15884:
15342:
15879:{\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}}}
22249:
27812:
25275:
21155:
1509:
10968:
10411:
9276:
1945:
10551:
9994:
28076:
8902:
26400:
20383:
3489:
1939:
10963:{\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}}}
10406:{\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}}}
19556:
18788:
26138:
2353:
8721:
14909:
9271:{\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).}
2957:
3248:
16390:
20014:
4190:
23115:
1308:
18267:
17518:
1748:
5266:
6221:(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
24138:
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
19308:
18625:
3261:, 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.
2149:
26395:{\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}}}
21601:
17016:
8520:
25909:
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
25098:
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,
24137:
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
14667:
3502:
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
17188:
5411:
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
21208:
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
2719:
3468:
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
3016:
16112:
5373:
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
1498:
914:
20378:{\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}}.}
3965:
25142:, 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
22939:
18085:
9959:
18908:.) 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.
5993:
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
5969:
23391:
16101:
13642:
17371:
17720:
1934:{\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.}
18074:
4607:
4326:
751:
13021:
2582:
21011:
5005:
3950:
3447:
355:
17915:
5709:
23983:
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.
1095:
19551:{\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)}
26126:
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
18783:{\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}}}
25193:, 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
17339:
11905:
13414:
7492:
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
22378:
16827:
18396:
14282:
6377:
6786:
6691:
6596:
6502:
2348:{\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}.}
23735:
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
21400:
11512:
8716:{\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}}}
21460:
16850:
14904:{\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}}}
13908:
6254:
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
12619:
11371:
21805:
19680:
17033:
13235:
2952:{\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}}.}
3243:{\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}}.}
16385:{\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}},}
11146:
16623:
12814:
11766:
19210:
9574:
7747:
5714:
2459:
1339:
762:
13526:
12455:
6215:
8895:
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
4185:{\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)}.}
1084:
23110:{\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}}}
12207:
8054:
190:
20516:, 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.
7590:
22705:
22162:
18802:. 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
18262:{\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)}
9496:
9644:
8890:
8475:
24086:
21279:
14158:
9836:
1004:
628:
26143:
18581:
13488:
21206:
23267:
17513:{\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)}}.}
15154:
15070:
11644:
24848:
24162:. 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
15929:
7233:
5366:
23903:
22022:
3686:
21145:
12277:
20815:
20615:
23788:
22521:
15310:
5537:
4789:
1624:, 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
22944:
18630:
15347:
10556:
9999:
7140:
5010:
17622:
4386:
17946:
4448:
4205:
640:
19844:
12825:
7239:
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
14484:
5261:{\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}}}
25089:
2464:
23599:
20922:
14002:
3749:
2643:
13950:
13721:
3828:
3294:
7985:
7942:
7899:
7813:
7676:
25187:
25136:
24895:
24452:
20670:
19722:
205:
19972:
17807:
17584:
16466:
22610:
18434:
17755:
9398:
7373:
24487:
14644:
6250:
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,
5534:
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):
5404:
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.
1303:{\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}}.}
5483:
4872:
27372:
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
18500:
7596:
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
26697:
10474:
104:
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
8335:
8217:
24205:
9333:
6840:
4979:
4655:
3598:
24399:
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
23529:
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23178:
23148:
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13781:
4432:
24986:
23957:
22094:
23730:
22643:
17247:
12672:
24714:
22533:
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
21929:
17606:
11775:
5407:
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
13246:
8081:
7840:
24234:
While the determinant can be computed directly using the
Leibniz rule this approach is extremely inefficient for large matrices, since that formula requires calculating
23259:
22294:
16699:
15890:
24304:
22195:
21968:
20841:
20745:
19904:
18529:
15238:
14069:
13521:
12497:
12354:
12320:
11175:
10520:
9824:
9798:
9772:
9746:
9678:
8750:
5509:
5398:
4693:
3546:
948:
22751:
22289:
21850:
18463:
18309:
14599:
14185:
6994:
6055:
25050:
24397:
21063:
19100:
13073:
9720:
8167:
7043:
6942:
6894:
6281:
6239:
3769:
3621:
2993:
2141:
25012:
24548:
23555:
19769:
7439:
6107:
6081:
23637:
23441:
19871:
19586:
19240:
18613:
11057:
6699:
6604:
6509:
6412:
4822:
3717:
24944:
21303:
20914:
20706:
19051:
13750:
21596:{\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} .}
20432:
19998:
18301:
17011:{\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}\,,}
12030:
11230:
11204:
8512:
3818:
3795:
24760:
24737:
24330:
24255:
24154:
is commutative). Nevertheless, various notions of non-commutative determinant have been formulated that preserve some of the properties of determinants, notably
11994:
11971:
395:
26464:
Camarero, Cristóbal (2018-12-05). "Simple, Fast and
Practicable Algorithms for Cholesky, LU and QR Decomposition Using Fast Rectangular Matrix Multiplication".
