15312:
15309:
15306:
15303:
15300:
15297:
15294:
15291:
15288:
15285:
15282:
15279:
15276:
15273:
15270:
15252:
15251:
15240:
15237:
15234:
15231:
15228:
15225:
15222:
15174:) denotes the
15168:
15167:
15156:
15153:
15150:
15147:
15144:
15141:
15138:
15135:
15132:
15129:
15126:
15123:
15120:
15117:
15114:
15111:
15108:
15105:
15102:
15084:
15083:
15072:
15069:
15066:
15063:
15060:
15057:
15054:
15051:
15048:
15045:
15042:
15039:
15036:
15033:
15030:
15027:
15024:
15021:
14990:
14987:
14974:
14954:
14934:
14923:
14922:
14909:
14901:
14898:
14895:
14891:
14887:
14885:
14882:
14878:
14875:
14872:
14868:
14864:
14860:
14857:
14854:
14850:
14846:
14845:
14842:
14839:
14837:
14834:
14832:
14829:
14827:
14824:
14823:
14818:
14815:
14812:
14808:
14804:
14802:
14799:
14795:
14792:
14789:
14785:
14781:
14777:
14774:
14771:
14767:
14763:
14762:
14757:
14754:
14751:
14747:
14743:
14741:
14738:
14734:
14731:
14728:
14724:
14720:
14716:
14713:
14710:
14706:
14702:
14701:
14699:
14694:
14689:
14685:
14658:
14657:
14646:
14643:
14640:
14637:
14634:
14629:
14625:
14601:
14577:
14568:of the matrix
14553:
14533:
14509:
14498:
14497:
14486:
14483:
14480:
14477:
14474:
14471:
14468:
14465:
14462:
14459:
14456:
14453:
14450:
14445:
14441:
14414:
14394:
14374:
14354:
14334:
14314:
14296:
14295:
14284:
14279:
14275:
14271:
14266:
14262:
14256:
14252:
14248:
14243:
14239:
14233:
14228:
14225:
14222:
14218:
14214:
14211:
14208:
14205:
14202:
14158:
14154:
14150:
14147:
14144:
14139:
14135:
14131:
14126:
14122:
14095:
14071:
14068:
14065:
14045:
14025:
14022:
14020:
14017:
14004:
14001:
13998:
13994:
13990:
13987:
13984:
13981:
13978:
13975:
13972:
13952:
13949:
13946:
13942:
13938:
13922:
13921:
13910:
13905:
13901:
13897:
13892:
13888:
13884:
13881:
13877:
13873:
13870:
13865:
13861:
13857:
13854:
13851:
13848:
13845:
13840:
13836:
13832:
13829:
13825:
13821:
13818:
13815:
13812:
13809:
13806:
13782:
13778:
13775:
13772:
13752:
13749:
13746:
13743:
13723:
13720:
13717:
13713:
13709:
13684:
13664:
13642:
13636:
13633:
13631:
13628:
13627:
13624:
13621:
13618:
13616:
13613:
13612:
13610:
13605:
13602:
13597:
13591:
13588:
13586:
13583:
13582:
13579:
13576:
13574:
13571:
13570:
13568:
13563:
13560:
13556:
13552:
13549:
13546:
13543:
13523:
13520:
13517:
13502:
13501:
13490:
13487:
13484:
13481:
13478:
13470:
13467:
13464:
13461:
13453:
13450:
13447:
13428:
13427:
13416:
13411:
13407:
13401:
13397:
13393:
13388:
13384:
13378:
13374:
13370:
13365:
13361:
13355:
13351:
13347:
13342:
13338:
13332:
13328:
13324:
13321:
13318:
13313:
13309:
13305:
13300:
13296:
13292:
13289:
13284:
13280:
13276:
13271:
13267:
13263:
13249:
13248:
13237:
13232:
13228:
13222:
13218:
13214:
13209:
13205:
13199:
13195:
13191:
13186:
13182:
13176:
13172:
13168:
13163:
13159:
13153:
13149:
13145:
13142:
13139:
13136:
13133:
13130:
13127:
13124:
13121:
13118:
13095:
13075:
13072:
13069:
13066:
13046:
13035:
13034:
13023:
13018:
13014:
13008:
13004:
13000:
12995:
12991:
12985:
12981:
12977:
12972:
12968:
12962:
12958:
12954:
12949:
12945:
12939:
12935:
12931:
12926:
12922:
12916:
12912:
12908:
12903:
12899:
12893:
12889:
12885:
12880:
12876:
12870:
12866:
12862:
12857:
12853:
12847:
12843:
12828:
12827:
12816:
12813:
12808:
12804:
12800:
12795:
12791:
12787:
12784:
12779:
12775:
12771:
12766:
12762:
12758:
12755:
12752:
12747:
12743:
12739:
12734:
12730:
12726:
12723:
12718:
12714:
12710:
12705:
12701:
12697:
12672:
12669:
12665:
12661:
12656:
12653:
12649:
12634:
12633:
12632:
12621:
12618:
12615:
12612:
12609:
12601:
12598:
12595:
12592:
12584:
12581:
12578:
12570:
12567:
12564:
12561:
12558:
12555:
12552:
12549:
12546:
12543:
12540:
12537:
12534:
12531:
12528:
12525:
12522:
12499:
12496:
12493:
12481:
12478:
12457:
12451:
12446:
12443:
12440:
12437:
12431:
12425:
12420:
12417:
12414:
12411:
12405:
12399:
12394:
12391:
12388:
12385:
12382:
12379:
12356:
12353:
12350:
12322:
12319:
12316:
12274:
12271:
12268:
12265:
12262:
12259:
12256:
12253:
12250:
12247:
12244:
12241:
12238:
12235:
12232:
12229:
12204:
12201:
12198:
12195:
12192:
12189:
12186:
12183:
12180:
12177:
12174:
12171:
12168:
12165:
12162:
12159:
12156:
12153:
12150:
12147:
12144:
12141:
12138:
12135:
12132:
12129:
12126:
12106:
12086:
12066:
12032:
12029:
12026:
12014:
12011:
12009:
12008:
11996:
11993:
11973:
11970:
11950:
11930:
11919:
11918:
11907:
11904:
11900:
11897:
11894:
11891:
11888:
11884:
11881:
11878:
11875:
11872:
11869:
11866:
11862:
11858:
11853:
11850:
11846:
11842:
11839:
11836:
11832:
11828:
11825:
11822:
11819:
11816:
11813:
11810:
11807:
11804:
11801:
11798:
11795:
11792:
11780:
11779:
11768:
11764:
11760:
11755:
11752:
11748:
11744:
11741:
11735:
11730:
11725:
11721:
11718:
11715:
11712:
11709:
11706:
11703:
11700:
11697:
11694:
11691:
11688:
11685:
11659:
11658:
11657:
11646:
11643:
11640:
11637:
11634:
11631:
11627:
11623:
11620:
11617:
11611:
11606:
11601:
11597:
11571:
11541:
11532:
11526:
11525:
11514:
11510:
11506:
11503:
11500:
11494:
11489:
11484:
11480:
11477:
11473:
11469:
11466:
11463:
11457:
11452:
11447:
11443:
11389:
11386:
11385:
11384:
11373:
11370:
11367:
11364:
11361:
11358:
11355:
11352:
11349:
11346:
11343:
11340:
11337:
11334:
11329:
11323:
11320:
11318:
11315:
11314:
11311:
11308:
11306:
11303:
11302:
11300:
11295:
11272:
11252:
11232:
11229:
11226:
11206:
11203:
11200:
11177:
11174:
11171:
11160:
11159:
11148:
11145:
11142:
11139:
11136:
11133:
11130:
11127:
11124:
11121:
11116:
11110:
11107:
11105:
11102:
11101:
11098:
11095:
11093:
11090:
11089:
11087:
11082:
11059:
11056:
11053:
11050:
11047:
11024:
11004:
10982:
10981:
10966:
10963:
10960:
10955:
10952:
10948:
10944:
10941:
10938:
10935:
10932:
10929:
10926:
10923:
10920:
10917:
10914:
10912:
10910:
10905:
10897:
10893:
10889:
10887:
10884:
10883:
10878:
10875:
10871:
10867:
10864:
10862:
10857:
10854:
10850:
10846:
10843:
10840:
10837:
10836:
10834:
10829:
10826:
10823:
10820:
10817:
10814:
10811:
10809:
10807:
10800:
10797:
10793:
10789:
10786:
10783:
10779:
10775:
10770:
10767:
10763:
10759:
10756:
10752:
10746:
10740:
10732:
10729:
10725:
10721:
10719:
10714:
10711:
10707:
10703:
10700:
10699:
10696:
10693:
10689:
10685:
10681:
10680:
10678:
10673:
10664:
10658:
10655:
10653:
10650:
10649:
10646:
10643:
10641:
10638:
10637:
10635:
10630:
10627:
10624:
10621:
10618:
10615:
10612:
10610:
10606:
10600:
10597:
10595:
10592:
10591:
10588:
10585:
10583:
10580:
10579:
10577:
10572:
10569:
10568:
10545:
10522:
10519:
10516:
10496:
10476:
10473:
10468:
10465:
10461:
10457:
10454:
10451:
10448:
10445:
10442:
10439:
10436:
10425:
10424:
10409:
10406:
10403:
10398:
10395:
10391:
10387:
10384:
10381:
10378:
10375:
10372:
10369:
10366:
10363:
10360:
10357:
10355:
10353:
10348:
10342:
10337:
10334:
10330:
10326:
10323:
10320:
10317:
10313:
10310:
10306:
10302:
10299:
10298:
10295:
10292:
10288:
10284:
10280:
10279:
10277:
10272:
10269:
10266:
10263:
10260:
10257:
10254:
10252:
10250:
10243:
10240:
10236:
10232:
10229:
10226:
10222:
10218:
10213:
10210:
10206:
10202:
10199:
10195:
10189:
10183:
10175:
10171:
10167:
10165:
10162:
10161:
10158:
10153:
10150:
10146:
10142:
10139:
10135:
10132:
10128:
10124:
10123:
10121:
10116:
10107:
10101:
10098:
10096:
10093:
10092:
10089:
10086:
10084:
10081:
10080:
10078:
10073:
10070:
10067:
10064:
10061:
10058:
10055:
10053:
10049:
10043:
10040:
10038:
10035:
10034:
10031:
10028:
10026:
10023:
10022:
10020:
10015:
10012:
10011:
9984:
9973:
9972:
9961:
9956:
9950:
9947:
9945:
9942:
9941:
9938:
9935:
9933:
9930:
9929:
9927:
9922:
9919:
9916:
9913:
9910:
9907:
9904:
9901:
9898:
9895:
9892:
9887:
9881:
9878:
9876:
9873:
9872:
9869:
9866:
9864:
9861:
9860:
9858:
9853:
9826:
9823:
9820:
9800:
9797:
9794:
9774:
9771:
9768:
9748:
9745:
9742:
9722:
9719:
9716:
9713:
9710:
9707:
9704:
9680:
9677:
9674:
9662:
9661:Block matrices
9659:
9658:
9657:
9646:
9643:
9640:
9637:
9631:
9628:
9624:
9619:
9614:
9611:
9607:
9588:
9587:
9576:
9573:
9569:
9566:
9563:
9560:
9557:
9554:
9551:
9548:
9545:
9542:
9539:
9536:
9533:
9530:
9527:
9524:
9510:
9509:
9498:
9493:
9490:
9486:
9480:
9477:
9474:
9470:
9466:
9463:
9460:
9457:
9452:
9449:
9446:
9442:
9438:
9435:
9432:
9429:
9426:
9423:
9400:
9397:
9394:
9391:
9388:
9373:
9370:
9331:
9326:
9323:
9318:
9278:
9274:
9268:
9264:
9260:
9255:
9251:
9246:
9240:
9237:
9234:
9231:
9228:
9225:
9222:
9218:
9214:
9209:
9201:
9198:
9195:
9190:
9186:
9182:
9180:
9177:
9173:
9170:
9167:
9162:
9158:
9154:
9150:
9147:
9144:
9139:
9135:
9131:
9127:
9124:
9121:
9116:
9112:
9108:
9107:
9104:
9101:
9099:
9096:
9094:
9091:
9089:
9086:
9084:
9081:
9080:
9075:
9070:
9066:
9062:
9060:
9057:
9053:
9048:
9044:
9040:
9036:
9031:
9027:
9023:
9019:
9014:
9010:
9006:
9005:
9000:
8996:
8992:
8990:
8987:
8983:
8979:
8975:
8971:
8967:
8963:
8959:
8955:
8951:
8950:
8947:
8944:
8942:
8939:
8937:
8934:
8932:
8929:
8927:
8924:
8923:
8921:
8904:
8903:
8892:
8887:
8884:
8881:
8877:
8871:
8868:
8865:
8861:
8855:
8852:
8849:
8845:
8841:
8838:
8835:
8830:
8825:
8822:
8819:
8815:
8811:
8808:
8805:
8802:
8799:
8774:
8752:
8749:
8746:
8735:
8734:
8721:
8715:
8712:
8710:
8707:
8706:
8703:
8700:
8698:
8695:
8694:
8692:
8687:
8684:
8679:
8673:
8670:
8668:
8665:
8664:
8661:
8658:
8656:
8653:
8652:
8650:
8645:
8642:
8637:
8631:
8628:
8626:
8623:
8622:
8619:
8616:
8614:
8611:
8610:
8608:
8603:
8600:
8595:
8589:
8586:
8584:
8581:
8579:
8576:
8575:
8572:
8569:
8567:
8564:
8562:
8559:
8558:
8555:
8552:
8550:
8547:
8545:
8542:
8541:
8539:
8514:
8511:
8508:
8489:
8488:
8477:
8472:
8469:
8466:
8462:
8456:
8453:
8450:
8446:
8440:
8437:
8434:
8430:
8426:
8423:
8420:
8415:
8410:
8407:
8404:
8400:
8396:
8393:
8390:
8387:
8384:
8361:
8348:is known as a
8335:
8332:
8329:
8325:
8319:
8316:
8313:
8309:
8305:
8302:
8299:
8279:
8259:
8239:
8219:
8216:
8213:
8210:
8207:
8204:
8201:
8198:
8195:
8192:
8189:
8167:
8164:
8161:
8157:
8129:
8114:
8111:
8081:
8077:
8056:
8053:
8050:
8047:
8042:
8038:
8033:
8029:
8026:
8023:
8020:
8015:
8011:
8000:quotient group
7987:
7984:
7981:
7978:
7973:
7969:
7944:
7941:
7938:
7935:
7930:
7926:
7901:
7898:
7895:
7892:
7887:
7883:
7862:
7840:
7836:
7815:
7812:
7809:
7806:
7801:
7797:
7749:
7746:
7743:
7740:
7735:
7731:
7727:
7724:
7721:
7718:
7715:
7710:
7706:
7678:
7675:
7672:
7669:
7664:
7660:
7636:
7616:
7605:
7604:
7590:
7587:
7583:
7579:
7576:
7573:
7570:
7567:
7564:
7558:
7555:
7552:
7549:
7545:
7540:
7536:
7531:
7528:
7524:
7520:
7516:
7484:
7461:
7441:
7438:
7418:
7398:
7387:
7386:
7375:
7372:
7369:
7366:
7363:
7360:
7357:
7354:
7351:
7348:
7345:
7342:
7339:
7336:
7322:matrix product
7309:
7289:
7273:
7270:
7248:
7247:
7235:
7232:
7229:
7226:
7223:
7219:
7208:
7204:
7200:
7173:
7157:
7154:
7142:
7139:
7136:
7133:
7130:
7127:
7124:
7121:
7118:
7115:
7112:
7109:
7106:
7103:
7099:
7095:
7091:
7087:
7084:
7080:
7076:
7072:
7058:
7057:
7044:
7040:
7036:
7032:
7028:
7024:
7020:
7008:
6995:
6991:
6987:
6983:
6980:
6976:
6972:
6968:
6956:
6943:
6939:
6935:
6931:
6927:
6923:
6919:
6907:
6895:
6891:
6887:
6883:
6879:
6875:
6871:
6860:
6856:
6855:
6840:
6837:
6832:
6820:
6815:
6810:
6805:
6801:
6800:
6786:
6780:
6777:
6774:
6772:
6769:
6767:
6764:
6763:
6760:
6757:
6755:
6752:
6750:
6747:
6746:
6743:
6740:
6738:
6735:
6732:
6730:
6727:
6726:
6724:
6719:
6716:
6705:
6691:
6685:
6682:
6679:
6677:
6674:
6672:
6669:
6668:
6665:
6662:
6660:
6657:
6655:
6652:
6651:
6648:
6645:
6643:
6640:
6637:
6635:
6632:
6631:
6629:
6624:
6621:
6610:
6596:
6590:
6587:
6584:
6582:
6579:
6577:
6574:
6573:
6570:
6567:
6565:
6562:
6560:
6557:
6556:
6553:
6550:
6548:
6545:
6543:
6540:
6537:
6536:
6534:
6529:
6526:
6515:
6502:
6496:
6493:
6490:
6488:
6485:
6483:
6480:
6479:
6476:
6473:
6471:
6468:
6466:
6463:
6462:
6459:
6456:
6454:
6451:
6448:
6446:
6443:
6440:
6439:
6437:
6432:
6429:
6419:
6403:
6391:
6390:
6379:
6374:
6368:
6365:
6362:
6360:
6357:
6355:
6352:
6349:
6348:
6345:
6342:
6340:
6337:
6335:
6332:
6331:
6328:
6325:
6323:
6320:
6317:
6315:
6312:
6309:
6308:
6306:
6301:
6298:
6275:
6258:
6255:
6254:
6253:
6241:
6217:
6212:
6209:
6205:
6199:
6194:
6191:
6188:
6184:
6180:
6175:
6172:
6168:
6164:
6159:
6155:
6149:
6145:
6141:
6138:
6135:
6132:
6129:
6109:
6106:
6103:
6083:
6080:
6077:
6057:
6054:
6049:
6046:
6042:
6017:
6006:
5998:
5987:
5971:
5967:
5961:
5957:
5953:
5950:
5947:
5942:
5938:
5934:
5929:
5925:
5921:
5916:
5912:
5908:
5903:
5899:
5894:
5890:
5886:
5880:
5876:
5872:
5869:
5866:
5861:
5857:
5853:
5848:
5844:
5840:
5835:
5831:
5827:
5822:
5818:
5813:
5809:
5806:
5802:
5796:
5792:
5788:
5785:
5780:
5776:
5772:
5767:
5763:
5759:
5754:
5750:
5746:
5741:
5737:
5732:
5711:
5707:
5701:
5697:
5693:
5690:
5687:
5682:
5678:
5674:
5671:
5668:
5663:
5659:
5655:
5652:
5649:
5644:
5640:
5635:
5631:
5628:
5624:
5618:
5614:
5610:
5607:
5604:
5599:
5595:
5591:
5588:
5583:
5579:
5575:
5572:
5569:
5564:
5560:
5555:
5543:
5531:
5511:
5508:
5505:
5485:
5482:
5479:
5476:
5471:
5467:
5463:
5460:
5457:
5454:
5451:
5448:
5428:
5425:
5420:standard basis
5400:
5397:
5394:
5382:
5381:
5380:
5379:
5368:
5365:
5361:
5355:
5351:
5347:
5344:
5341:
5338:
5335:
5332:
5329:
5326:
5323:
5320:
5317:
5312:
5308:
5303:
5283:
5281:
5280:
5279:
5263:
5257:
5253:
5249:
5246:
5243:
5240:
5237:
5234:
5231:
5226:
5222:
5217:
5213:
5209:
5203:
5199:
5195:
5192:
5189:
5186:
5183:
5180:
5175:
5171:
5166:
5162:
5159:
5156:
5153:
5151:
5149:
5145:
5139:
5135:
5131:
5128:
5125:
5120:
5117:
5114:
5110:
5106:
5103:
5100:
5097:
5094:
5091:
5088:
5083:
5080:
5077:
5073:
5069:
5066:
5063:
5058:
5054:
5048:
5043:
5040:
5038:
5035:
5031:
5027:
5023:
5022:
4994:column vectors
4981:
4978:
4975:
4972:
4969:
4966:
4961:
4957:
4933:
4912:
4910:
4894:
4874:
4871:
4867:
4864:
4861:
4857:
4847:
4822:
4818:
4803:
4802:
4791:
4786:
4779:
4775:
4771:
4768:
4765:
4760:
4756:
4750:
4745:
4742:
4719:
4695:
4692:
4689:
4677:
4674:
4672:
4669:
4657:
4654:
4651:
4648:
4645:
4642:
4639:
4636:
4621:
4620:
4609:
4602:
4598:
4594:
4591:
4587:
4583:
4575:
4571:
4567:
4564:
4560:
4552:
4548:
4544:
4539:
4535:
4530:
4522:
4518:
4514:
4511:
4508:
4503:
4499:
4495:
4490:
4486:
4481:
4477:
4474:
4471:
4468:
4465:
4434:
4431:
4428:
4425:
4422:
4419:
4416:
4384:
4380:
4376:
4373:
4370:
4365:
4361:
4356:
4341:
4340:
4327:
4321:
4318:
4315:
4312:
4309:
4306:
4302:
4296:
4291:
4288:
4285:
4281:
4277:
4274:
4271:
4268:
4265:
4261:
4253:
4249:
4245:
4242:
4238:
4234:
4231:
4228:
4225:
4222:
4204:
4203:
4192:
4187:
4184:
4181:
4178:
4175:
4172:
4168:
4164:
4159:
4156:
4153:
4150:
4147:
4144:
4140:
4136:
4133:
4130:
4127:
4124:
4117:
4113:
4109:
4106:
4102:
4098:
4093:
4085:
4082:
4079:
4075:
4071:
4066:
4063:
4060:
4056:
4052:
4051:
4048:
4044:
4041:
4040:
4035:
4032:
4029:
4025:
4021:
4016:
4013:
4010:
4006:
4002:
4001:
3999:
3994:
3991:
3988:
3985:
3982:
3968:sigma notation
3964:
3963:
3952:
3947:
3939:
3936:
3933:
3929:
3925:
3920:
3917:
3914:
3910:
3906:
3905:
3902:
3898:
3895:
3894:
3889:
3886:
3883:
3879:
3875:
3870:
3867:
3864:
3860:
3856:
3855:
3853:
3848:
3845:
3820:
3817:
3797:
3794:
3791:
3771:
3751:
3748:
3745:
3742:
3739:
3717:
3713:
3688:
3685:
3682:
3679:
3676:
3673:
3670:
3667:
3664:
3661:
3658:
3655:
3652:
3649:
3646:
3643:
3623:
3600:
3597:
3594:
3591:
3588:
3585:
3582:
3579:
3576:
3548:
3545:
3542:
3530:
3520:
3511:rule of Sarrus
3504:Rule of Sarrus
3461:
3460:
3446:
3443:
3440:
3437:
3434:
3431:
3428:
3425:
3422:
3419:
3416:
3413:
3410:
3407:
3404:
3401:
3398:
3395:
3392:
3389:
3386:
3383:
3380:
3377:
3374:
3369:
3363:
3360:
3358:
3355:
3353:
3350:
3349:
3346:
3343:
3341:
3338:
3336:
3333:
3332:
3329:
3326:
3324:
3321:
3319:
3316:
3315:
3313:
3288:
3287:3 × 3 matrices
3285:
3280:Main article:
3277:
3274:
3262:
3261:
3250:
3245:
3237:
3234:
3231:
3227:
3223:
3221:
3218:
3214:
3211:
3208:
3204:
3200:
3196:
3193:
3190:
3186:
3182:
3181:
3178:
3175:
3173:
3170:
3168:
3165:
3163:
3160:
3159:
3154:
3151:
3148:
3144:
3140:
3138:
3135:
3131:
3128:
3125:
3121:
3117:
3113:
3110:
3107:
3103:
3099:
3098:
3093:
3090:
3087:
3083:
3079:
3077:
3074:
3070:
3067:
3064:
3060:
3056:
3052:
3049:
3046:
3042:
3038:
3037:
3035:
2993:
2990:
2987:
2983:
2971:
2970:
2959:
2954:
2946:
2943:
2940:
2936:
2932:
2930:
2927:
2923:
2920:
2917:
2913:
2909:
2905:
2902:
2899:
2895:
2891:
2890:
2887:
2884:
2882:
2879:
2877:
2874:
2872:
2869:
2868:
2863:
2860:
2857:
2853:
2849:
2847:
2844:
2840:
2837:
2834:
2830:
2826:
2822:
2819:
2816:
2812:
2808:
2807:
2802:
2799:
2796:
2792:
2788:
2786:
2783:
2779:
2776:
2773:
2769:
2765:
2761:
2758:
2755:
2751:
2747:
2746:
2744:
2739:
2736:
2705:
2702:
2645:
2642:
2639:
2636:
2628:
2625:
2622:
2619:
2616:
2613:
2584:
2580:
2576:
2573:
2567:
2564:
2559:
2555:
2551:
2548:
2545:
2540:
2535:
2528:
2524:
2520:
2517:
2514:
2509:
2504:
2497:
2493:
2488:
2484:
2481:
2461:
2456:
2451:
2446:
2443:
2440:
2435:
2430:
2425:
2420:
2415:
2389:maps the unit
2378:
2367:
2366:
2355:
2350:
2345:
2340:
2335:
2329:
2326:
2325:
2322:
2319:
2318:
2315:
2312:
2311:
2308:
2305:
2304:
2302:
2297:
2293:
2290:
2286:
2281:
2276:
2271:
2266:
2260:
2257:
2256:
2253:
2250:
2249:
2246:
2243:
2242:
2239:
2236:
2235:
2233:
2228:
2224:
2219:
2214:
2209:
2204:
2198:
2195:
2194:
2191:
2188:
2187:
2184:
2181:
2180:
2177:
2174:
2173:
2171:
2166:
2142:
2133:
2128:
2123:
2121:
2118:
2114:
2109:
2104:
2100:
2095:
2090:
2089:
2086:
2082:
2079:
1961:parallelepiped
1953:
1952:
1941:
1938:
1935:
1932:
1929:
1926:
1923:
1918:
1912:
1909:
1908:
1905:
1902:
1901:
1899:
1894:
1889:
1883:
1880:
1879:
1876:
1873:
1870:
1869:
1867:
1862:
1858:
1855:
1850:
1845:
1841:
1837:
1831:
1826:
1821:
1816:
1812:
1809:
1805:
1800:
1795:
1790:
1784:
1779:
1774:
1770:
1741:scalar product
1697:for the angle
1537:standard basis
1516:
1513:
1512:
1511:
1500:
1495:
1489:
1486:
1484:
1481:
1480:
1477:
1474:
1472:
1469:
1468:
1466:
1461:
1458:
1455:
1452:
1449:
1446:
1443:
1440:
1437:
1434:
1431:
1428:
1425:
1422:
1419:
1416:
1413:
1410:
1407:
1404:
1399:
1393:
1390:
1388:
1385:
1382:
1379:
1378:
1375:
1372:
1370:
1367:
1364:
1361:
1360:
1358:
1333:
1322:
1321:
1310:
1305:
1298:
1295:
1291:
1289:
1286:
1285:
1281:
1278:
1274:
1272:
1269:
1268:
1266:
1261:
1256:
1250:
1247:
1245:
1242:
1241:
1238:
1235:
1233:
1230:
1229:
1227:
1222:
1219:
1216:
1212:
1209:
1205:
1202:
1199:
1196:
1193:
1189:
1186:
1182:
1179:
1176:
1173:
1170:
1165:
1158:
1155:
1151:
1148:
1145:
1143:
1140:
1139:
1135:
1132:
1128:
1125:
1122:
1120:
1117:
1116:
1114:
1098:
1097:
1086:
1083:
1080:
1077:
1074:
1071:
1068:
1065:
1060:
1054:
1051:
1049:
1046:
1045:
1042:
1039:
1037:
1034:
1033:
1031:
1004:
998:
995:
993:
990:
989:
986:
983:
981:
978:
977:
975:
950:
947:
944:
932:
929:
928:
927:
916:
913:
910:
907:
904:
901:
898:
895:
892:
889:
886:
883:
880:
877:
874:
871:
868:
865:
860:
853:
850:
846:
844:
841:
840:
837:
834:
832:
829:
828:
826:
821:
816:
810:
807:
804:
802:
799:
798:
795:
792:
790:
787:
786:
784:
779:
765:
764:
753:
750:
747:
744:
741:
738:
735:
730:
724:
721:
719:
716:
715:
712:
709:
707:
704:
703:
701:
696:
691:
685:
682:
680:
677:
676:
673:
670:
668:
665:
664:
662:
657:
628:
622:
619:
617:
614:
613:
610:
607:
605:
602:
601:
599:
581:
578:
542:parallelepiped
472:
471:
468:
465:
458:
397:
394:
369:
368:
357:
354:
351:
348:
345:
342:
339:
336:
333:
330:
327:
324:
321:
318:
315:
312:
309:
306:
303:
300:
297:
294:
291:
288:
285:
280:
274:
271:
269:
266:
264:
261:
260:
257:
254:
252:
249:
247:
244:
243:
240:
237:
235:
232:
230:
227:
226:
224:
204:
203:
192:
189:
186:
183:
180:
177:
174:
169:
163:
160:
158:
155:
154:
151:
148:
146:
143:
142:
140:
104:the matrix is
102:if and only if
26:
9:
6:
4:
3:
2:
28139:
28128:
28125:
28123:
28120:
28118:
28117:Matrix theory
28115:
28113:
28110:
28109:
28107:
28092:
28084:
28083:
28080:
28074:
28071:
28069:
28068:Sparse matrix
28066:
28064:
28061:
28059:
28056:
28054:
28051:
28050:
28048:
28046:
28042:
28036:
28033:
28031:
28028:
28026:
28023:
28021:
28018:
28016:
28013:
28011:
28008:
28007:
28005:
28003:constructions
28002:
27998:
27992:
27991:Outermorphism
27989:
27987:
27984:
27982:
27979:
27977:
27974:
27972:
27969:
27967:
27964:
27962:
27959:
27957:
27954:
27952:
27951:Cross product
27949:
27947:
27944:
27943:
27941:
27939:
27935:
27929:
27926:
27924:
27921:
27919:
27918:Outer product
27916:
27914:
27911:
27909:
27906:
27904:
27901:
27899:
27898:Orthogonality
27896:
27895:
27893:
27891:
27887:
27881:
27878:
27876:
27875:Cramer's rule
27873:
27871:
27868:
27866:
27863:
27861:
27858:
27856:
27853:
27851:
27848:
27846:
27845:Decomposition
27843:
27841:
27838:
27837:
27835:
27833:
27829:
27824:
27814:
27811:
27809:
27806:
27804:
27801:
27799:
27796:
27794:
27791:
27789:
27786:
27784:
27781:
27779:
27776:
27774:
27771:
27769:
27766:
27764:
27761:
27759:
27756:
27754:
27751:
27749:
27746:
27744:
27741:
27739:
27736:
27734:
27731:
27729:
27726:
27724:
27721:
27720:
27718:
27714:
27708:
27705:
27703:
27700:
27699:
27696:
27692:
27685:
27680:
27678:
27673:
27671:
27666:
27665:
27662:
27656:
27653:
27651:
27648:
27645:
27642:
27639:
27635:
27632:
27629:
27627:
27624:
27621:
27617:
27616:
27611:
27607:
27602:
27597:
27596:
27591:
27590:"Determinant"
27588:
27583:
27580:
27576:
27575:
27570:
27569:"Determinant"
27565:
27564:
27554:
27548:
27540:
27536:
27532:
27528:
27525:
27521:
27517:
27513:
27509:
27505:
27501:
27497:
27493:
27490:
27488:9780801873966
27484:
27480:
27476:
27471:
27468:
27464:
27460:
27458:0-03-029558-0
27454:
27450:
27445:
27442:
27438:
27434:
27429:
27425:
27421:
27416:
27412:
27411:
27405:
27402:
27398:
27394:
27392:0-486-67766-4
27388:
27384:
27379:
27376:
27374:3-540-19376-6
27370:
27366:
27362:
27358:
27357:Meldrum, John
27354:
27349:
27348:
27339:
27333:
27329:
27324:
27315:on 2007-02-01
27311:
27307:
27303:
27299:
27293:
27288:
27283:
27279:
27272:
27267:
27263:
27258:
27254:
27248:
27244:
27239:
27236:
27234:9780387964126
27230:
27226:
27221:
27218:
27216:9780387962054
27212:
27208:
27203:
27199:
27193:
27189:
27185:
27181:
27177:
27173:
27171:
27168:
27164:
27160:
27157:
27154:
27152:0-534-99845-3
27148:
27144:
27139:
27135:
27131:
27127:
27124:on 2009-10-31
27123:
27119:
27113:
27109:
27108:
27102:
27099:
27097:0-387-98403-8
27093:
27089:
27085:
27084:
27079:
27075:
27072:
27070:9789401799447
27066:
27062:
27057:
27054:
27048:
27044:
27039:
27036:
27034:9780521883894
27030:
27026:
27025:
27020:
27015:
27012:
27008:
27004:
26998:
26994:
26990:
26986:
26981:
26978:
26976:9780128010495
26972:
26968:
26963:
26956:
26951:
26946:
26942:
26938:
26931:
26926:
26923:
26919:
26915:
26911:
26907:
26903:
26898:
26893:
26889:
26885:
26880:
26876:
26871:
26866:
26861:
26857:
26853:
26852:
26846:
26843:
26839:
26835:
26833:9780471452348
26829:
26825:
26820:
26815:
26810:
26805:
26800:
26796:
26792:
26791:
26786:
26781:
26778:
26776:9783540642435
26772:
26768:
26763:
26756:
26752:
26748:
26744:
26740:
26736:
26732:
26725:
26721:
26720:de Boor, Carl
26717:
26710:
26706:
26702:
26698:
26694:
26690:
26686:
26679:
26674:
26670:
26664:
26660:
26656:
26652:
26648:
26644:
26639:
26638:
26636:
26619:
26614:
26610:
26606:
26602:
26595:
26589:
26585:
26580:
26566:on 2011-08-07
26562:
26558:
26556:0-89791-875-4
26552:
26548:
26544:
26540:
26533:
26526:
26520:
26515:
26508:
26503:
26497:
26492:
26483:
26478:
26471:
26464:
26459:
26452:
26446:
26438:
26432:
26428:
26427:
26419:
26396:
26393:
26390:
26387:
26382:
26376:
26371:
26364:
26359:
26353:
26348:
26345:
26342:
26337:
26331:
26326:
26319:
26314:
26308:
26303:
26300:
26295:
26289:
26284:
26277:
26272:
26266:
26261:
26259:
26249:
26243:
26238:
26231:
26226:
26220:
26215:
26212:
26207:
26201:
26196:
26189:
26184:
26178:
26173:
26170:
26167:
26165:
26160:
26157:
26145:
26141:
26134:
26128:
26123:
26116:
26115:Bourbaki 1998
26112:
26107:
26100:
26099:Bourbaki 1998
26095:
26088:
26087:
26082:
26081:Mac Lane 1998
26077:
26070:
26065:
26058:
26053:
26045:
26038:
26031:
26026:
26019:
26014:
26007:
26002:
25996:
25991:
25985:
25979:
25972:
25967:
25961:, p. 494
25960:
25955:
25949:
25943:
25937:
25931:
25924:
25917:
25910:
25905:
25899:
25895:
25891:
25885:
25879:, p. 59)
25878:
25873:
25866:
25861:
25845:
25841:
25834:
25828:, p. 405
25827:
25822:
25813:
25811:
25809:
25802:
25800:
25793:
25786:
25781:
25774:
25769:
25761:
25757:
25753:
25749:
25745:
25741:
25738:(1): 99–115.