25211:
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24654:
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24614:
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24510:
24275:
23692:
23668:
23491:
23414:
23226:
23198:
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22911:
22858:
22838:
22818:
22791:
22771:
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22551:
22418:
22398:
22239:
22215:
22066:
22042:
21907:
20881:
20861:
20765:
20635:
20406:
19789:
19300:
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19071:
17938:
17799:
17779:
17523:
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
14972:
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14932:
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14507:
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14372:
14352:
14332:
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13662:
13093:
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12104:
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12064:
11948:
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11022:
11002:
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10494:
9982:
8772:
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8237:
8127:
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7614:
7482:
7459:
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7171:
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6015:
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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.
2376:
1331:
18943:
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
12359:
6112:
13789:
25139:
17183:{\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}\,,}
12505:
11278:
5986:
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.
21665:
23396:
This definition agrees with the more concrete coordinate-dependent definition. This can be shown using the uniqueness of a multilinear alternating form on
19598:
12109:
26520:
13101:
27259:
23988:
provides different extensions of the determinant for such infinite-dimensional situations, which however only work for particular kinds of operators.
11065:
16502:
19007:. Of the textbooks on the subject Spottiswoode's was the first. In America, Hanus (1886), Weld (1893), and Muir/Metzler (1933) published treatises.
12680:
1493:{\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}}.}
909:{\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.}
27541:
11668:
6217:
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
19112:
13523:
matrix algebras. For example, consider the complex numbers as a matrix algebra. The complex numbers have a representation as matrices of the form
9507:
7688:
1512:
The area of the parallelogram is the absolute value of the determinant of the matrix formed by the vectors representing the parallelogram's sides.
24227:, where for applications such as checking invertibility and finding eigenvalues the determinant has largely been supplanted by other techniques.
2396:
9283:
25828:
20480:, 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
2995:
etc. are, for many purposes, real or complex numbers. As discussed below, the determinant is also defined for matrices whose entries are in a
1613:
is in general a different parallelogram, but since the determinant is symmetric with respect to rows and columns, the area will be the same.)
18867:. He introduced the word "determinant" (Laplace had used "resultant"), though not in the present signification, but rather as applied to the
12212:
11575:
components, the formula allows quick calculation of the determinant of a matrix that differs from the identity matrix by a matrix of rank 1:
1012:
558:
1690:
between the vectors, which is simply base times height, the length of one vector times the perpendicular component of the other. Due to the
24355:. 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
21886:
The above definition of the determinant using the
Leibniz rule holds works more generally when the entries of the matrix are elements of a
7993:
121:
26438:"... we mention that the determinant, though a convenient notion theoretically, rarely finds a useful role in numerical algorithms.", see
7499:
26666:
24332:. The Laplace expansion is similarly inefficient. Therefore, more involved techniques have been developed for calculating determinants.
22659:
22112:
12499:
matrices with complex entries, the determinant of the sum can be written in terms of determinants and traces in the following identity:
26943:
21935:
of real or complex numbers. Moreover, the characterization of the determinant as the unique alternating multilinear map that satisfies
9954:{\displaystyle \det {\begin{pmatrix}A&0\\C&D\end{pmatrix}}=\det(A)\det(D)=\det {\begin{pmatrix}A&B\\0&D\end{pmatrix}}.}
9406:
9589:
8782:
8367:
24005:
21231:
18804:
5964:{\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}|.}
3452:
In this expression, each term has one factor from each row, all in different columns, arranged in increasing row order. For example,
27622:
23386:{\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}.}
14104:
956:
580:
27670:
27070:
18534:
13425:
21161:
16096:{\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)~.}
13637:{\displaystyle aI+b\mathbf {i} :=a{\begin{pmatrix}1&0\\0&1\end{pmatrix}}+b{\begin{pmatrix}0&-1\\1&0\end{pmatrix}}}
27603:
15085:
28003:
18808:(九章算術, Chinese scholars, around the 3rd century BCE). In Europe, solutions of linear systems of two equations were expressed by
15004:
11580:
9826:, respectively. The easiest such formula, which can be proven using either the Leibniz formula or a factorization involving the
7461:
is the identity matrix. The above-mentioned unique characterization of alternating multilinear maps therefore shows this claim.
2077:
463:
The above properties relating to rows (properties 2–4) may be replaced by the corresponding statements with respect to columns.
28061:
24788:
7183:
5285:
23800:
21976:
3626:
27324:
27284:
27239:
27184:
27104:
27094:
27039:
26989:
26655:
26423:
21068:
950:-matrices, and that continue to hold for determinants of larger matrices. They are as follows: first, the determinant of the
20770:
20570:
424:
Determinants can also be defined by some of their properties. Namely, the determinant is the unique function defined on the
23742:
22426:
17715:{\displaystyle {\frac {d\det(A)}{d\alpha }}=\operatorname {tr} \left(\operatorname {adj} (A){\frac {dA}{d\alpha }}\right).}
15253:
5431:
4725:
25633:
18900:
reduces to the multiplication theorem. On the same day (November 30, 1812) that Binet presented his paper to the Academy,
4669:
The determinant can be characterized by the following three key properties. To state these, it is convenient to regard an
25255:
25146:, i.e., how many bits of accuracy are needed to store intermediate values occurring in the computation. For example, the
18069:{\displaystyle {\frac {\partial \det(A)}{\partial A_{ij}}}=\operatorname {adj} (A)_{ji}=\det(A)\left(A^{-1}\right)_{ji}.}
7054:
4602:{\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}},}
4321:{\displaystyle \det(A)=\sum _{\sigma \in S_{n}}\left(\operatorname {sgn}(\sigma )\prod _{i=1}^{n}a_{i,\sigma (i)}\right)}
1952:
is the absolute value of the determinant of the matrix formed by the columns constructed from the vectors r1, r2, and r3.
746:{\displaystyle \det {\begin{pmatrix}a&b\\c&d\end{pmatrix}}={\begin{vmatrix}a&b\\c&d\end{vmatrix}}=ad-bc.}
27488:
25936:
18875:. Gauss also arrived at the notion of reciprocal (inverse) determinants, and came very near the multiplication theorem.
13016:{\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}.}
4338:
25092:
24112:
20434:
are linearly independent. If it can be shown that the Wronskian is zero everywhere on an interval then, in the case of
11381:
10974:
15894:
3797:
if the permutation can be obtained with an even number of transpositions (exchanges of two entries); otherwise, it is
2577:{\displaystyle P=\left\{c_{1}\mathbf {a} _{1}+\cdots +c_{n}\mathbf {a} _{n}\mid 0\leq c_{i}\leq 1\ \forall i\right\}.}
505:), although other methods of solution are computationally much more efficient. Determinants are used for defining the
27475:
27445:
27379:
27361:
27221:
27203:
27139:
27084:
27057:
27021:
26963:
26820:
26763:
26743:
26543:
21006:{\displaystyle \operatorname {volume} (f(S))={\sqrt {\det \left(A^{\textsf {T}}A\right)}}\operatorname {volume} (S).}
19794:
1333:(i.e., all entries in that column are multiplied by that number), the determinant is also multiplied by that number:
26837:
Fisikopoulos, Vissarion; Peñaranda, Luis (2016), "Faster geometric algorithms via dynamic determinant computation",
18924:
for 1841 he specially treats this subject, as well as the class of alternating functions which Sylvester has called
14423:
3945:{\displaystyle A={\begin{bmatrix}a_{1,1}\ldots a_{1,n}\\\vdots \qquad \vdots \\a_{n,1}\ldots a_{n,n}\end{bmatrix}},}
3442:{\displaystyle {\begin{vmatrix}a&b&c\\d&e&f\\g&h&i\end{vmatrix}}=aei+bfg+cdh-ceg-bdi-afh.\ }
28115:
28051:
26839:
26647:
25055:
18798:
Historically, determinants were used long before matrices: A determinant was originally defined as a property of a
3270:
3258:
371:
27598:
23572:
13955:
3722:
2596:
350:{\displaystyle {\begin{vmatrix}a&b&c\\d&e&f\\g&h&i\end{vmatrix}}=aei+bfg+cdh-ceg-bdi-afh.}
28013:
27949:
27614:
24108:
18915:
17910:{\displaystyle {\frac {d\det(A)}{d\alpha }}=\det(A)\operatorname {tr} \left(A^{-1}{\frac {dA}{d\alpha }}\right).}
15898:
13916:
13687:
475:
7951:
7908:
7865:
7779:
7642:
27467:
26549:
26413:
25153:
25102:
24861:
24676:
can be quickly calculated, since they are the products of the respective diagonal entries. The determinant of
24418:
20640:
19688:
3010:), or it can be denoted directly in terms of the matrix entries by writing enclosing bars instead of brackets:
27298:
23501:
of vector spaces. Minors of a matrix can also be cast in this setting, by considering lower alternating forms
19913:
17554:
16411:
5704:{\displaystyle |a_{1},\dots ,a_{j},\dots a_{i},\dots ,a_{n}|=-|a_{1},\dots ,a_{i},\dots ,a_{j},\dots ,a_{n}|.}
27567:
27151:
25348:
25294:
24143:
22588:
18412:
17728:
9371:
7319:
24457:
22645:. Adding yet another layer of abstraction, this is captured by saying that the determinant is a morphism of
14607:
27791:
27663:
27632:
26074:
20439:
19733:
18972:
18879:
4840:
25666:
23473:(as opposed to the determinant associated to an endomorphism) is sometimes also called the determinant of
18471:
1544:
under the mapping. The parallelogram defined by the rows of the above matrix is the one with vertices at
27896:
27746:
27562:
27118:
27012:
21410:
18988:
17547:
The Leibniz formula shows that the determinant of real (or analogously for complex) square matrices is a
17355:
12281:
10419:
1605:
is the area of the parallelogram, and thus represents the scale factor by which areas are transformed by
924:
The determinant has several key properties that can be proved by direct evaluation of the definition for
24127:
For matrices over non-commutative rings, multilinearity and alternating properties are incompatible for
1956:
Thus the determinant gives the scaling factor and the orientation induced by the mapping represented by
27801:
27695:
27260:"Division-free algorithms for the determinant and the Pfaffian: algebraic and combinatorial approaches"
27176:
26778:
25611:
Lin, Minghua; Sra, Suvrit (2014). "Completely strong superadditivity of generalized matrix functions".