25737:
25733:
25725:
25718:
25713:
25706:
25702:
25697:
25689:
25685:
25682:
25672:
25663:
25658:
25654:
25650:
25646:
25639:
25630:
25625:
25618:
25612:
25606:
25598:
25594:
25590:
25586:
25581:
25576:
25572:
25568:
25561:
25553:
25549:
25545:
25541:
25537:
25533:
25529:
25525:
25521:
25514:
25507:
25502:
25496:, p. 306
25495:
25490:
25483:
25478:
25471:
25470:functoriality
25467:
25466:Bourbaki 1998
25461:
25454:
25449:
25442:
25438:
25433:
25426:
25421:
25413:
25408:
25407:
25398:
25383:
25379:
25373:
25365:
25361:
25357:
25356:
25348:
25341:
25336:
25332:
25322:
25319:
25317:
25314:
25312:
25309:
25307:
25304:
25302:
25299:
25298:
25293:
25287:
25282:
25275:
25273:
25269:
25268:
25263:
25262:Lewis Carroll
25258:
25256:
25252:
25247:
25243:
25238:
25234:
25230:
25225:
25211:
25203:
25182:
25178:
25171:
25160:
25156:
25152:
25131:
25127:
25120:
25107:
25105:
25084:
25080:
25073:
25044:
25038:
25032:
25012:
25009:
25006:
24984:
24980:
24976:
24970:
24964:
24941:
24935:
24915:
24890:
24886:
24879:
24848:
24842:
24833:
24827:
24818:
24815:
24809:
24796:
24795:
24794:
24780:
24760:
24757:
24737:
24734:
24714:
24694:
24674:
24654:
24634:
24614:
24594:
24574:
24567:
24548:
24545:
24542:
24539:
24536:
24533:
24526:
24525:
24524:
24523:as a product
24510:
24501:
24484:
24481:
24475:
24447:
24443:
24436:
24425:
24421:
24417:
24413:
24394:
24374:
24371:
24369:
24361:
24357:
24353:
24344:
24330:
24327:
24320:
24304:
24301:
24298:
24290:
24275:
24255:
24252:
24243:
24241:
24237:
24227:
24225:
24221:
24203:
24188:
24187:supermatrices
24184:
24180:
24176:
24172:
24168:
24164:
24160:
24156:
24151:bilinear form
24146:
24142:
24131:
24129:
24125:
24124:tracial state
24121:
24117:
24107:
24105:
24086:
24074:
24071:
24068:
24062:
24059:
24053:
24050:
24044:
24041:
24038:
24032:
24029:
24026:
24013:
24012:
24011:
24009:
24005:
24000:
23998:
23987:
23985:
23981:
23977:
23973:
23957:
23951:
23948:
23943:
23939:
23935:
23931:
23923:
23901:
23897:
23893:
23888:
23884:
23880:
23875:
23871:
23867:
23862:
23858:
23854:
23848:
23845:
23842:
23839:
23836:
23833:
23830:
23827:
23824:
23821:
23808:
23807:
23806:
23804:
23785:
23779:
23774:
23771:
23768:
23764:
23760:
23757:
23750:
23730:
23727:
23721:
23718:
23708:
23707:
23706:
23692:
23684:
23668:
23661:
23651:
23635:
23631:
23623:
23619:
23615:
23596:
23590:
23587:
23579:
23569:
23555:
23552:
23549:
23529:
23524:
23520:
23511:
23507:
23491:
23471:
23466:
23462:
23439:
23435:
23414:
23391:
23386:
23382:
23378:
23375:
23372:
23367:
23363:
23359:
23353:
23344:
23340:
23334:
23330:
23326:
23323:
23320:
23315:
23311:
23306:
23301:
23297:
23292:
23288:
23283:
23275:
23274:
23273:
23259:
23256:
23251:
23247:
23226:
23218:
23214:
23213:Bourbaki 1998
23198:
23178:
23173:
23169:
23148:
23143:
23139:
23111:
23106:
23102:
23098:
23095:
23092:
23089:
23084:
23080:
23076:
23073:
23068:
23064:
23060:
23055:
23048:
23044:
23040:
23037:
23034:
23029:
23025:
23021:
23016:
23012:
23004:
22999:
22995:
22989:
22984:
22979:
22975:
22971:
22968:
22963:
22959:
22947:
22946:
22945:
22931:
22911:
22891:
22886:
22882:
22874:
22858:
22838:
22818:
22811:
22807:
22791:
22771:
22751:
22745:
22742:
22739:
22729:
22705:
22700:
22685:
22681:
22677:
22667:
22666:
22665:
22663:
22659:
22641:
22633:
22608:
22604:
22595:
22591:
22587:
22571:
22551:
22543:
22538:
22510:
22507:
22504:
22500:
22493:
22481:
22467:
22464:
22461:
22457:
22441:
22434:
22433:
22432:
22418:
22398:
22375:
22369:
22364:
22360:
22350:
22344:
22339:
22335:
22331:
22325:
22319:
22314:
22310:
22289:
22283:
22280:
22277:
22270:
22261:
22257:
22255:
22239:
22231:
22215:
22195:
22192:
22189:
22181:
22162:
22157:
22153:
22143:
22137:
22132:
22128:
22120:
22119:
22118:
22117:
22112:
22110:
22089:
22086:
22066:
22058:
22042:
22019:
22013:
22008:
22005:
22002:
21998:
21994:
21991:
21982:
21968:
21965:
21959:
21945:
21907:
21900:
21890:
21888:
21884:
21880:
21876:
21872:
21869:
21850:
21844:
21841:
21838:
21831:
21830:
21829:
21828:
21824:
21805:
21799:
21790:
21784:
21773:
21770:
21762:
21750:
21741:
21735:
21723:
21712:
21709:
21701:
21692:
21686:
21673:
21672:
21671:
21668:
21665:
21661:
21656:
21652:
21648:
21644:
21629:
21627:
21622:
21620:
21601:
21593:
21588:
21570:
21567:
21554:
21536:
21530:
21525:
21521:
21517:
21509:
21494:
21486:
21480:
21476:
21468:
21467:
21466:
21463:
21459:
21455:
21450:
21446:
21442:
21438:
21434:
21431:
21427:
21423:
21419:
21400:
21395:
21392:
21389:
21386:
21383:
21380:
21377:
21372:
21364:
21360:
21349:
21345:
21335:
21330:
21324:
21318:
21311:
21310:
21309:
21308:
21303:
21299:
21279:
21274:
21259:
21249:
21246:
21239:
21238:
21237:
21235:
21231:
21227:
21204:
21189:
21179:
21176:
21167:
21163:
21161:
21160:spanning tree
21137:
21134:
21131:
21128:
21125:
21122:
21119:
21116:
21113:
21110:
21107:
21093:
21088:
21085:
21063:
21060:
21057:
21054:
21051:
21048:
21045:
21038:
21034:
21030:
21011:
21005:
20999:
20996:
20990:
20986:
20975:
20970:
20961:
20952:
20946:
20940:
20937:
20930:
20929:
20928:
20927:is given by:
20911:
20905:
20897:
20881:
20861:
20841:
20838:
20835:
20813:
20798:
20788:
20785:
20765:
20737:
20703:
20697:
20689:
20686:
20668:
20658:
20655:
20635:
20613:
20598:
20588:
20585:
20577:
20573:
20563:
20561:
20557:
20545:
20541:
20537:
20534:is positive,
20533:
20528:
20526:
20522:
20518:
20514:
20510:
20506:
20502:
20498:
20494:
20490:
20486:
20482:
20476:
20466:
20464:
20460:
20456:
20452:
20448:
20432:
20429:
20426:
20406:
20383:
20378:
20369:
20358:
20355:
20352:
20344:
20340:
20334:
20326:
20315:
20312:
20309:
20301:
20297:
20288:
20277:
20274:
20271:
20263:
20259:
20251:
20246:
20241:
20236:
20226:
20219:
20215:
20211:
20205:
20197:
20190:
20186:
20182:
20173:
20166:
20162:
20158:
20147:
20139:
20135:
20129:
20121:
20113:
20109:
20100:
20092:
20088:
20081:
20076:
20070:
20059:
20055:
20051:
20048:
20045:
20040:
20036:
20029:
20022:
20021:
20020:
20018:
20014:
19998:
19995:
19992:
19969:
19961:
19957:
19953:
19950:
19947:
19941:
19933:
19929:
19920:
19904:
19901:
19898:
19890:
19887:
19869:
19865:
19842:
19832:
19827:
19823:
19819:
19814:
19810:
19789:
19769:
19758:
19748:
19746:
19742:
19738:
19717:
19713:
19706:
19680:
19675:
19671:
19663:
19654:
19651:
19644:
19638:
19635:
19632:
19626:
19620:
19617:
19613:
19606:
19605:
19604:
19602:
19584:
19580:
19553:
19542:
19538:
19534:
19529:
19521:
19517:
19511:
19504:
19501:
19498:
19494:
19486:
19482:
19474:
19471:
19468:
19464:
19458:
19451:
19447:
19440:
19430:
19426:
19420:
19415:
19412:
19409:
19405:
19401:
19396:
19388:
19384:
19378:
19373:
19368:
19361:
19357:
19350:
19342:
19334:
19330:
19316:
19315:
19314:
19300:
19280:
19260:
19238:
19234:
19210:
19207:
19204:
19201:
19198:
19195:
19192:
19189:
19186:
19183:
19180:
19170:
19154:
19150:
19137:
19132:
19128:
19120:
19119:
19118:
19116:
19115:Cramer's rule
19097:
19071:
19051:
19048:
19045:
19042:
19034:
19027:Cramer's rule
19019:
19017:
19013:
19009:
19005:
19001:
18997:
18993:
18989:
18985:
18981:
18977:
18973:
18969:
18965:
18961:
18957:
18952:
18950:
18946:
18942:
18938:
18934:
18933:
18928:
18924:
18923:Jacobi (1841)
18920:
18918:
18914:
18909:
18905:
18900:
18896:
18892:
18887:
18885:
18881:
18877:
18873:
18869:
18867:
18863:
18859:
18855:
18851:
18847:
18845:
18841:
18840:Bézout (1779)
18837:
18836:Cramer (1750)
18833:
18829:
18828:Seki Takakazu
18824:
18822:
18818:
18817:
18812:
18784:
18776:
18768:
18766:
18758:
18728:
18720:
18718:
18710:
18680:
18672:
18670:
18662:
18633:
18632:
18631:
18629:
18628:cross product
18613:
18610:
18607:
18604:
18601:
18579:
18573:
18568:
18563:
18557:
18552:
18549:
18529:
18526:
18523:
18514:
18498:
18479:
18463:
18460:
18457:
18432:
18428:
18419:
18415:
18396:
18392:
18387:
18383:
18379:
18375:
18372:
18369:
18363:
18357:
18354:
18351:
18348:
18345:
18339:
18336:
18333:
18330:
18317:
18316:
18315:
18301:
18298:
18295:
18287:
18266:
18261:
18257:
18253:
18249:
18246:
18243:
18239:
18235:
18230:
18227:
18223:
18218:
18214:
18211:
18205:
18196:
18192:
18187:
18183:
18179:
18175:
18172:
18169:
18163:
18157:
18151:
18148:
18142:
18139:
18136:
18130:
18121:
18115:
18112:
18109:
18106:
18093:
18092:
18091:
18074:
18069:
18066:
18061:
18056:
18053:
18049:
18045:
18037:
18028:
18023:
18020:
18012:
18006:
18003:
18000:
17992:
17989:
17985:
17973:
17954:
17953:
17952:
17938:
17915:
17911:
17904:
17901:
17896:
17893:
17885:
17882:
17878:
17873:
17869:
17866:
17860:
17851:
17845:
17842:
17834:
17825:
17815:
17814:
17813:
17799:
17779:
17771:
17752:
17746:
17743:
17720:
17716:
17709:
17706:
17701:
17698:
17689:
17683:
17680:
17676:
17672:
17669:
17666:
17660:
17657:
17649:
17640:
17630:
17629:
17628:
17626:
17622:
17582:
17579:
17576:
17561:
17551:
17549:
17545:
17541:
17537:
17536:harmonic mean
17518:
17512:
17507:
17503:
17499:
17495:
17492:
17487:
17484:
17477:
17471:
17465:
17462:
17457:
17454:
17449:
17443:
17440:
17431:
17422:
17415:
17410:
17407:
17403:
17399:
17395:
17392:
17388:
17379:
17378:
17377:
17374:
17372:
17368:
17363:
17359:
17336:
17333:
17330:
17324:
17321:
17318:
17312:
17303:
17300:
17297:
17293:
17287:
17284:
17280:
17276:
17273:
17269:
17265:
17262:
17255:
17254:
17253:
17250:
17239:
17235:
17231:
17224:
17220:
17207:
17188:
17182:
17177:
17172:
17167:
17163:
17159:
17155:
17152:
17147:
17141:
17137:
17131:
17123:
17120:
17104:
17101:
17098:
17094:
17090:
17086:
17078:
17075:
17071:
17059:
17056:
17053:
17049:
17041:
17040:
17039:
17036:
17016:
17010:
17005:
17000:
16995:
16991:
16987:
16983:
16980:
16975:
16969:
16961:
16958:
16942:
16939:
16936:
16932:
16928:
16924:
16916:
16913:
16909:
16897:
16894:
16891:
16887:
16883:
16877:
16874:
16871:
16858:
16857:
16856:
16854:
16850:
16827:
16822:
16817:
16813:
16807:
16803:
16799:
16793:
16787:
16784:
16781:
16776:
16771:
16767:
16761:
16756:
16752:
16746:
16742:
16738:
16733:
16728:
16720:
16717:
16707:
16706:
16705:
16701:
16691:
16684:
16680:
16673:
16663:
16656:
16652:
16646:
16623:
16615:
16611:
16607:
16604:
16601:
16596:
16592:
16588:
16583:
16579:
16570:
16566:
16559:
16556:
16549:
16541:
16538:
16529:
16523:
16510:
16509:
16508:
16506:
16502:
16497:
16493:
16489:
16485:
16466:
16463:
16460:
16455:
16451:
16447:
16442:
16437:
16434:
16431:
16427:
16419:
16418:
16417:
16413:
16390:
16383:
16379:
16373:
16368:
16364:
16360:
16355:
16352:
16346:
16341:
16337:
16329:
16325:
16320:
16312:
16309:
16304:
16300:
16291:
16288:
16277:
16272:
16269:
16266:
16262:
16253:
16250:
16245:
16241:
16237:
16234:
16231:
16228:
16223:
16219:
16215:
16212:
16207:
16203:
16195:
16192:
16187:
16183:
16179:
16176:
16173:
16168:
16164:
16160:
16155:
16151:
16143:
16139:
16133:
16120:
16119:
16118:
16101:
16092:
16089:
16086:
16083:
16080:
16067:
16062:
16058:
16054:
16050:
16047:
16042:
16039:
16036:
16033:
16030:
16026:
16020:
16015:
16012:
16009:
16005:
15999:
15996:
15991:
15988:
15983:
15980:
15977:
15973:
15960:
15957:
15954:
15949:
15945:
15937:
15936:
15935:
15933:
15925:
15921:
15911:
15907:
15903:
15880:
15876:
15871:
15866:
15862:
15858:
15854:
15851:
15848:
15845:
15839:
15833:
15830:
15823:
15818:
15814:
15810:
15806:
15803:
15800:
15797:
15792:
15787:
15782:
15777:
15773:
15769:
15765:
15762:
15758:
15753:
15750:
15745:
15740:
15733:
15727:
15724:
15720:
15714:
15709:
15705:
15701:
15697:
15694:
15691:
15688:
15683:
15678:
15671:
15665:
15662:
15658:
15652:
15646:
15643:
15638:
15636:
15628:
15615:
15611:
15606:
15601:
15597:
15593:
15589:
15586:
15583:
15580:
15576:
15571:
15567:
15563:
15559:
15556:
15547:
15541:
15538:
15535:
15532:
15527:
15522:
15515:
15509:
15506:
15502:
15496:
15490:
15487:
15482:
15480:
15472:
15459:
15455:
15450:
15445:
15441:
15437:
15433:
15430:
15427:
15422:
15417:
15410:
15404:
15401:
15397:
15391:
15385:
15382:
15377:
15375:
15367:
15350:
15349:
15348:
15344:
15337:
15330:
15310:
15301:
15295:
15292:
15286:
15283:
15280:
15274:
15261:
15260:
15259:
15238:
15235:
15229:
15223:
15220:
15213:
15212:
15211:
15201:
15177:
15154:
15142:
15136:
15133:
15121:
15118:
15115:
15109:
15103:
15100:
15093:
15092:
15091:
15064:
15058:
15055:
15049:
15046:
15043:
15034:
15028:
15025:
15012:
15011:
15010:
15000:
14996:
14986:
14972:
14952:
14932:
14907:
14899:
14896:
14893:
14889:
14883:
14876:
14873:
14870:
14866:
14858:
14855:
14852:
14848:
14840:
14835:
14830:
14825:
14816:
14813:
14810:
14806:
14800:
14793:
14790:
14787:
14783:
14775:
14772:
14769:
14765:
14755:
14752:
14749:
14745:
14739:
14732:
14729:
14726:
14722:
14714:
14711:
14708:
14704:
14697:
14692:
14687:
14683:
14675:
14674:
14673:
14671:
14667:
14663:
14644:
14641:
14635:
14627:
14623:
14615:
14614:
14613:
14599:
14591:
14575:
14567:
14551:
14531:
14523:
14522:indeterminate
14507:
14484:
14478:
14475:
14472:
14469:
14466:
14457:
14451:
14443:
14439:
14431:
14430:
14429:
14426:
14412:
14392:
14372:
14352:
14332:
14312:
14303:
14301:
14282:
14277:
14273:
14269:
14264:
14260:
14254:
14250:
14246:
14241:
14237:
14231:
14226:
14223:
14220:
14216:
14212:
14206:
14193:
14192:
14191:
14189:
14174:
14156:
14152:
14148:
14145:
14142:
14137:
14133:
14129:
14124:
14120:
14112:
14109:
14093:
14085:
14082:-matrix with
14069:
14066:
14063:
14043:
14035:
14031:
14016:
14002:
13999:
13982:
13976:
13950:
13947:
13908:
13903:
13899:
13895:
13890:
13886:
13882:
13863:
13859:
13855:
13849:
13838:
13834:
13830:
13819:
13816:
13813:
13810:
13797:
13796:
13795:
13776:
13773:
13770:
13750:
13747:
13744:
13741:
13721:
13718:
13682:
13662:
13640:
13634:
13629:
13622:
13619:
13614:
13608:
13603:
13600:
13595:
13589:
13584:
13577:
13572:
13566:
13561:
13558:
13550:
13547:
13544:
13541:
13521:
13518:
13515:
13505:
13488:
13482:
13479:
13468:
13462:
13448:
13433:
13432:
13431:
13409:
13405:
13399:
13395:
13391:
13386:
13382:
13376:
13372:
13368:
13363:
13359:
13353:
13349:
13345:
13340:
13336:
13330:
13326:
13319:
13311:
13307:
13303:
13298:
13294:
13282:
13278:
13274:
13269:
13265:
13254:
13253:
13252:
13235:
13230:
13226:
13220:
13216:
13212:
13207:
13203:
13197:
13193:
13189:
13184:
13180:
13174:
13170:
13166:
13161:
13157:
13151:
13147:
13143:
13137:
13128:
13122:
13109:
13108:
13107:
13093:
13070:
13044:
13021:
13016:
13012:
13006:
13002:
12998:
12993:
12989:
12983:
12979:
12975:
12970:
12966:
12960:
12956:
12952:
12947:
12943:
12937:
12933:
12929:
12924:
12920:
12914:
12910:
12906:
12901:
12897:
12891:
12887:
12883:
12878:
12874:
12868:
12864:
12860:
12855:
12851:
12845:
12841:
12833:
12832:
12831:
12814:
12806:
12802:
12798:
12793:
12789:
12777:
12773:
12769:
12764:
12760:
12753:
12745:
12741:
12737:
12732:
12728:
12716:
12712:
12708:
12703:
12699:
12688:
12687:
12686:
12670:
12667:
12663:
12659:
12654:
12651:
12647:
12619:
12613:
12610:
12599:
12593:
12579:
12568:
12562:
12553:
12547:
12538:
12532:
12529:
12526:
12513:
12512:
12511:
12497:
12494:
12491:
12477:
12475:
12455:
12449:
12441:
12429:
12423:
12415:
12403:
12397:
12389:
12386:
12383:
12354:
12351:
12348:
12320:
12317:
12314:
12306:
12302:
12294:
12290:
12269:
12260:
12254:
12245:
12239:
12236:
12233:
12199:
12196:
12193:
12184:
12178:
12175:
12172:
12163:
12157:
12148:
12142:
12139:
12136:
12133:
12130:
12104:
12084:
12064:
12057:
12054:However, for
12052:
12050:
12046:
12030:
12027:
12024:
11994:
11991:
11971:
11968:
11948:
11928:
11920:
11905:
11902:
11895:
11889:
11886:
11882:
11879:
11873:
11864:
11860:
11856:
11851:
11848:
11844:
11840:
11837:
11834:
11830:
11820:
11811:
11805:
11802:
11799:
11796:
11783:
11781:
11766:
11762:
11758:
11753:
11750:
11746:
11742:
11739:
11728:
11723:
11713:
11704:
11698:
11695:
11692:
11689:
11676:
11674:
11669:
11665:
11660:
11644:
11641:
11638:
11635:
11632:
11629:
11625:
11621:
11618:
11615:
11604:
11599:
11588:
11587:
11585:
11581:
11577:
11573:
11572:
11570:
11567:
11564:
11560:
11554:
11550:
11544:
11540:
11535:
11531:
11512:
11508:
11504:
11501:
11498:
11487:
11482:
11475:
11471:
11467:
11464:
11461:
11450:
11445:
11434:
11433:
11432:
11430:
11426:
11421:
11417:
11412:
11407:
11403:
11398:
11394:
11371:
11365:
11362:
11359:
11347:
11344:
11341:
11332:
11327:
11321:
11316:
11309:
11304:
11298:
11286:
11285:
11284:
11270:
11250:
11230:
11227:
11224:
11204:
11201:
11198:
11189:
11175:
11172:
11169:
11146:
11140:
11137:
11134:
11131:
11128:
11119:
11114:
11108:
11103:
11096:
11091:
11085:
11073:
11072:
11071:
11057:
11054:
11051:
11048:
11045:
11037:
11022:
11002:
10994:
10989:
10987:
10964:
10958:
10953:
10950:
10946:
10942:
10939:
10936:
10924:
10915:
10913:
10903:
10895:
10891:
10885:
10876:
10873:
10869:
10865:
10860:
10855:
10852:
10848:
10844:
10841:
10838:
10832:
10821:
10812:
10810:
10798:
10795:
10787:
10777:
10768:
10765:
10761:
10750:
10744:
10738:
10730:
10727:
10723:
10717:
10712:
10709:
10705:
10701:
10694:
10687:
10683:
10676:
10662:
10656:
10651:
10644:
10639:
10633:
10622:
10613:
10611:
10604:
10598:
10593:
10586:
10581:
10575:
10559:
10558:
10557:
10543:
10534:
10520:
10517:
10514:
10494:
10471:
10466:
10463:
10459:
10455:
10452:
10449:
10440:
10407:
10401:
10396:
10393:
10389:
10385:
10382:
10379:
10367:
10358:
10356:
10346:
10340:
10335:
10332:
10328:
10324:
10321:
10318:
10311:
10308:
10304:
10300:
10293:
10286:
10282:
10275:
10264:
10255:
10253:
10241:
10238:
10230:
10220:
10211:
10208:
10204:
10193:
10187:
10181:
10173:
10169:
10163:
10156:
10151:
10148:
10144:
10140:
10133:
10130:
10126:
10119:
10105:
10099:
10094:
10087:
10082:
10076:
10065:
10056:
10054:
10047:
10041:
10036:
10029:
10024:
10018:
10002:
10001:
10000:
9998:
9982:
9959:
9954:
9948:
9943:
9936:
9931:
9925:
9917:
9911:
9899:
9890:
9885:
9879:
9874:
9867:
9862:
9856:
9844:
9843:
9842:
9840:
9824:
9821:
9818:
9798:
9795:
9792:
9772:
9769:
9766:
9746:
9743:
9740:
9733:of dimension
9720:
9717:
9714:
9711:
9708:
9705:
9702:
9694:
9678:
9675:
9672:
9644:
9641:
9638:
9635:
9629:
9622:
9617:
9612:
9609:
9605:
9597:
9596:
9595:
9593:
9574:
9571:
9564:
9561:
9558:
9552:
9549:
9546:
9543:
9540:
9537:
9534:
9528:
9515:
9514:
9513:
9496:
9491:
9488:
9484:
9478:
9475:
9472:
9464:
9461:
9455:
9450:
9447:
9444:
9433:
9427:
9424:
9414:
9413:
9412:
9395:
9389:
9386:
9379:
9369:
9368:) submatrix.
9367:
9363:
9359:
9356:
9352:
9348:
9324:
9321:
9304:
9300:
9296:
9292:
9276:
9272:
9266:
9262:
9258:
9253:
9249:
9244:
9238:
9235:
9232:
9229:
9226:
9223:
9220:
9216:
9212:
9207:
9199:
9196:
9193:
9188:
9184:
9178:
9171:
9168:
9165:
9160:
9156:
9148:
9145:
9142:
9137:
9133:
9125:
9122:
9119:
9114:
9110:
9102:
9097:
9092:
9087:
9082:
9073:
9068:
9064:
9058:
9051:
9046:
9042:
9034:
9029:
9025:
9017:
9012:
9008:
8998:
8994:
8988:
8981:
8977:
8969:
8965:
8957:
8953:
8945:
8940:
8935:
8930:
8925:
8919:
8910:
8890:
8885:
8882:
8879:
8875:
8869:
8866:
8863:
8859:
8853:
8850:
8847:
8839:
8836:
8828:
8823:
8820:
8817:
8813:
8809:
8803:
8790:
8789:
8788:
8786:
8772:
8750:
8747:
8744:
8719:
8713:
8708:
8701:
8696:
8690:
8685:
8682:
8677:
8671:
8666:
8659:
8654:
8648:
8643:
8640:
8635:
8629:
8624:
8617:
8612:
8606:
8601:
8598:
8593:
8587:
8582:
8577:
8570:
8565:
8560:
8553:
8548:
8543:
8537:
8528:
8527:
8526:
8512:
8509:
8506:
8498:
8475:
8470:
8467:
8464:
8460:
8454:
8451:
8448:
8444:
8438:
8435:
8432:
8424:
8421:
8413:
8408:
8405:
8402:
8398:
8394:
8388:
8375:
8374:
8373:
8359:
8351:
8333:
8330:
8327:
8323:
8317:
8314:
8311:
8303:
8300:
8277:
8257:
8237:
8214:
8211:
8208:
8202:
8196:
8193:
8190:
8165:
8162:
8159:
8155:
8146:
8142:
8127:
8119:
8110:
8108:
8104:
8100:
8095:
8079:
8075:
8051:
8045:
8040:
8036:
8031:
8024:
8018:
8013:
8009:
8001:
7982:
7976:
7971:
7967:
7958:
7939:
7933:
7928:
7924:
7915:
7896:
7890:
7885:
7881:
7860:
7838:
7834:
7810:
7804:
7799:
7795:
7786:
7781:
7779:
7775:
7771:
7767:
7763:
7744:
7738:
7733:
7729:
7725:
7719:
7713:
7708:
7704:
7696:
7692:
7673:
7667:
7662:
7658:
7650:
7634:
7627:over a field
7614:
7588:
7585:
7574:
7562:
7553:
7543:
7538:
7534:
7529:
7526:
7522:
7518:
7507:
7506:
7505:
7502:
7498:
7482:
7473:
7459:
7439:
7416:
7396:
7370:
7358:
7349:
7343:
7340:
7327:
7326:
7325:
7323:
7307:
7287:
7279:
7269:
7267:
7263:
7258:
7254:
7230:
7221:
7217:
7206:
7202:
7191:
7190:
7189:
7187:
7171:
7163:
7153:
7140:
7137:
7128:
7125:
7119:
7116:
7113:
7110:
7104:
7101:
7093:
7085:
7082:
7074:
7056:
7038:
7030:
7022:
7009:
7007:
6989:
6981:
6978:
6970:
6957:
6955:
6937:
6929:
6921:
6908:
6889:
6881:
6873:
6861:
6858:
6857:
6854:
6838:
6835:
6830:
6821:
6819:
6816:
6814:
6811:
6809:
6806:
6803:
6802:
6799:
6784:
6778:
6775:
6770:
6765:
6758:
6753:
6748:
6741:
6736:
6733:
6728:
6722:
6717:
6714:
6706:
6704:
6689:
6683:
6680:
6675:
6670:
6663:
6658:
6653:
6646:
6641:
6638:
6633:
6627:
6622:
6619:
6611:
6609:
6594:
6588:
6585:
6580:
6575:
6568:
6563:
6558:
6551:
6546:
6541:
6538:
6532:
6527:
6524:
6516:
6500:
6494:
6491:
6486:
6481:
6474:
6469:
6464:
6457:
6452:
6449:
6444:
6441:
6435:
6430:
6427:
6420:
6417:
6416:
6401:
6377:
6372:
6366:
6363:
6358:
6353:
6350:
6343:
6338:
6333:
6326:
6321:
6318:
6313:
6310:
6304:
6299:
6296:
6289:
6288:
6287:
6273:
6264:
6239:
6231:
6215:
6210:
6207:
6203:
6197:
6192:
6189:
6186:
6182:
6178:
6173:
6170:
6166:
6162:
6157:
6153:
6147:
6143:
6139:
6133:
6107:
6104:
6101:
6081:
6078:
6075:
6055:
6052:
6047:
6044:
6040:
6031:
6015:
6007:
6003:
5999:
5996:
5992:
5988:
5985:
5969:
5959:
5955:
5951:
5948:
5945:
5940:
5936:
5932:
5927:
5923:
5919:
5914:
5910:
5906:
5901:
5897:
5888:
5878:
5874:
5870:
5867:
5864:
5859:
5855:
5851:
5846:
5842:
5838:
5833:
5829:
5825:
5820:
5816:
5807:
5804:
5794:
5790:
5786:
5783:
5778:
5774:
5770:
5765:
5761:
5757:
5752:
5748:
5744:
5739:
5735:
5709:
5699:
5695:
5691:
5688:
5685:
5680:
5676:
5672:
5669:
5666:
5661:
5657:
5653:
5650:
5647:
5642:
5638:
5629:
5626:
5616:
5612:
5608:
5605:
5602:
5597:
5593:
5589:
5586:
5581:
5577:
5573:
5570:
5567:
5562:
5558:
5544:
5529:
5509:
5506:
5503:
5480:
5469:
5465:
5461:
5455:
5452:
5438:
5434:
5433:
5432:
5424:
5421:
5416:
5414:
5398:
5395:
5392:
5366:
5363:
5353:
5349:
5345:
5342:
5339:
5336:
5333:
5330:
5327:
5324:
5321:
5318:
5315:
5310:
5306:
5293:
5292:
5290:
5289:
5282:
5255:
5251:
5247:
5244:
5241:
5238:
5235:
5232:
5229:
5224:
5220:
5211:
5201:
5197:
5193:
5190:
5187:
5184:
5181:
5178:
5173:
5169:
5160:
5157:
5154:
5152:
5137:
5133:
5129:
5126:
5123:
5118:
5115:
5112:
5108:
5104:
5101:
5098:
5095:
5092:
5089:
5086:
5081:
5078:
5075:
5071:
5067:
5064:
5061:
5056:
5052:
5041:
5039:
5029:
5013:
5012:
5010:
5006:
5003:and a number
5002:
4998:
4995:
4979:
4976:
4973:
4970:
4967:
4964:
4959:
4955:
4947:
4931:
4923:
4919:
4918:
4911:
4908:
4892:
4872:
4869:
4865:
4862:
4859:
4846:
4845:
4844:
4842:
4838:
4820:
4816:
4808:
4807:column vector
4789:
4777:
4773:
4769:
4766:
4763:
4758:
4754:
4743:
4740:
4733:
4732:
4731:
4717:
4709:
4693:
4690:
4687:
4668:
4655:
4649:
4646:
4643:
4640:
4637:
4607:
4600:
4596:
4592:
4589:
4585:
4581:
4573:
4569:
4565:
4562:
4558:
4550:
4546:
4542:
4537:
4533:
4528:
4520:
4516:
4512:
4509:
4506:
4501:
4497:
4493:
4488:
4484:
4479:
4475:
4469:
4456:
4455:
4454:
4452:
4429:
4426:
4423:
4420:
4417:
4406:
4382:
4378:
4374:
4371:
4368:
4363:
4359:
4354:
4346:
4325:
4316:
4310:
4307:
4304:
4300:
4294:
4289:
4286:
4283:
4279:
4272:
4266:
4263:
4259:
4251:
4247:
4243:
4240:
4236:
4232:
4226:
4213:
4212:
4211:
4209:
4190:
4182:
4176:
4173:
4170:
4166:
4162:
4154:
4148:
4145:
4142:
4138:
4131:
4125:
4122:
4115:
4111:
4107:
4104:
4100:
4096:
4091:
4083:
4080:
4077:
4073:
4069:
4064:
4061:
4058:
4054:
4046:
4042:
4033:
4030:
4027:
4023:
4019:
4014:
4011:
4008:
4004:
3997:
3992:
3986:
3973:
3972:
3971:
3970:for the sum,
3969:
3950:
3945:
3937:
3934:
3931:
3927:
3923:
3918:
3915:
3912:
3908:
3900:
3896:
3887:
3884:
3881:
3877:
3873:
3868:
3865:
3862:
3858:
3851:
3846:
3843:
3836:
3835:
3834:
3831:
3818:
3815:
3795:
3792:
3789:
3769:
3746:
3740:
3737:
3715:
3711:
3702:
3683:
3677:
3674:
3671:
3668:
3662:
3656:
3653:
3647:
3641:
3621:
3614:
3595:
3592:
3589:
3586:
3583:
3580:
3577:
3566:
3562:
3546:
3543:
3540:
3528:
3524:
3519:
3512:
3505:
3501:
3497:
3495:
3491:
3487:
3483:
3478:
3474:
3470:
3466:
3444:
3441:
3438:
3435:
3432:
3429:
3426:
3423:
3420:
3417:
3414:
3411:
3408:
3405:
3402:
3399:
3396:
3393:
3390:
3387:
3384:
3381:
3378:
3375:
3372:
3367:
3361:
3356:
3351:
3344:
3339:
3334:
3327:
3322:
3317:
3311:
3302:
3301:
3300:
3294:
3283:
3273:
3271:
3267:
3248:
3243:
3235:
3232:
3229:
3225:
3219:
3212:
3209:
3206:
3202:
3194:
3191:
3188:
3184:
3176:
3171:
3166:
3161:
3152:
3149:
3146:
3142:
3136:
3129:
3126:
3123:
3119:
3111:
3108:
3105:
3101:
3091:
3088:
3085:
3081:
3075:
3068:
3065:
3062:
3058:
3050:
3047:
3044:
3040:
3033:
3024:
3023:
3022:
3020:
3016:
3011:
3009:
2991:
2988:
2985:
2981:
2957:
2952:
2944:
2941:
2938:
2934:
2928:
2921:
2918:
2915:
2911:
2903:
2900:
2897:
2893:
2885:
2880:
2875:
2870:
2861:
2858:
2855:
2851:
2845:
2838:
2835:
2832:
2828:
2820:
2817:
2814:
2810:
2800:
2797:
2794:
2790:
2784:
2777:
2774:
2771:
2767:
2759:
2756:
2753:
2749:
2742:
2737:
2734:
2727:
2726:
2725:
2723:
2719:
2715:
2714:square matrix
2711:
2701:
2699:
2695:
2691:
2687:
2683:
2680:is less than
2679:
2675:
2671:
2667:
2663:
2659:
2643:
2637:
2626:
2623:
2617:
2603:
2600:
2595:
2582:
2578:
2574:
2565:
2562:
2557:
2553:
2549:
2546:
2543:
2538:
2526:
2522:
2518:
2515:
2512:
2507:
2495:
2491:
2486:
2482:
2479:
2459:
2454:
2444:
2441:
2438:
2433:
2423:
2418:
2403:
2402:parallelotope
2400:-dimensional
2399:
2395:
2393:
2376:
2353:
2348:
2338:
2333:
2327:
2320:
2313:
2306:
2300:
2295:
2291:
2288:
2284:
2279:
2269:
2264:
2258:
2251:
2244:
2237:
2231:
2226:
2222:
2217:
2207:
2202:
2196:
2189:
2182:
2175:
2169:
2164:
2157:
2156:
2155:
2140:
2131:
2119:
2112:
2098:
2084:
2080:
2077:
2069:
2065:
2061:
2057:
2051:
2048:
2044:
2039:
2033:
2029:
2025:
2021:
2013:
2009:
2001:
1997:
1987:
1983:
1982:
1976:
1974:
1970:
1962:
1957:
1939:
1936:
1933:
1930:
1927:
1924:
1921:
1916:
1910:
1903:
1897:
1892:
1887:
1881:
1874:
1871:
1865:
1860:
1856:
1853:
1848:
1843:
1835:
1829:
1824:
1814:
1810:
1807:
1803:
1768:
1756:
1755:
1754:
1751:
1747:
1742:
1737:
1733:
1729:
1721:
1717:
1713:
1708:
1704:
1700:
1695:
1691:
1687:
1679:
1675:
1671:
1664:
1660:
1656:
1650:
1646:
1642:To show that
1640:
1638:
1634:
1630:
1625:
1614:
1610:
1604:
1600:
1596:
1588:
1584:
1580:
1576:
1568:
1564:
1554:
1550:
1549:parallelogram
1538:
1534:
1521:
1498:
1493:
1487:
1482:
1475:
1470:
1464:
1459:
1456:
1453:
1447:
1444:
1441:
1438:
1435:
1429:
1426:
1423:
1420:
1417:
1414:
1411:
1408:
1405:
1402:
1397:
1391:
1386:
1383:
1380:
1373:
1368:
1365:
1362:
1356:
1347:
1346:
1345:
1331:
1308:
1303:
1296:
1293:
1287:
1279:
1276:
1270:
1264:
1259:
1254:
1248:
1243:
1236:
1231:
1225:
1220:
1217:
1210:
1207:
1203:
1200:
1194:
1187:
1184:
1180:
1177:
1171:
1168:
1163:
1156:
1153:
1149:
1146:
1141:
1133:
1130:
1126:
1123:
1118:
1112:
1103:
1102:
1101:
1084:
1081:
1078:
1075:
1072:
1069:
1066:
1063:
1058:
1052:
1047:
1040:
1035:
1029:
1020:
1019:
1018:
1002:
996:
991:
984:
979:
973:
964:
948:
945:
942:
914:
911:
908:
902:
899:
896:
890:
881:
878:
872:
869:
863:
858:
851:
848:
842:
835:
830:
824:
819:
814:
808:
805:
800:
793:
788:
782:
770:
769:
768:
767:For example,
751:
748:
745:
742:
739:
736:
733:
728:
722:
717:
710:
705:
699:
694:
689:
683:
678:
671:
666:
660:
648:
647:
646:
626:
620:
615:
608:
603:
597:
577:
575:
571:
567:
563:
559:
551:
547:
543:
540:-dimensional
535:
532:-dimensional
528:, the signed
527:
523:
519:
515:
514:Cramer's rule
511:
507:
502:
500:
496:
492:
488:
484:
480:
475:
469:
466:
459:
452:
448:
447:
446:
443:
439:
433:
431:
427:
423:
419:
411:
395:
392:
384:
379:
375:
355:
352:
349:
346:
343:
340:
337:
334:
331:
328:
325:
322:
319:
316:
313:
310:
307:
304:
301:
298:
295:
292:
289:
286:
283:
278:
272:
267:
262:
255:
250:
245:
238:
233:
228:
222:
213:
212:
211:
190:
187:
184:
181:
178:
175:
172:
167:
161:
156:
149:
144:
138:
129:
128:
127:
120:
118:
113:
111:
107:
103:
99:
95:
88:
80:
72:
65:
60:
59:square matrix
56:
52:
48:
44:
37:
33:
19:
28112:Determinants
28001:Vector space
27945:
27733:Vector space
27613:
27593:
27572:
27538:
27507:
27503:
27474:
27448:
27432:
27423:
27419:
27409:
27382:
27359:, Springer,
27352:
27327:
27317:, retrieved
27310:the original
27277:
27261:
27242:
27224:
27206:
27183:
27142:
27133:
27130:Muir, Thomas
27122:the original
27106:
27087:
27082:
27063:, Springer,
27060:
27042:
27022:
26984:
26969:, Elsevier,
26966:
26940:
26936:
26897:math/0203276
26887:
26883:
26855:
26849:
26823:
26794:
26788:
26769:, Springer,
26766:
26734:
26730:
26688:
26684:
26654:
26642:
26608:
26604:
26594:
26588:Bareiss 1968
26579:
26568:. Retrieved
26561:the original
26538:
26525:
26514:
26502:
26491:
26470:
26465:, §1.1, §4.3
26458:
26453:, Lecture 1.
26445:
26425:
26418:
26143:
26139:
26133:
26122:
26106:
26094:
26084:
26076:
26064:
26052:
26043:
26037:
26025:
26013:
26001:
25990:
25978:
25966:
25954:
25942:
25930:
25922:
25916:
25909:Kleiner 2007
25904:
25889:
25884:
25872:
25867:, p. 80
25865:Kleiner 2007
25860:
25848:. Retrieved
25844:the original
25833:
25821:
25798:
25792:
25780:
25768:
25735:
25731:
25724:
25712:
25707:, Def. 1.2.3
25696:
25688:MathOverflow
25687:
25680:
25671:
25652:
25648:
25638:
25617:
25605:
25570:
25566:
25560:
25527:
25523:
25513:
25501:
25489:
25477:
25460:
25448:
25440:
25432:
25420:
25405:
25397:
25385:. Retrieved
25381:
25372:
25354:
25347:
25335:
25265:
25259:
25254:
25250:
25245:
25241:
25232:
25228:
25226:
25108:
24868:
24563:
24502:
24380:
24372:
24350:
24244:
24233:
24220:graded rings
24174:
24162:
24158:
24147:
24140:
24137:
24113:
24101:
24001:
23993:
23980:reduced norm
23919:
23745:
23657:
23575:
23406:
23216:
23130:
22808:of (finite)
22731:
22539:
22536:
22266:
22232:of units in
22178:between the
22177:
22113:
21983:
21896:
21886:
21882:
21878:
21870:
21868:vector space
21865:
21820:
21666:
21663:
21659:
21654:
21650:
21646:
21640:
21623:
21616:
21465:is given by
21461:
21457:
21453:
21448:
21444:
21440:
21436:
21432:
21425:
21415:
21301:
21297:
21294:
21233:
21223:
21026:
20719:is given by
20569:
20559:
20543:
20535:
20531:
20529:
20516:
20508:
20496:
20492:
20488:
20484:
20478:
20398:
19884:lies in the
19754:
19695:
19600:
19571:
19225:
19030:
19022:Applications
18980:Spottiswoode
18964:persymmetric
18953:
18936:
18930:
18921:
18907:
18903:
18898:
18897:columns and
18894:
18888:
18880:discriminant
18870:
18848:
18844:plane curves
18825:
18814:
18808:
18515:
18414:Lie algebras
18411:
18283:
18089:
17951:, these are
17930:
17768:denotes the
17735:
17557:
17533:
17375:
17369:between two
17361:
17357:
17354:
17248:
17245:
17233:
17229:
17222:
17218:
17206:power series
17203:
17034:
17031:
16852:
16842:
16696:
16689:
16682:
16678:
16668:
16661:
16654:
16650:
16641:
16638:
16504:
16500:
16495:
16491:
16487:
16481:
16408:
16405:
16116:
15923:
15919:
15899:
15342:
15335:
15328:
15325:
15258:is given by
15253:
15169:
15085:
14998:
14992:
14924:
14659:
14499:
14427:
14304:
14297:
14187:
14107:
14027:
13923:
13695:real. Since
13507:
13503:
13429:
13250:
13036:
12829:
12638:
12483:
12291:
12053:
12048:
12044:
12016:
11672:
11667:
11663:
11583:
11582:, each with
11579:
11575:
11568:
11562:
11558:
11552:
11548:
11542:
11538:
11533:
11529:
11527:
11428:
11424:
11419:
11415:
11410:
11409:matrix, and
11405:
11401:
11396:
11391:
11190:
11161:
10992:
10990:
10983:
10535:
10426:
9974:
9693:block matrix
9664:
9589:
9511:
9375:
9365:
9361:
9354:
9350:
9302:
9298:
9297:to write an
9290:
8905:
8764:
8736:
8492:
8490:
8352:. For every
8147:. The minor
8116:
8102:
8096:
7782:
7769:
7762:matrix group
7606:
7474:
7388:
7277:
7275:
7265:
7261:
7256:
7252:
7249:
7185:
7159:
7061:
7010:
6958:
6909:
6822:
6817:
6812:
6807:
6707:
6612:
6517:
6260:
6005:alternating.
6001:
5990:
5430:
5417:
5412:
5383:
5286:
5008:
5004:
5000:
4996:
4921:
4915:
4843:-th column.
4840:
4836:
4804:
4707:
4679:
4622:
4450:
4342:
4205:
3965:
3832:
3561:permutations
3532:
3526:
3522:
3508:
3493:
3489:
3485:
3481:
3476:
3472:
3468:
3464:
3462:
3292:
3290:
3265:
3263:
3018:
3014:
3012:
2973:The entries
2972:
2721:
2717:
2709:
2707:
2689:
2681:
2677:
2673:
2665:
2664:produced by
2657:
2601:
2596:
2397:
2391:
2368:
2067:
2059:
2055:
2052:
2046:
2042:
2037:
2031:
2027:
2023:
2019:
2011:
2007:
1999:
1995:
1985:
1979:
1977:
1968:
1966:
1749:
1745:
1735:
1731:
1727:
1719:
1715:
1711:
1698:
1693:
1689:
1685:
1677:
1673:
1669:
1662:
1658:
1654:
1648:
1644:
1641:
1626:
1612:
1608:
1605:
1598:
1594:
1586:
1582:
1578:
1574:
1566:
1562:
1526:
1323:
1099:
934:
766:
583:
546:endomorphism
506:coefficients
503:
494:
483:endomorphism
476:
473:
441:
437:
434:
377:
373:
370:
205:
121:
114:
86:
78:
70:
63:
46:
40:
27981:Multivector
27946:Determinant
27903:Dot product
27748:Linear span
25971:Cajori 1993
25796:Cajori, F.