20489:
20463:
19021:
18799:
18466:
8282:
8172:
7398:, both sides of the equation are alternating and multilinear as a function depending on the columns of
2661:.) In particular, if the determinant is zero, then this parallelotope has volume zero and is not fully
2658:
2587:
550:
498:
26871:
Garibaldi, Skip (2004), "The characteristic polynomial and determinant are not ad hoc constructions",
24181:
18615:
are column vectors of length 3, then the gradient over one of the three vectors may be written as the
17334:{\displaystyle \operatorname {tr} \left(I-A^{-1}\right)\leq \log \det(A)\leq \operatorname {tr} (A-I)}
12032:
of two square matrices of the same size is not in general expressible in terms of the determinants of
9297:
6815:
4939:
4619:
3559:
28041:
27690:
25719:
Kondratyuk, L. A.; Krivoruchenko, M. I. (1992). "Superconducting quark matter in SU(2) color group".
23504:
23446:
23153:
23123:
22866:
21862:
is defined to be the determinant of the matrix describing it, with respect to an arbitrary choice of
18996:
18980:
15920:
14022:
13755:
11900:{\displaystyle \det(X+cr)=\det(X)\det \left(1+rX^{-1}c\right)=\det(X)+r\,\operatorname {adj} (X)\,c.}
8338:
7902:
5412:
in defining the determinant, since without it the existence of an appropriate function is not clear.
4399:
506:
24949:
23915:
22071:
18964:
13409:{\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})}
4438:
if two of the integers are equal, and otherwise as the signature of the permutation defined by the
28033:
27916:
27608:
27484:
25832:
25309:
25225:
24412:
24224:
23700:
22615:
22373:{\displaystyle \operatorname {GL} _{n}(f):\operatorname {GL} _{n}(R)\to \operatorname {GL} _{n}(S)}
21607:
20408:) in a specified interval if and only if the given functions and all their derivatives up to order
18905:
16822:{\displaystyle (AB)_{J}^{I}=\sum _{K}A_{K}^{I}B_{J}^{K},\operatorname {tr} (A)=\sum _{I}A_{I}^{I}.}
14658:
14654:
14510:
12631:
12044:
8087:
7754:
2674:
27644:
Determinants explained in an easy fashion in the 4th chapter as a part of a Linear Algebra course.
25366:
25299:
24897:
reached by decomposition methods has been improved by different methods. If two matrices of order
24699:
24159:
22716:
21912:
17589:
28110:
28079:
28008:
27786:
27656:
27557:
26402:
a contradiction. There is no useful notion of multi-linear functions over a non-commutative ring.
25786:
25352:
24104:
23566:
21214:
20001:
19724:
time, which is comparable to more common methods of solving systems of linear equations, such as
18992:
18391:{\displaystyle \det(I+\epsilon X)=1+\operatorname {tr} (X)\epsilon +O\left(\epsilon ^{2}\right).}
14277:{\displaystyle \det(A)=\prod _{i=1}^{n}\lambda _{i}=\lambda _{1}\lambda _{2}\cdots \lambda _{n}.}
8059:
7818:
5972:
3553:
421:
with the same determinant, equal to the product of the diagonal entries of the row echelon form.
25138:
is based on the following idea: one replaces permutations (as in the Leibniz rule) by so-called
23231:
21630:
The above identities concerning the determinant of products and inverses of matrices imply that
6372:{\displaystyle A={\begin{bmatrix}-2&-1&2\\2&1&4\\-3&3&-1\end{bmatrix}}.}
493:
Determinants occur throughout mathematics. For example, a matrix is often used to represent the
28105:
27843:
27776:
27766:
26623:
25910:
Institute de France in Paris on November 30, 1812, and which was subsequently published in the
25400:
24408:
24362:
So, the determinant can be computed for almost free from the result of a Gaussian elemination.
24283:
24228:
24092:
23972:
22582:
22174:
21938:
21870:. By the similarity invariance, this determinant is independent of the choice of the basis for
21863:
20820:
20711:
20544:
19883:
18929:
18508:
15205:
14983:
14161:
14048:
13500:
12476:
12333:
12299:
11154:
10499:
9803:
9777:
9751:
9725:
9657:
9346:
8729:
5488:
5377:
4672:
3525:
927:
86:
43:
27638:
25189:, but the bit length of intermediate values can become exponentially long. By comparison, the
22724:
22262:
21823:
18439:
14584:
6950:
6781:{\displaystyle E={\begin{bmatrix}18&-3&2\\0&3&4\\0&0&-1\end{bmatrix}}}
6686:{\displaystyle D={\begin{bmatrix}5&-3&2\\13&3&4\\0&0&-1\end{bmatrix}}}
6591:{\displaystyle C={\begin{bmatrix}-3&5&2\\3&13&4\\0&0&-1\end{bmatrix}}}
6497:{\displaystyle B={\begin{bmatrix}-3&-1&2\\3&1&4\\0&3&-1\end{bmatrix}}}
6024:
28100:
27858:
27853:
27848:
27781:
27726:
27168:
26590:"Dodgson condensation: The historical and mathematical development of an experimental method"
25805:
Campbell, H: "Linear Algebra With Applications", pages 111–112. Appleton Century Crofts, 1971
25017:
24373:
24223:
Determinants are mainly used as a theoretical tool. They are rarely calculated explicitly in
22553:
of the determinant of such a matrix is equal to the determinant of the matrix reduced modulo
21815:
21811:
21395:{\displaystyle D(f)=\left({\frac {\partial f_{i}}{\partial x_{j}}}\right)_{1\leq i,j\leq n}.}
21030:
20528:
19907:
19906:-matrix consisting of the three vectors is zero. The same idea is also used in the theory of
19076:
18850:
17241:, the trace operator gives the following tight lower and upper bounds on the log determinant
13049:
9687:
8139:
7766:
7002:
6901:
6853:
6224:
3754:
3606:
2965:
2650:
2390:
2062:
26918:
25554:
Sothanaphan, Nat (January 2017). "Determinants of block matrices with noncommuting blocks".
24991:
24518:
23534:
19771:
is zero if and only if the column vectors (or, equivalently, the row vectors) of the matrix
19751:
7421:
6086:
6060:
3257:, i.e. one with the same number of rows and columns: the determinant can be defined via the
27868:
27833:
27820:
27711:
27519:
27455:
27389:
27294:
27158:
26999:
26910:
25728:
25599:
25260:
25147:
24340:
23996:
23992:
23615:
23602:
23419:
22650:
22218:
22168:
21406:
19849:
19564:
19218:
18968:
18860:
18586:
14578:
12462:
11030:
9335:
7683:
7637:
6251:
5983:
5425:
4800:
3695:
3503:
written beside it as in the illustration. This scheme for calculating the determinant of a
554:
479:
414:
39:
26919:"A condensation-based application of Cramer's rule for solving large-scale linear systems"
25886:
24920:
22248:
20890:
20682:
19027:
13726:
11507:{\displaystyle \det \left(I_{\mathit {m}}+AB\right)=\det \left(I_{\mathit {n}}+BA\right),}
8:
28046:
27926:
27901:
27751:
27619:
27594:
25508:
24344:
24167:
24116:
23985:
23648:
21932:
20411:
19977:
19745:
18920:
18882:(1811, 1812), who formally stated the theorem relating to the product of two matrices of
18853:(1773) treated determinants of the second and third order and applied it to questions of
18280:
17613:
17359:
12009:
11420:
have dimensions allowing them to be multiplied in either order forming a square matrix):
11209:
11183:
8491:
7485:
3800:
3774:
2682:
25732:
24742:
24719:
24312:
24237:
21617:, the determinant can be used to measure the rate of expansion of a map near the poles.
14017:
The determinant is closely related to two other central concepts in linear algebra, the
13903:{\displaystyle \det(aI+b\mathbf {i} )=a^{2}\det(I)+b^{2}\det(\mathbf {i} )=a^{2}+b^{2}.}
11976:
11953:
377:
27756:
27508:
27397:
27345:
26898:
26880:
26848:
26735:
26689:
26465:
25744:
25612:
25581:
25563:
25536:
25528:
25393:
25304:
25289:
25280:
25196:
24900:
24765:
24679:
24659:
24639:
24619:
24599:
24579:
24559:
24554:
24495:
24260:
23677:
23653:
23476:
23399:
23211:
23183:
22916:
22896:
22843:
22823:
22803:
22776:
22756:
22574:
22556:
22536:
22403:
22383:
22242:
22224:
22200:
22104:
22051:
22045:
22027:
21892:
21295:
20866:
20846:
20750:
20620:
20496:. In the case of an orthogonal basis, the magnitude of the determinant is equal to the
20451:
20391:
19774:
19285:
19265:
19245:
19056:
18854:
17923:
17784:
17764:
15164:
14957:
14937:
14917:
14560:
14536:
14516:
14492:
14397:
14377:
14357:
14337:
14317:
14297:
14288:
14078:
14028:
13667:
13647:
13078:
13029:
12614:{\displaystyle \det(A+B)=\det(A)+\det(B)+{\text{tr}}(A){\text{tr}}(B)-{\text{tr}}(AB).}
12089:
12069:
12049:
11933:
11913:
11366:{\displaystyle \det {\begin{pmatrix}A&B\\B&A\end{pmatrix}}=\det(A-B)\det(A+B).}
11255:
11235:
11007:
10987:
10528:
10479:
9967:
9580:
8897:
8757:
8344:
8262:
8242:
8222:
8112:
7845:
7773:
7619:
7599:
7467:
7444:
7401:
7381:
7292:
7272:
7156:
6386:
6258:
6000:
5514:
4934:
4916:
4877:
4702:
4333:
2686:
2361:
1316:
410:
98:
27110:
26792:
26773:
21800:{\displaystyle \det(A)=\det(X)^{-1}\det(B)\det(X)=\det(B)\det(X)^{-1}\det(X)=\det(B).}
21154:
11177:
blocks, again under appropriate commutativity conditions among the individual blocks.