25703:, §VIII.2,
25655:: 332–341.
25573:: 202–218.
25425:Harris 2014
24230:Calculation
22806:free module
21626:Cartography
21430:open subset
21029:tetrahedron
20896:dimensional
20874:, then the
20501:orientation
20487:vectors in
20459:polynomials
19004:Christoffel
18949:Cayley 1841
18943:(1839) and
18850:Vandermonde
18542:-matrix as
15210:satisfying
14566:eigenvalues
14111:eigenvalues
14030:eigenvalues
9295:generalized
8141:recursively
8103:rectangular
7776:), and the
7693:called the
6859:Determinant
6804:Obtained by
6068:, whenever
5288:alternating
4917:multilinear
4208:pi notation
2670:orientation
2472:the region
2038:signed area
1765:Signed area
1629:signed area
1553:unit square
1533:linear maps
550:orientation
522:eigenvalues
495:determinant
110:isomorphism
47:determinant
43:mathematics
32:Risk factor
28106:Categories
28015:Direct sum
27850:Invertible
27753:Linear map
27319:2020-06-04
26943:: 98–109.
26737:(2): 3–7,
26633:See also:
26629:References
26570:2011-01-22
26482:1812.02056
25898:37.0181.02
25850:24 January
25801:p. 80
25580:1805.06027
25437:Serge Lang
24869:The order
24183:Berezinian
23803:quaternion
22924:. The map
22726:See also:
22228:) and the
22109:unimodular
21657:such that
21033:skew lines
20898:volume of
20685:measurable
18996:Wronskians
18972:circulants
18937:alternants
18516:Writing a
18418:Lie groups
17560:polynomial
17554:Derivative
16490:arguments
14612:such that
12468:since the
9997:invertible
8785:-th column
7768:(which if
7501:invertible
4835:(for each
4805:where the
3565:signatures
3563:and their
2704:Definition
2698:one-to-one
1973:equi-areal
1724:, so that
210:matrix is
126:matrix is
106:invertible
94:linear map
28045:Numerical
27808:Transpose
27595:MathWorld
27579:EMS Press
27524:123637858
27426:: 267–271
27385:, Dover,
27182:(2018) .
27165:54:75–90
27132:(1960) ,
26865:1206.7067
26842:248917264
26814:1813/6003
26519:Rote 2001
26030:Lang 1985
26018:Lang 2002
26006:Lang 1985
25959:Eves 1990
25826:Eves 1990
25760:120467300
25701:Lang 1985
25629:1410.1958
25597:119272194
25524:Math. Gaz
25508:, §0.8.2.
25453:Lang 1987
25340:Lang 1985
25172:
25121:
25074:
24999:for some
24977:≥
24880:
24834:⋅
24819:ε
24758:−
24715:ε
24476:
24437:
24302:×
24289:factorial
24063:
24054:
24045:
23955:→
23780:
23772:×
23725:→
23618:character
23597:σ
23591:
23578:permanent
23521:⋀
23463:⋀
23379:∧
23376:⋯
23373:∧
23360:⋅
23327:∧
23324:⋯
23321:∧
23289:⋀
23257:∈
23170:⋀
23140:⋀
23096:∧
23093:⋯
23090:∧
23074:∧
23058:↦
23041:∧
23038:⋯
23035:∧
23022:∧
22996:⋀
22992:→
22976:⋀
22960:⋀
22883:⋀
22749:→
22691:→
22642:×
22634:−
22370:
22357:→
22345:
22320:
22287:→
22193:×
22158:×
22150:→
22138:
22014:
22006:×
21995:∈
21984:A matrix
21848:→
21771:−
21710:−
21568:
21531:ϕ
21522:∫
21495:ϕ
21477:∫
21393:≤
21381:≤
21357:∂
21342:∂
21265:→
21195:→
21180::
21135:−
21123:−
21111:−
21094:⋅
21000:
20941:
20839:×
20804:→
20659:⊂
20604:→
20455:resultant
20430:−
20356:−
20335:⋯
20313:−
20275:−
20252:⋮
20247:⋱
20242:⋮
20237:⋮
20206:⋯
20130:⋯
20049:…
20017:Wronskian
19996:−
19951:…
19902:×
19833:∈
19759:vectors:
19707:
19639:
19621:
19512:…
19472:−
19459:…
19406:∑
19379:…
19369:…
19205:…
19008:Frobenius
18988:Pfaffians
18941:Sylvester
18834:in 1693.
18777:×
18742:∇
18729:×
18694:∇
18681:×
18646:∇
18527:×
18384:ϵ
18370:ϵ
18358:
18337:ϵ
18258:ϵ
18244:ϵ
18228:−
18215:
18184:ϵ
18170:ϵ
18152:
18143:
18122:−
18113:ϵ
18054:−
18007:
17982:∂
17965:∂
17905:α
17883:−
17870:
17846:α
17747:
17710:α
17684:
17673:
17661:α
17580:×
17496:
17478:≤
17466:
17450:≤
17423:≤
17408:−
17396:
17334:−
17325:
17319:≤
17304:
17298:≤
17285:−
17277:−
17266:
17156:
17121:−
17110:∞
17095:∑
17091:−
17065:∞
17050:∑
16984:
16959:−
16948:∞
16933:∑
16929:−
16903:∞
16888:∑
16804:∑
16788:
16743:∑
16605:…
16539:−
16503:– 1)! tr(
16428:∑
16356:
16289:−
16263:∏
16232:⋯
16193:≥
16177:…
16144:∑
16090:≤
16084:≤
16051:
16034:−
16006:∑
15992:−
15981:−
15908:, or the
15855:
15846:−
15834:
15807:
15766:
15728:
15698:
15689:−
15666:
15590:
15560:
15542:
15533:−
15510:
15434:
15428:−
15405:
15296:
15287:
15224:
15200:logarithm
15194:) of exp(
15170:Here exp(
15137:
15122:
15104:
15059:
15050:
15029:
14884:⋯
14841:⋮
14836:⋱
14831:⋮
14826:⋮
14801:⋯
14740:⋯
14636:λ
14624:χ
14600:λ
14476:−
14470:⋅
14440:χ
14274:λ
14270:⋯
14261:λ
14251:λ
14238:λ
14217:∏
14153:λ
14146:…
14134:λ
14121:λ
14067:×
13734:, taking
13620:−
13519:×
13469:−
13320:−
13213:−
13190:−
12999:−
12976:−
12953:−
12930:−
12754:−
12600:−
12495:×
12404:≥
12352:×
12318:×
12305:Hermitian
12246:≥
12164:≥
11890:
11849:−
11751:−
11345:−
11173:×
11135:−
10951:−
10940:−
10874:−
10853:−
10842:−
10796:−
10766:−
10745:⏟
10728:−
10710:−
10702:−
10533:-matrix.
10518:×
10464:−
10453:−
10394:−
10383:−
10333:−
10322:−
10309:−
10239:−
10209:−
10188:⏟
10149:−
10141:−
10131:−
9822:×
9796:×
9770:×
9744:×
9676:×
9639:
9610:−
9562:
9547:
9462:−
9428:
9390:
9358:submatrix
9259:−
9236:≤
9224:≤
9217:∏
9197:−
9179:⋯
9169:−
9146:−
9123:−
9103:⋮
9098:⋱
9093:⋮
9088:⋮
9083:⋮
9059:⋯
8989:⋯
8941:⋯
8837:−
8814:∑
8748:×
8641:−
8422:−
8399:∑
8301:−
8212:−
8203:×
8194:−
8080:×
8046:
8019:
7977:
7934:
7891:
7839:×
7805:
7739:
7726:⊂
7714:
7668:
7586:−
7527:−
7475:A matrix
7162:transpose
7156:Transpose
7126:−
7120:⋅
7114:⋅
7105:−
7086:−
6982:−
6831:−
6776:−
6734:−
6681:−
6639:−
6586:−
6539:−
6492:−
6450:−
6442:−
6364:−
6351:−
6319:−
6311:−
6240:σ
6183:∏
6163:⋯
5949:…
5868:…
5808:−
5784:…
5689:…
5670:…
5651:…
5630:−
5606:…
5590:…
5571:…
5507:×
5396:×
5343:…
5331:…
5319:…
5245:…
5233:…
5194:…
5182:…
5161:⋅
5127:…
5093:⋅
5079:−
5065:…
4971:⋅
4920:: if the
4767:…
4691:×
4644:…
4582:⋯
4543:⋯
4529:ε
4510:…
4480:∑
4424:…
4372:…
4355:ε
4311:σ
4280:∏
4273:σ
4267:
4244:∈
4241:σ
4237:∑
4177:σ
4163:⋯
4149:σ
4132:σ
4126:
4108:∈
4105:σ
4101:∑
4070:…
4047:⋮
4043:⋮
4020:…
3924:…
3901:⋮
3897:⋮
3874:…
3816:−
3770:σ
3747:σ
3741:
3678:σ
3672:…
3657:σ
3642:σ
3622:σ
3590:…
3544:×
3433:−
3421:−
3409:−
3220:⋯
3177:⋮
3172:⋱
3167:⋮
3162:⋮
3137:⋯
3076:⋯
2929:⋯
2886:⋮
2881:⋱
2876:⋮
2871:⋮
2846:⋯
2785:⋯
2720:rows and
2627:±
2572:∀
2563:≤
2550:≤
2544:∣
2516:⋯
2442:…
2321:⋮
2289:…
2252:⋮
2190:⋮
2120:⋯
2036:) is the
1931:−
1893:⋅
1872:−
1854:θ
1825:⊥
1808:θ
1460:⋅
1442:−
1415:−
1384:⋅
1366:⋅
1195:−
1073:−
946:×
912:−
900:⋅
891:−
879:−
873:⋅
849:−
806:−
743:−
410:factorial
344:−
332:−
320:−
182:−
28091:Category
28030:Subspace
28025:Quotient
27976:Bivector
27890:Bilinear
27832:Matrices
27707:Glossary
27634:Archived
27547:citation
27533:(1772),
27498:(1841),
27080:(1998),
27021:(2009),
26955:Archived
26858:: 1–16,
26755:archived
26751:62780452
26722:(1990),
26709:archived
26659:Springer
26653:(2015).
26586:, §1.1,
26117:, §III.5
26113:, §5.2,
26101:, §III.8
26008:, §VII.3
25552:41879675
25484:, §0.8.7
25387:16 March
25364:Archived
25342:, §VII.1
25278:See also
25249:, where
24957:, where
24319:of order
24169:and the
23976:Pfaffian
23681:that is
23658:For any
23614:immanant
22267:Given a
21456: :
21037:vertices
20854:-matrix
20556:rotation
20481:sequence
20220:′
20191:′
20167:′
19012:Hessians
18984:Glaisher
18956:Lebesgue
18927:Jacobian
18862:Lagrange
17770:adjugate
14945:between
14032:and the
11546:are the
11070:), then
8350:cofactor
7691:subgroup
5496:(for an
5439:, i.e.,
5411:-matrix
4885:, where
4706:-matrix
3529:matrices
1981:bivector
1857:′
1736:θ′
1297:′
1280:′
1211:′
1188:′
1157:′
1134:′
564:and the
558:calculus
552:and the
526:geometry
55:function
53:-valued
27702:Outline
27467:1104435
27413:, Paris
27401:3363427
27306:1911585
27243:Algebra
27170:0019078
27011:2347309
26922:2104048
26914:4145188
26705:2004533
26059:, §11.4
26020:, §IV.8
25740:Bibcode
25544:3620776
24418:or the
24153:with a
23620:of the
21428:and an
20683:is any
20509:product
20015:), the
19889:spanned
18976:Catalan
18884:quantic
18832:Leibniz
18821:Cardano
18805:History
16695:, ...,
16667:, ...,
14520:is the
14188:product
14178:occurs
14084:complex
12047:and of
11671:matrix
11038:(i.e.,
11036:commute
6257:Example
6032:, i.e.
6002:another
5522:matrix
4992:of two
2684:. This
2396:to the
2154:, then
2066:matrix
588:matrix
36:Epitope
27986:Tensor
27798:Kernel
27728:Vector
27723:Scalar
27522:
27485:
27465:
27455:
27399:
27389:
27371:
27334:
27304:
27294:
27249:
27231:
27213:
27194:
27149:
27114:
27094:
27067:
27049:
27031:
27009:
26999:
26973:
26920:
26912:
26840:
26830:
26773:
26749:
26703:
26665:
26553:
26509:, §1.1
26433:
25911:, §5.2
25896:
25787:, §6.6
25758:
25595:
25550:
25542:
25427:, §4.7
24414:, the
24116:factor
23217:define
22764:of an
22079:. For
21236:. For
20997:volume
20938:volume
20688:subset
20648:, and
20011:times
19226:where
18968:Hankel
18945:Cayley
18913:Cauchy
18858:minors
18594:where
18284:using
17736:where
17376:Also,
17032:where
16099:
16075:
16072:
15970:
15967:
15964:
15922:= (−1)
15828:
15554:
15340:, and
14500:Here,
14056:be an
11528:where
8497:th row
8145:minors
6418:Matrix
4905:is an
4405:tuples
4206:Using
3492:gives
3448:
2599:signed
2569:
2053:If an
1990:(0, 0)
1734:| cos
1707:cosine
1692:| sin
1591:, and
1557:(0, 0)
534:volume
90:|
84:|
51:scalar
45:, the
27855:Minor
27840:Block
27778:Basis
27520:S2CID
27313:(PDF)
27274:(PDF)
26958:(PDF)
26933:(PDF)
26910:JSTOR
26892:arXiv
26860:arXiv
26758:(PDF)
26747:S2CID
26727:(PDF)
26712:(PDF)
26701:JSTOR
26681:(PDF)
26564:(PDF)
26535:(PDF)
26477:arXiv
25756:S2CID
25624:arXiv
25593:S2CID
25575:arXiv
25548:S2CID
25540:JSTOR
25412:10–17
25328:Notes
25085:2.376
24564:of a
24422:(for
24126:on a
23982:of a
23970:of a
23542:with
21944:field
21875:basis
20552:3 × 3
20548:2 × 2
20493:basis
19886:plane
19743:, or
19016:Trudi
19002:) by
18960:Hesse
18891:Binet
18882:of a
18872:Gauss
17221:>
16507:) as
14995:trace
14989:Trace
14590:roots
13655:with
11413:, an
11399:, an
10507:is a
10487:when
9841:, is
9364:) x (
9347:terms
7955:is a
7787:from
7497:field
7452:when
6028:is a
5991:other
3611:is a
3516:3 × 3
3297:3 × 3
2716:with
2712:be a
2688:that
2686:means
2394:-cube
2026:) ∧ (
586:2 × 2
560:with
536:of a
524:. In
508:in a
491:basis
485:of a
408:(the
208:3 × 3
124:2 × 2
98:basis
82:, or
49:is a
28010:Dual
27865:Rank
27553:link
27508:1841
27483:ISBN
27453:ISBN
27387:ISBN
27369:ISBN
27332:ISBN
27292:ISBN
27247:ISBN
27229:ISBN
27211:ISBN
27192:ISBN
27147:ISBN
27112:ISBN
27092:ISBN
27065:ISBN
27047:ISBN
27029:ISBN
26997:ISBN
26971:ISBN
26838:OCLC
26828:ISBN
26771:ISBN
26663:ISBN
26551:ISBN
26431:ISBN
25852:2012
25389:2018
25253:and
25010:>
24667:and
24002:The
23922:norm
23920:the
23553:<
22871:-th
22810:rank
22623:and
21649:and
20542:(if
20495:for
20463:root
19006:and
19000:Muir
17228:det(
17216:for
16676:and
16499:= −(
15900:cf.