5711:
This formula can be applied iteratively when several columns are swapped. For example
27954:
27911:
27838:
27731:
27575:
27535:
27523:
27512:
27471:
27441:
27375:
27357:
27320:
27280:
27235:
27217:
27199:
27180:
27135:
27100:
27080:
27066:
27053:
27035:
27017:
26985:
26959:
26826:
26816:
26759:
26651:
26539:
26528:
Proceedings of the 1997 international symposium on Symbolic and algebraic computation
26419:
25937:
http://www-history.mcs.st-and.ac.uk/history/HistTopics/Matrices_and_determinants.html
25748:
25585:
25274:
25190:
24356:
22530:
22257:
22097:
21631:
21025:
20509:
20501:
20435:
19675:{\displaystyle A\,\operatorname {adj} (A)=\operatorname {adj} (A)\,A=\det(A)\,I_{n}.}
18864:
18863:(1801) made the next advance. Like Lagrange, he made much use of determinants in the
18820:
9985:
8106:
7489:
6018:
5971:
Yet more generally, any permutation of the columns multiplies the determinant by the
1694:
this already is the signed area, yet it may be expressed more conveniently using the
1508:
562:
487:
467:
406:
105:
94:
26739:
25540:
25263:. Unfortunately this interesting method does not always work in its original form.
13230:{\displaystyle \det(A)+\det(B)+A_{11}B_{22}+B_{11}A_{22}-A_{12}B_{21}-B_{12}A_{21}.}
1960:. When the determinant is equal to one, the linear mapping defined by the matrix is
27959:
27863:
27716:
27500:
27425:
27349:
27270:
27007:
26977:
26933:
26890:
26858:
26797:
26787:
26727:
26681:
26601:
26531:
25882:
25736:
25645:
25573:
25520:
24404:
24400:
24348:
24155:
23606:
22861:
22798:
21887:
20673:
19874:
19729:
19725:
19103:
18809:
17536:
15188:
14650:
12293:
12289:
9827:
8098:
whose entries are the determinants of all quadratic submatrices of a given matrix.
5276:
2996:
1728:
becomes the signed area in question, which can be determined by the pattern of the
502:
418:
27643:
27629:
Compute determinants of matrices up to order 6 using Laplace expansion you choose.
28018:
27811:
27771:
27761:
27626:
27578:
27464:
Companion Encyclopedia of the History and Philosophy of the Mathematical Sciences
27451:
27385:
27290:
27155:
26995:
26906:
26863:
26708:
23960:
23610:
22646:
22578:
22024:
is invertible (in the sense that there is an inverse matrix whose entries are in
21218:
20513:
18845:
gave the general method of expanding a determinant in terms of its complementary
17532:
16840:
expansion of the logarithm when the expansion converges. If every eigenvalue of
16837:
16472:
9366:
8095:
7945:
7762:
6218:
4905:
4895:
3689:
1625:
1621:
951:
439:
11141:{\displaystyle \det {\begin{pmatrix}A&B\\C&D\end{pmatrix}}=\det(AD-BC).}
28023:
27944:
27679:
27147:
25143:
24307:
24212:
23737:
20564:
20560:
20493:
19000:
18956:
18872:
18846:
18274:
17609:
17528:
16618:{\displaystyle \det(A)={\frac {(-1)^{n}}{n!}}B_{n}(s_{1},s_{2},\ldots ,s_{n}).}
14072:
8133:
7988:
7310:
5408:
3956:
3499:
3492:
3253:
There are various equivalent ways to define the determinant of a square matrix
1949:
1729:
1525:
530:
90:
27353:
26981:
26938:
26606:
25577:
25342:
24762:
for an odd number of permutations). Once such a LU decomposition is known for
21970:
still holds, as do all the properties that result from that characterization.
12809:{\displaystyle (A_{11}+B_{11})(A_{22}+B_{22})-(A_{12}+B_{12})(A_{21}+B_{21}).}
9680:-matrix above continues to hold, under appropriate further assumptions, for a
7772:
Because the determinant respects multiplication and inverses, it is in fact a
28094:
28056:
27979:
27939:
27906:
27886:
27504:
27269:, Lecture Notes in Comput. Sci., vol. 2122, Springer, pp. 119–135,
26830:
26639:
25458:
25250:
24175:
23498:
23494:
21148:
18933:
18816:
18616:
17524:
14007:
11761:{\displaystyle \det(X+AB)=\det(X)\det \left(I_{\mathit {n}}+BX^{-1}A\right),}
11024:
4982:
4795:
4442:
tuple of integers. With the Levi-Civita symbol, the Leibniz formula becomes
2702:
1961:
1944:
1537:
538:
47:
27275:
26667:"Sylvester's Identity and Multistep Integer-Preserving Gaussian Elimination"
25650:
19791:
are linearly dependent. For example, given two linearly independent vectors
19205:{\displaystyle x_{i}={\frac {\det(A_{i})}{\det(A)}}\qquad i=1,2,3,\ldots ,n}
9569:{\displaystyle (\det A)I=A\operatorname {adj} A=(\operatorname {adj} A)\,A.}
7742:{\displaystyle \operatorname {SL} _{n}(K)\subset \operatorname {GL} _{n}(K)}
1671:
representing the parallelogram's sides. The signed area can be expressed as
545:-dimensional volume are transformed under the endomorphism. This is used in
459:
Adding a multiple of one row to another row does not change the determinant.
27989:
27878:
27828:
27721:
24306:-matrix. Thus, the number of required operations grows very quickly: it is
23968:
21856:
18952:
18868:
17194:
14099:
9681:
7750:
2454:{\displaystyle \mathbf {a} _{1},\mathbf {a} _{2},\ldots ,\mathbf {a} _{n},}
534:
471:
26731:
26712:
26535:
25972:
8094:
matrices. This formula can also be recast as a multiplicative formula for
27969:
27891:
27736:
24351:. One can restrict the computation to elementary matrices of determinant
24208:
23790:, but also includes several further cases including the determinant of a
22794:
21614:
21418:
21017:
19877:
18838:
18832:
18402:
16693:. The product and trace of such matrices are defined in a natural way as
12450:{\displaystyle {\sqrt{\det(A+B)}}\geq {\sqrt{\det(A)}}+{\sqrt{\det(B)}},}
9988:, then it follows with results from the section on multiplicativity that
8752:-matrices gives back the Leibniz formula mentioned above. Similarly, the
6210:{\displaystyle \det(A)=a_{11}a_{22}\cdots a_{nn}=\prod _{i=1}^{n}a_{ii}.}
4196:
3688:
exhausting the entire set. The set of all such permutations, called the
3549:
2052:
1617:
1616:
The absolute value of the determinant together with the sign becomes the
1541:
494:
31:
20:
19:
This article is about mathematics. For determinants in epidemiology, see
27429:
19302:. This follows immediately by column expansion of the determinant, i.e.
18999:; compound determinants by Sylvester, Reiss, and Picquet; Jacobians and
18940:
introduced the modern notation for the determinant using vertical bars.
5280:: whenever two columns of a matrix are identical, its determinant is 0:
456:
Multiplying a row by a number multiplies the determinant by this number.
370:
matrix can be defined in several equivalent ways, the most common being
27998:
27741:
27164:
26902:
26693:
25740:
25532:
25425:
24171:
23910:
23791:
22096:, this means that the determinant is +1 or −1. Such a matrix is called
20447:
18948:
18849:: Vandermonde had already given a special case. Immediately following,
17548:
14994:
and also equals the sum of the eigenvalues. Thus, for complex matrices
14661:
asserts that this is equivalent to the determinants of the submatrices
14554:
14018:
7749:. More generally, the word "special" indicates the subgroup of another
1521:
1079:{\displaystyle {\begin{vmatrix}a&b\\a&b\end{vmatrix}}=ab-ba=0.}
510:
82:
26774:"Triangular Factorization and Inversion by Fast Matrix Multiplication"
26530:. ISSAC '97. Kihei, Maui, Hawaii, United States: ACM. pp. 28–31.
20492:
of the basis is consistent with or opposite to the orientation of the
11996:
have the same characteristic polynomials (hence the same eigenvalues).
27796:
27583:
26885:
26802:
24277:
24231:, however, does frequently use calculations related to determinants.
24134:, so there is no good definition of the determinant in this setting.
23671:
22241:. Since it respects the multiplication in both groups, this map is a
21021:
20884:
20443:
20005:
18984:
18960:
18406:
12202:{\displaystyle \det(A+B+C)+\det(C)\geq \det(A+C)+\det(B+C){\text{,}}}
9579:
Thus the adjugate matrix can be used for expressing the inverse of a
8129:
8049:{\displaystyle \operatorname {GL} _{n}(K)/\operatorname {SL} _{n}(K)}
7150:
3601:
2379:
1641:
is the signed area, one may consider a matrix containing two vectors
398:
185:{\displaystyle {\begin{vmatrix}a&b\\c&d\end{vmatrix}}=ad-bc,}
26894:
26685:
26589:
25524:
14553:. By means of this polynomial, determinants can be used to find the
11151:
This formula has been generalized to matrices composed of more than
27964:
27407:
Cayley, Arthur (1841), "On a theorem in the geometry of position",
26470:
25924:
25568:
23964:
23569:
of a matrix is defined as the determinant, except that the factors
22526:
holds. In other words, the displayed commutative diagram commutes.
20488:. In that case, the sign of the determinant determines whether the
20469:
18976:
18944:
17758:
7679:
7585:{\displaystyle \det \left(A^{-1}\right)={\frac {1}{\det(A)}}=^{-1}}
6109:, then its determinant equals the product of the diagonal entries:
3522:
Generalizing the above to higher dimensions, the determinant of an
2657:. (The sign shows whether the transformation preserves or reverses
1969:
546:
514:
486:
of a linear endomorphism, which does not depend on the choice of a
26853:
25634:"Inequalities of Generalized Matrix Functions via Tensor Products"
25617:
22700:{\displaystyle \det :\operatorname {GL} _{n}\to \mathbb {G} _{m}.}
22157:{\displaystyle \operatorname {GL} _{n}(R)\rightarrow R^{\times },}
16628:
This formula can also be used to find the determinant of a matrix
16471:
The formula can be expressed in terms of the complete exponential
7378:
This key fact can be proven by observing that, for a fixed matrix
27648:
27125:, Revised and enlarged by William H. Metzler, New York, NY: Dover
25879:
The Theory of Determinants in the historical Order of Development
24359:, its determinant is the product of the entries of its diagonal.