14993:The
14965:and
13963:and
13763:and
13675:and
12337:and
12097:and
11984:and
11941:and
11556:and
11537:and
11427:and
11263:and
11217:and
11191:For
11015:and
10993:same
9811:and
9376:The
9289:The
9230:<
8097:The
7300:and
6823:add
6105:<
6079:>
5984:sign
4999:and
4343:The
3509:The
3494:dbi,
3467:has
3291:The
2708:Let
2696:nor
2694:onto
2064:real
2004:and
1714:= (−
1703:sine
1667:and
1633:area
77:det
69:det(
27512:doi
27437:doi
27361:doi
27282:doi
26989:doi
26945:doi
26902:doi
26888:111
26870:doi
26809:hdl
26799:doi
26739:doi
26693:doi
26613:doi
26609:429
26543:doi
25894:JFM
25748:doi
25736:344
25657:doi
25585:doi
25571:512
25532:doi
25264:of
24837:det
24822:det
24804:det
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24222:).
24185:on
24157:of
24143:≥ 2
24060:log
24042:exp
24021:det
23816:det
23765:Mat
23716:det
23588:sgn
23508:or
23348:det
23272:):
23131:As
22904:of
22675:det
22485:det
22448:det
22059:in
21999:Mat
21954:det
21877:in
21794:det
21779:det
21757:det
21745:det
21730:det
21718:det
21696:det
21681:det
21559:det
21435:of
21232:of
21102:det
20967:det
20732:det
20550:or
20523:in
20483:of
19767:det
19658:det
19636:adj
19618:adj
19548:det
19436:det
19346:det
19324:det
19165:det
19144:det
19092:det
18974:by
18753:det
18705:det
18657:det
18455:det
18325:det
18200:det
18149:adj
18125:det
18101:det
18032:det
18004:adj
17968:det
17855:det
17829:det
17772:of
17744:adj
17681:adj
17644:det
17597:to
17426:det
17307:det
17301:log
17208:in
16866:det
16681:= (
16653:= (
16518:det
16486:of
16414:≥ 0
16128:det
15918:det
15623:det
15467:det
15362:det
15345:= 4
15338:= 3
15331:= 2
15284:exp
15269:det
15221:exp
15202:of
15186:of
15178:of
15134:exp
15128:det
15119:log
15047:exp
15026:exp
15020:det
14997:tr(
14664:is
14461:det
14201:det
13986:det
13971:det
13869:det
13844:det
13805:det
13132:det
13117:det
13065:det
12557:det
12542:det
12521:det
12436:det
12410:det
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12264:det
12249:det
12228:det
12188:det
12167:det
12152:det
12125:det
12051:.
12013:Sum
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11868:det
11827:det
11815:det
11791:det
11720:det
11708:det
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11596:det
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11442:det
11354:det
11336:det
11294:det
11123:det
11081:det
10931:det
10919:det
10828:det
10816:det
10785:det
10755:det
10672:det
10629:det
10617:det
10571:det
10435:det
10374:det
10362:det
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10259:det
10228:det
10198:det
10115:det
10072:det
10060:det
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9975:If
9921:det
9906:det
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9636:adj
9627:det
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8798:det
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7569:det
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7437:det
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7225:det
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6128:det
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3482:bdi
3465:bdi
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2612:det
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1804:sin
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1657:≡ (
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915:19.
778:det
656:det
643:det
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453:is
412:of
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41:In
28108::
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27592:.
27577:,
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22129:GL
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21889:.
21667:BX
21662:=
21621:.
21460:→
21300:×
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20465:.
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19741:QR
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19737:LU
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18429:SL
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25681:n
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19581:a
19557:)
19554:A
19551:(
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19535:=
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19160:)
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18762:)
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18721:=
18714:)
18711:A
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18580:]
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18376:O
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18367:)
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18334:+
18331:I
18328:(
18302:I
18299:=
18296:A
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18267:)
18262:2
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18250:O
18247:+
18240:)
18236:X
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18224:A
18219:(
18209:)
18206:A
18203:(
18197:=
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18188:2
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18176:O
18173:+
18167:)
18164:X
18161:)
18158:A
18155:(
18146:(
18137:=
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18131:A
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18119:)
18116:X
18110:+
18107:A
18104:(
18075:.
18070:i
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18062:)
18057:1
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18041:)
18038:A
18035:(
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18024:i
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18017:)
18013:A
18010:(
18001:=
17993:j
17990:i
17986:A
17977:)
17974:A
17971:(
17939:A
17916:.
17912:)
17902:d
17897:A
17894:d
17886:1
17879:A
17874:(
17864:)
17861:A
17858:(
17852:=
17843:d
17838:)
17835:A
17832:(
17826:d
17800:A
17780:A
17756:)
17753:A
17750:(
17721:.
17717:)
17707:d
17702:A
17699:d
17693:)
17690:A
17687:(
17677:(
17667:=
17658:d
17653:)
17650:A
17647:(
17641:d
17606:R
17583:n
17577:n
17572:R
17519:.
17513:)
17508:2
17504:A
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17488:n
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17475:)
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17469:(
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17444:n
17441:1
17436:)
17432:A
17429:(
17416:)
17411:1
17404:A
17400:(
17389:n
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17358:A
17340:)
17337:I
17331:A
17328:(
17316:)
17313:A
17310:(
17294:)
17288:1
17281:A
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17249:A
17236:)
17230:I
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17128:)
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17102:=
17099:j
17087:(
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17057:=
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17017:,
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17006:)
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16988:(
16976:j
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16966:)
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16940:=
16937:j
16925:(
16917:!
16914:k
16910:1
16898:0
16895:=
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16881:)
16878:A
16875:+
16872:I
16869:(
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16845:n
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16823:I
16818:I
16814:A
16808:I
16800:=
16797:)
16794:A
16791:(
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16777:K
16772:J
16768:B
16762:I
16757:K
16753:A
16747:K
16739:=
16734:I
16729:J
16725:)
16721:B
16718:A
16715:(
16702:)
16699:r
16697:j
16693:2
16690:j
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16674:)
16671:r
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16655:i
16651:I
16644:J
16642:A
16624:.
16621:)
16616:n
16612:s
16608:,
16602:,
16597:2
16593:s
16589:,
16584:1
16580:s
16576:(
16571:n
16567:B
16560:!
16557:n
16550:n
16546:)
16542:1
16536:(
16530:=
16527:)
16524:A
16521:(
16505:A
16501:l
16496:l
16492:s
16488:n
16467:.
16464:n
16461:=
16456:l
16452:k
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16443:n
16438:1
16435:=
16432:l
16411:l
16409:k
16391:,
16384:l
16380:k
16374:)
16369:l
16365:A
16361:(
16347:!
16342:l
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16330:l
16326:k
16321:l
16313:1
16310:+
16305:l
16301:k
16296:)
16292:1
16286:(
16278:n
16273:1
16270:=
16267:l
16254:n
16251:=
16246:n
16242:k
16238:n
16235:+
16229:+
16224:2
16220:k
16216:2
16213:+
16208:1
16204:k
16196:0
16188:n
16184:k
16180:,
16174:,
16169:2
16165:k
16161:,
16156:1
16152:k
16140:=
16137:)
16134:A
16131:(
16102:.
16096:)
16093:n
16087:m
16081:1
16078:(
16068:)
16063:k
16059:A
16055:(
16043:k
16040:+
16037:m
16031:n
16027:c
16021:m
16016:1
16013:=
16010:k
16000:m
15997:1
15989:=
15984:m
15978:n
15974:c
15961:;
15958:1
15955:=
15950:n
15946:c
15927:0
15924:c
15920:A
15914:n
15881:.
15877:)
15872:)
15867:4
15863:A
15859:(
15849:6
15843:)
15840:A
15837:(
15824:)
15819:3
15815:A
15811:(
15801:8
15798:+
15793:2
15788:)
15783:)
15778:2
15774:A
15770:(
15759:(
15754:3
15751:+
15746:2
15741:)
15737:)
15734:A
15731:(
15721:(
15715:)
15710:2
15706:A
15702:(
15692:6
15684:4
15679:)
15675:)
15672:A
15669:(
15659:(
15653:(
15644:1
15639:=
15632:)
15629:A
15626:(
15616:,
15612:)
15607:)
15602:3
15598:A
15594:(
15584:2
15581:+
15577:)
15572:2
15568:A
15564:(
15551:)
15548:A
15545:(
15536:3
15528:3
15523:)
15519:)
15516:A
15513:(
15503:(
15497:(
15491:6
15488:1
15483:=
15476:)
15473:A
15470:(
15460:,
15456:)
15451:)
15446:2
15442:A
15438:(
15423:2
15418:)
15414:)
15411:A
15408:(
15398:(
15392:(
15386:2
15383:1
15378:=
15371:)
15368:A
15365:(
15343:n
15336:n
15329:n
15311:.
15308:)
15305:)
15302:L
15299:(
15290:(
15281:=
15278:)
15275:A
15272:(
15256:A
15239:A
15236:=
15233:)
15230:L
15227:(
15208:L
15204:A
15196:A
15192:λ
15188:A
15184:λ
15180:A
15172:A
15155:.
15152:)
15149:)
15146:)
15143:A
15140:(
15131:(
15125:(
15116:=
15113:)
15110:A
15107:(
15088:A
15071:)
15068:)
15065:A
15062:(
15053:(
15044:=
15041:)
15038:)
15035:A
15032:(
15023:(
15007:A
15003:A
14999:A
14973:n
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14933:k
14908:]
14900:k
14897:,
14894:k
14890:a
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14874:,
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14746:a
14733:2
14730:,
14727:1
14723:a
14715:1
14712:,
14709:1
14705:a
14698:[
14688:k
14684:A
14642:=
14639:)
14633:(
14628:A
14576:A
14552:A
14532:I
14508:t
14485:.
14482:)
14479:A
14473:I
14467:t
14464:(
14458:=
14455:)
14452:t
14449:(
14444:A
14413:A
14393:0
14373:A
14353:A
14333:0
14313:A
14283:.
14278:n
14265:2
14255:1
14247:=
14242:i
14232:n
14227:1
14224:=
14221:i
14213:=
14210:)
14207:A
14204:(
14184:A
14180:μ
14176:μ
14157:n
14149:,
14143:,
14138:2
14130:,
14125:1
14108:n
14094:A
14070:n
14064:n
14044:A
14003:1
14000:=
13997:)
13993:i
13989:(
13983:=
13980:)
13977:I
13974:(
13951:0
13948:=
13945:)
13941:i
13937:(
13909:.
13904:2
13900:b
13896:+
13891:2
13887:a
13883:=
13880:)
13876:i
13872:(
13864:2
13860:b
13856:+
13853:)
13850:I
13847:(
13839:2
13835:a
13831:=
13828:)
13824:i
13820:b
13817:+
13814:I
13811:a
13808:(
13781:i
13777:b
13774:=
13771:B
13751:I
13748:a
13745:=
13742:A
13722:0
13719:=
13716:)
13712:i
13708:(
13683:b
13663:a
13641:)
13635:0
13630:1
13623:1
13615:0
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13604:b
13601:+
13596:)
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13562:a
13555:i
13551:b
13548:+
13545:I
13542:a
13522:2
13516:2
13489:.
13486:)
13483:B
13480:A
13477:(
13466:)
13463:B
13460:(
13452:)
13449:A
13446:(
13415:)
13406:B
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13392:+
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13369:+
13360:B
13350:A
13346:+
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13323:(
13317:)
13308:B
13304:+
13295:B
13291:(
13288:)
13279:A
13275:+
13266:A
13262:(
13236:.
13227:A
13217:B
13204:B
13194:A
13181:A
13171:B
13167:+
13158:B
13148:A
13144:+
13141:)
13138:B
13135:(
13129:+
13126:)
13123:A
13120:(
13094:B
13074:)
13071:A
13068:(
13045:A
13022:.
13013:B
13003:B
12990:B
12980:A
12967:A
12957:B
12944:A
12934:A
12921:B
12911:B
12907:+
12898:B
12888:A
12884:+
12875:A
12865:B
12861:+
12852:A
12842:A
12815:.
12812:)
12803:B
12799:+
12790:A
12786:(
12783:)
12774:B
12770:+
12761:A
12757:(
12751:)
12742:B
12738:+
12729:A
12725:(
12722:)
12713:B
12709:+
12700:A
12696:(
12671:j
12668:i
12664:B
12660:,
12655:j
12652:i
12648:A
12620:.
12617:)
12614:B
12611:A
12608:(
12597:)
12594:B
12591:(
12583:)
12580:A
12577:(
12569:+
12566:)
12563:B
12560:(
12554:+
12551:)
12548:A
12545:(
12539:=
12536:)
12533:B
12530:+
12527:A
12524:(
12498:2
12492:2
12470:n
12456:,
12450:n
12445:)
12442:B
12439:(
12430:+
12424:n
12419:)
12416:A
12413:(
12398:n
12393:)
12390:B
12387:+
12384:A
12381:(
12355:n
12349:n
12339:B
12335:A
12321:n
12315:n
12297:n
12277:.
12273:)
12270:B
12267:(
12261:+
12258:)
12255:A
12252:(
12243:)
12240:B
12237:+
12234:A
12231:(
12207:,
12203:)
12200:C
12197:+
12194:B
12191:(
12185:+
12182:)
12179:C
12176:+
12173:A
12170:(
12161:)
12158:C
12155:(
12149:+
12146:)
12143:C
12140:+
12137:B
12134:+
12131:A
12128:(
12105:C
12085:B
12065:A
12049:B
12045:A
12031:B
12028:+
12025:A
11995:A
11992:B
11972:B
11969:A
11949:B
11929:A
11906:.
11903:c
11899:)
11896:X
11893:(
11883:r
11880:+
11877:)
11874:X
11871:(
11865:=
11861:)
11857:c
11852:1
11845:X
11841:r
11838:+
11835:1
11831:(
11824:)
11821:X
11818:(
11812:=
11809:)
11806:r
11803:c
11800:+
11797:X
11794:(
11767:,
11763:)
11759:A
11754:1
11747:X
11743:B
11740:+
11734:n
11729:I
11724:(
11717:)
11714:X
11711:(
11705:=
11702:)
11699:B
11696:A
11693:+
11690:X
11687:(
11673:X
11668:m
11664:m
11645:.
11642:c
11639:r
11636:+
11633:1
11630:=
11626:)
11622:r
11619:c
11616:+
11610:m
11605:I
11600:(
11584:m
11580:r
11576:c
11563:n
11559:n
11553:m
11549:m
11543:n
11539:I
11534:m
11530:I
11513:,
11509:)
11505:A
11502:B
11499:+
11493:n
11488:I
11483:(
11476:=
11472:)
11468:B
11465:A
11462:+
11456:m
11451:I
11446:(
11429:B
11425:A
11420:m
11416:n
11411:B
11406:n
11402:m
11397:A
11372:.
11369:)
11366:B
11363:+
11360:A
11357:(
11351:)
11348:B
11342:A
11339:(
11333:=
11328:)
11322:A
11317:B
11310:B
11305:A
11299:(
11271:B
11251:A
11231:C
11228:=
11225:B
11205:D
11202:=
11199:A
11176:2
11170:2
11147:.
11144:)
11141:C
11138:B
11132:D
11129:A
11126:(
11120:=
11115:)
11109:D
11104:C
11097:B
11092:A
11086:(
11058:C
11055:D
11052:=
11049:D
11046:C
11023:D
11003:C
10965:.
10962:)
10959:C
10954:1
10947:D
10943:B
10937:A
10934:(
10928:)
10925:D
10922:(
10916:=
10904:)
10896:n
10892:I
10886:0
10877:1
10870:D
10866:B
10861:C
10856:1
10849:D
10845:B
10839:A
10833:(
10825:)
10822:D
10819:(
10813:=
10799:1
10792:)
10788:D
10782:(
10778:=
10774:)
10769:1
10762:D
10758:(
10751:=
10739:)
10731:1
10724:D
10718:C
10713:1
10706:D
10695:0
10688:m
10684:I
10677:(
10663:)
10657:D
10652:C
10645:B
10640:A
10634:(
10626:)
10623:D
10620:(
10614:=
10605:)
10599:D
10594:C
10587:B
10582:A
10576:(
10544:D
10521:1
10515:1
10495:D
10475:)
10472:B
10467:1
10460:A
10456:C
10450:D
10447:(
10444:)
10441:A
10438:(
10408:,
10405:)
10402:B
10397:1
10390:A
10386:C
10380:D
10377:(
10371:)
10368:A
10365:(
10359:=
10347:)
10341:B
10336:1
10329:A
10325:C
10319:D
10312:1
10305:A
10301:C
10294:0
10287:m
10283:I
10276:(
10268:)
10265:A
10262:(
10256:=
10242:1
10235:)
10231:A
10225:(
10221:=
10217:)
10212:1
10205:A
10201:(
10194:=
10182:)
10174:n
10170:I
10164:0
10157:B
10152:1
10145:A
10134:1
10127:A
10120:(
10106:)
10100:D
10095:C
10088:B
10083:A
10077:(
10069:)
10066:A
10063:(
10057:=
10048:)
10042:D
10037:C
10030:B
10025:A
10019:(
9983:A
9960:.
9955:)
9949:D
9944:0
9937:B
9932:A
9926:(
9918:=
9915:)
9912:D
9909:(
9903:)
9900:A
9897:(
9891:=
9886:)
9880:D
9875:C
9868:0
9863:A
9857:(
9825:n
9819:n
9799:m
9793:n
9773:n
9767:m
9747:m
9741:m
9721:D
9718:,
9715:C
9712:,
9709:B
9706:,
9703:A
9679:2
9673:2
9645:.
9642:A
9630:A
9623:1
9618:=
9613:1
9606:A
9575:.
9572:A
9568:)
9565:A
9556:(
9553:=
9550:A
9541:A
9538:=
9535:I
9532:)
9529:A
9523:(
9497:.
9492:i
9489:j
9485:M
9479:j
9476:+
9473:i
9469:)
9465:1
9459:(
9456:=
9451:j
9448:,
9445:i
9441:)
9437:)
9434:A
9431:(
9422:(
9399:)
9396:A
9393:(
9355:k
9351:k
9330:)
9325:k
9322:n
9317:(
9303:n
9299:n
9291:n
9277:.
9273:)
9267:i
9263:x
9254:j
9250:x
9245:(
9239:n
9233:j
9227:i
9221:1
9213:=
9208:|
9200:1
9194:n
9189:n
9185:x
9172:1
9166:n
9161:3
9157:x
9149:1
9143:n
9138:2
9134:x
9126:1
9120:n
9115:1
9111:x
9074:2
9069:n
9065:x
9052:2
9047:3
9043:x
9035:2
9030:2
9026:x
9018:2
9013:1
9009:x
8999:n
8995:x
8982:3
8978:x
8970:2
8966:x
8958:1
8954:x
8946:1
8936:1
8931:1
8926:1
8920:|
8891:.
8886:j
8883:,
8880:i
8876:M
8870:j
8867:,
8864:i
8860:a
8854:j
8851:+
8848:i
8844:)
8840:1
8834:(
8829:n
8824:1
8821:=
8818:i
8810:=
8807:)
8804:A
8801:(
8773:j
8751:2
8745:2
8720:|
8714:h
8709:g
8702:e
8697:d
8691:|
8686:c
8683:+
8678:|
8672:i
8667:g
8660:f
8655:d
8649:|
8644:b
8636:|
8630:i
8625:h
8618:f
8613:e
8607:|
8602:a
8599:=
8594:|
8588:i
8583:h
8578:g
8571:f
8566:e
8561:d
8554:c
8549:b
8544:a
8538:|
8513:1
8510:=
8507:i
8495:i
8476:,
8471:j
8468:,
8465:i
8461:M
8455:j
8452:,
8449:i
8445:a
8439:j
8436:+
8433:i
8429:)
8425:1
8419:(
8414:n
8409:1
8406:=
8403:j
8395:=
8392:)
8389:A
8386:(
8360:i
8334:j
8331:,
8328:i
8324:M
8318:j
8315:+
8312:i
8308:)
8304:1
8298:(
8278:j
8258:i
8238:A
8218:)
8215:1
8209:n
8206:(
8200:)
8197:1
8191:n
8188:(
8166:j
8163:,
8160:i
8156:M
8128:A
8076:K
8055:)
8052:K
8049:(
8041:n
8032:/
8028:)
8025:K
8022:(
8014:n
7986:)
7983:K
7980:(
7972:n
7943:)
7940:K
7937:(
7929:n
7900:)
7897:K
7894:(
7886:n
7861:K
7835:K
7814:)
7811:K
7808:(
7800:n
7770:n
7748:)
7745:K
7742:(
7734:n
7723:)
7720:K
7717:(
7709:n
7677:)
7674:K
7671:(
7663:n
7635:K
7615:n
7603:.
7589:1
7582:]
7578:)
7575:A
7572:(
7566:[
7563:=
7557:)
7554:A
7551:(
7544:1
7539:=
7535:)
7530:1
7523:A
7519:(
7483:A
7460:A
7440:B
7417:A
7397:B
7374:)
7371:B
7368:(
7362:)
7359:A
7356:(
7350:=
7347:)
7344:B
7341:A
7338:(
7308:B
7288:A
7266:n
7262:n
7257:n
7253:n
7246:.
7234:)
7231:A
7228:(
7222:=
7218:)
7212:T
7207:A
7203:(
7186:A
7172:A
7138:=
7135:)
7132:)
7129:1
7123:(
7117:3
7108:(
7102:=
7098:|
7094:E
7090:|
7083:=
7079:|
7075:A
7071:|
7043:|
7039:D
7035:|
7031:=
7027:|
7023:E
7019:|
6994:|
6990:C
6986:|
6979:=
6975:|
6971:D
6967:|
6942:|
6938:C
6934:|
6930:=
6926:|
6922:B
6918:|
6894:|
6890:B
6886:|
6882:=
6878:|
6874:A
6870:|
6839:3
6785:]
6779:1
6771:0
6766:0
6759:4
6754:3
6749:0
6742:2
6737:3
6723:[
6718:=
6715:E
6690:]
6684:1
6676:0
6671:0
6664:4
6659:3
6647:2
6642:3
6634:5
6628:[
6623:=
6620:D
6595:]
6589:1
6581:0
6576:0
6569:4
6559:3
6552:2
6547:5
6542:3
6533:[
6528:=
6525:C
6501:]
6495:1
6487:3
6482:0
6475:4
6470:1
6465:3
6458:2
6453:1
6445:3
6436:[
6431:=
6428:B
6402:A
6378:.
6373:]
6367:1
6359:3
6354:3
6344:4
6339:1
6334:2
6327:2
6322:1
6314:2
6305:[
6300:=
6297:A
6274:A
6216:.
6211:i
6208:i
6204:a
6198:n
6193:1
6190:=
6187:i
6179:=
6174:n
6171:n
6167:a
6154:a
6144:a
6140:=
6137:)
6134:A
6131:(
6108:j
6102:i
6082:j
6076:i
6056:0
6053:=
6048:j
6045:i
6041:a
6016:A
5970:.
5966:|
5960:n
5956:a
5952:,
5946:,
5941:4
5937:a
5933:,
5928:3
5924:a
5920:,
5915:2
5911:a
5907:,
5902:1
5898:a
5893:|
5889:=
5885:|
5879:n
5875:a
5871:,
5865:,
5860:4
5856:a
5852:,
5847:2
5843:a
5839:,
5834:3
5830:a
5826:,
5821:1
5817:a
5812:|
5805:=
5801:|
5795:n
5791:a
5787:,
5779:4
5775:a
5771:,
5766:2
5762:a
5758:,
5753:1
5749:a
5745:,
5740:3
5736:a
5731:|
5710:.
5706:|
5700:n
5696:a
5692:,
5686:,
5681:j
5677:a
5673:,
5667:,
5662:i
5658:a
5654:,
5648:,
5643:1
5639:a
5634:|
5627:=
5623:|
5617:n
5613:a
5609:,
5603:,
5598:i
5594:a
5587:,
5582:j
5578:a
5574:,
5568:,
5563:1
5559:a
5554:|
5530:A
5510:n
5504:n
5484:)
5481:A
5478:(
5470:n
5466:c
5462:=
5459:)
5456:A
5453:c
5450:(
5413:A
5399:n
5393:n
5364:=
5360:|
5354:n
5350:a
5346:,
5340:,
5337:v
5334:,
5328:,
5325:v
5322:,
5316:,
5311:1
5307:a
5302:|
5262:|
5256:n
5252:a
5248:,
5242:,
5239:w
5236:,
5230:,
5225:1
5221:a
5216:|
5212:+
5208:|
5202:n
5198:a
5191:,
5188:v
5185:,
5179:,
5174:1
5170:a
5165:|
5158:r
5155:=
5144:|
5138:n
5134:a
5130:,
5124:,
5119:1
5116:+
5113:j
5109:a
5105:,
5102:w
5099:+
5096:v
5090:r
5087:,
5082:1
5076:j
5072:a
5068:,
5062:,
5057:1
5053:a
5047:|
5042:=
5034:|
5030:A
5026:|
5009:A
5005:r
5001:w
4997:v
4980:w
4977:+
4974:v
4968:r
4965:=
4960:j
4956:a
4932:A
4922:j
4909:.
4893:I
4873:1
4870:=
4866:)
4863:I
4860:(
4841:i
4837:i
4821:i
4817:a
4790:,
4785:)
4778:n
4774:a
4770:,
4764:,
4759:1
4755:a
4749:(
4744:=
4741:A
4718:n
4708:A
4694:n
4688:n
4656:.
4653:}
4650:n
4647:,
4641:,
4638:1
4635:{
4625:n
4608:,
4601:n
4597:i
4593:,
4590:n
4586:a
4574:1
4570:i
4566:,
4563:1
4559:a
4551:n
4547:i
4538:1
4534:i
4521:n
4517:i
4513:,
4507:,
4502:2
4498:i
4494:,
4489:1
4485:i
4476:=
4473:)
4470:A
4467:(
4447:0
4433:}
4430:n
4427:,
4421:,
4418:1
4415:{
4403:-
4401:n
4383:n
4379:i
4375:,
4369:,
4364:1
4360:i
4339:.
4326:)
4320:)
4317:i
4314:(
4308:,
4305:i
4301:a
4295:n
4290:1
4287:=
4284:i
4276:)
4270:(
4260:(
4252:n
4248:S
4233:=
4230:)
4227:A
4224:(
4191:.
4186:)
4183:n
4180:(
4174:,
4171:n
4167:a
4158:)
4155:1
4152:(
4146:,
4143:1
4139:a
4135:)
4129:(
4116:n
4112:S
4097:=
4092:|
4084:n
4081:,
4078:n
4074:a
4065:1
4062:,
4059:n
4055:a
4034:n
4031:,
4028:1
4024:a
4015:1
4012:,
4009:1
4005:a
3998:|
3993:=
3990:)
3987:A
3984:(
3951:,
3946:]
3938:n
3935:,
3932:n
3928:a
3919:1
3916:,
3913:n
3909:a
3888:n
3885:,
3882:1
3878:a
3869:1
3866:,
3863:1
3859:a
3852:[
3847:=
3844:A
3796:,
3793:1
3790:+
3750:)
3744:(
3716:n
3712:S
3687:)
3684:n
3681:(
3675:,
3669:,
3666:)
3663:2
3660:(
3654:,
3651:)
3648:1
3645:(
3599:}
3596:n
3593:,
3587:,
3584:2
3581:,
3578:1
3575:{
3547:n
3541:n
3527:n
3523:n
3477:i
3473:d
3469:b
3445:.
3442:h
3439:f
3436:a
3430:i
3427:d
3424:b
3418:g
3415:e
3412:c
3406:h
3403:d
3400:c
3397:+
3394:g
3391:f
3388:b
3385:+
3382:i
3379:e
3376:a
3373:=
3368:|
3362:i
3357:h
3352:g
3345:f
3340:e
3335:d
3328:c
3323:b
3318:a
3312:|
3266:A
3249:.
3244:|
3236:n
3233:,
3230:n
3226:a
3213:2
3210:,
3207:n
3203:a
3195:1
3192:,
3189:n
3185:a
3153:n
3150:,
3147:2
3143:a
3130:2
3127:,
3124:2
3120:a
3112:1
3109:,
3106:2
3102:a
3092:n
3089:,
3086:1
3082:a
3069:2
3066:,
3063:1
3059:a
3051:1
3048:,
3045:1
3041:a
3034:|
3019:A
3015:A
2992:1
2989:,
2986:1
2982:a
2958:.
2953:]
2945:n
2942:,
2939:n
2935:a
2922:2
2919:,
2916:n
2912:a
2904:1
2901:,
2898:n
2894:a
2862:n
2859:,
2856:2
2852:a
2839:2
2836:,
2833:2
2829:a
2821:1
2818:,
2815:2
2811:a
2801:n
2798:,
2795:1
2791:a
2778:2
2775:,
2772:1
2768:a
2760:1
2757:,
2754:1
2750:a
2743:[
2738:=
2735:A
2722:n
2718:n
2710:A
2690:A
2682:n
2678:A
2674:n
2666:A
2658:n
2644:,
2641:)
2638:P
2635:(
2624:=
2621:)
2618:A
2615:(
2602:n
2583:.
2579:}
2575:i
2566:1
2558:i
2554:c
2547:0
2539:n
2534:a
2527:n
2523:c
2519:+
2513:+
2508:1
2503:a
2496:1
2492:c
2487:{
2483:=
2480:P
2460:,
2455:n
2450:a
2445:,
2439:,
2434:2
2429:a
2424:,
2419:1
2414:a
2398:n
2392:n
2377:A
2354:.
2349:n
2344:a
2339:=
2334:)
2328:1
2314:0
2307:0
2301:(
2296:A
2292:,
2285:,
2280:2
2275:a
2270:=
2265:)
2259:0
2245:1
2238:0
2232:(
2227:A
2223:,
2218:1
2213:a
2208:=
2203:)
2197:0
2183:0
2176:1
2170:(
2165:A
2141:]
2132:n
2127:a
2113:2
2108:a
2099:1
2094:a
2085:[
2081:=
2078:A
2068:A
2060:n
2056:n
2034:)
2032:d
2028:c
2024:b
2020:a
2018:(
2014:)
2012:d
2008:c
2006:(
2002:)
2000:b
1996:a
1994:(
1969:A
1940:.
1937:c
1934:b
1928:d
1925:a
1922:=
1917:)
1911:d
1904:c
1898:(
1888:)
1882:a
1875:b
1866:(
1861:=
1844:|
1840:v
1836:|
1830:|
1820:u
1815:|
1811:=
1799:|
1794:v
1789:|
1783:|
1778:u
1773:|
1769:=
1732:v
1728:u
1726:|
1722:)
1720:a
1716:b
1712:u
1699:θ
1694:θ
1690:v
1686:u
1684:|
1680:)
1678:d
1674:c
1670:v
1665:)
1663:b
1659:a
1655:u
1622:A
1618:A
1601:)
1599:d
1595:c
1593:(
1589:)
1587:d
1583:b
1579:c
1575:a
1573:(
1569:)
1567:b
1563:a
1561:(
1545:A
1541:A
1529:A
1499:.
1494:|
1488:d
1483:c
1476:b
1471:a
1465:|
1457:r
1454:=
1451:)
1448:c
1445:b
1439:d
1436:a
1433:(
1430:r
1427:=
1424:c
1421:r
1418:b
1412:d
1409:a
1406:r
1403:=
1398:|
1392:d
1387:c
1381:r
1374:b
1369:a
1363:r
1357:|
1332:r
1309:.
1304:|
1294:d
1288:c
1277:b
1271:a
1265:|
1260:+
1255:|
1249:d
1244:c
1237:b
1232:a
1226:|
1221:=
1218:c
1215:)
1208:b
1204:+
1201:b
1198:(
1192:)
1185:d
1181:+
1178:d
1175:(
1172:a
1169:=
1164:|
1154:d
1150:+
1147:d
1142:c
1131:b
1127:+
1124:b
1119:a
1113:|
1082:=
1079:a
1076:b
1070:b
1067:a
1064:=
1059:|
1053:b
1048:a
1041:b
1036:a
1030:|
1003:)
997:1
992:0
985:0
980:1
974:(
949:2
943:2
909:=
906:)
903:1
897:7
894:(
888:)
885:)
882:4
876:(
870:3
867:(
864:=
859:|
852:4
843:1
836:7
831:3
825:|
820:=
815:)
809:4
801:1
794:7
789:3
783:(
752:.
749:c
746:b
740:d
737:a
734:=
729:|
723:d
718:c
711:b
706:a
700:|
695:=
690:)
684:d
679:c
672:b
667:a
661:(
627:)
621:d
616:c
609:b
604:a
598:(
554:n
538:n
530:n
464:.
457:.
455:1
442:n
438:n
414:n
396:!
393:n
378:n
374:n
356:.
353:h
350:f
347:a
341:i
338:d
335:b
329:g
326:e
323:c
317:h
314:d
311:c
308:+
305:g
302:f
299:b
296:+
293:i
290:e
287:a
284:=
279:|
273:i
268:h
263:g
256:f
251:e
246:d
239:c
234:b
229:a
223:|
191:,
188:c
185:b
179:d
176:a
173:=
168:|
162:d
157:c
150:b
145:a
139:|
87:A
79:A
73:)
71:A
64:A
38:.
20:)
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