22840:
can be formulated in a coordinate-free manner by considering the
20563:
of the determinant of real vectors is equal to the volume of the
20468:
The determinant can be thought of as assigning a number to every
18914:
used the functional determinant which Sylvester later called the
14294:
From this, one immediately sees that the determinant of a matrix
9491:{\displaystyle (\operatorname {adj} (A))_{i,j}=(-1)^{i+j}M_{ji}.}
2665:-dimensional, which indicates that the dimension of the image of
24:
25095:. This exponent has been further lowered, as of 2016, to 2.373.
24596:
in each column, and otherwise zeros), a lower triangular matrix
9639:{\displaystyle A^{-1}={\frac {1}{\det A}}\operatorname {adj} A.}
8885:{\displaystyle \det(A)=\sum _{i=1}^{n}(-1)^{i+j}a_{i,j}M_{i,j}.}
8470:{\displaystyle \det(A)=\sum _{j=1}^{n}(-1)^{i+j}a_{i,j}M_{i,j},}
6241:
which gives a non-zero contribution is the identity permutation.
1089:
This holds similarly if the two columns are the same. Moreover,
1006:
is 1. Second, the determinant is zero if two rows are the same:
81:. Its value characterizes some properties of the matrix and the
27974:
25014:, then there is an algorithm computing the determinant in time
24107:, one may define a positive real-valued determinant called the
24081:{\displaystyle \det(I+A)=\exp(\operatorname {tr} (\log(I+A))).}
23642:
21274:{\displaystyle f:\mathbf {R} ^{n}\rightarrow \mathbf {R} ^{n},}
20676:
20438:, this implies the given functions are linearly dependent. See
18901:
18841:(1771) first recognized determinants as independent functions.
15893:. Such expressions are deducible from combinatorial arguments,
5978:
If some column can be expressed as a linear combination of the
3488:
1695:
1620:
of the parallelogram. The signed area is the same as the usual
533:
is expressed by a determinant, and the determinant of a linear
522:
405:) signed products of matrix entries. It can be computed by the
27524:"Recherches sur le calcul intégral et sur le systéme du monde"
26521:"On the worst-case complexity of integer Gaussian elimination"
24215:
form the class closest to matrices with commutative elements.
23180:
is given by multiplying with some scalar, i.e., an element in
14153:{\displaystyle \lambda _{1},\lambda _{2},\ldots ,\lambda _{n}}
1973:
is related to these ideas. In 2D, it can be interpreted as an
999:{\displaystyle {\begin{pmatrix}1&0\\0&1\end{pmatrix}}}
623:{\displaystyle {\begin{pmatrix}a&b\\c&d\end{pmatrix}}}
482:
does not depend on the chosen basis. This allows defining the
27216:, Undergraduate Texts in Mathematics (3 ed.), Springer,
27198:, Undergraduate Texts in Mathematics (2 ed.), Springer,
19004:
18576:{\displaystyle A={\begin{bmatrix}a&b&c\end{bmatrix}}}
13483:{\displaystyle {\text{tr}}(A){\text{tr}}(B)-{\text{tr}}(AB).}
4393:
25091:
algorithm for computing the determinant exists based on the
24122:
21201:{\displaystyle f\colon \mathbf {R} ^{2}\to \mathbf {R} ^{2}}
21147:, or any other combination of pairs of vertices that form a
19880:
by the former two vectors exactly if the determinant of the
14012:
24111:
using the canonical trace. In fact, corresponding to every
18827:
stated, without proof, Cramer's rule. Both Cramer and also
18815:
Determinants proper originated separately from the work of
17354:. This relationship can be derived via the formula for the
15149:{\displaystyle \operatorname {tr} (A)=\log(\det(\exp(A))).}
14171:
times in this list.) Then, it turns out the determinant of
8132:
in terms of determinants of smaller matrices, known as its
1698:
of the complementary angle to a perpendicular vector, e.g.
1691:
634:" or by vertical bars around the matrix, and is defined as
89:, by the matrix. In particular, the determinant is nonzero
25259:
fame) invented a method for computing determinants called
24091:
Another infinite-dimensional notion of determinant is the
23565:
Determinants as treated above admit several variants: the
22420:. The determinant respects these maps, i.e., the identity
21659:. Indeed, repeatedly applying the above identities yields
15065:{\displaystyle \det(\exp(A))=\exp(\operatorname {tr} (A))}
14287:
The product of all non-zero eigenvalues is referred to as
14008:
Properties of the determinant in relation to other notions
11639:{\displaystyle \det \left(I_{\mathit {m}}+cr\right)=1+rc.}
9400:
is the transpose of the matrix of the cofactors, that is,
8488:. For example, the Laplace expansion along the first row (
1536:. In either case, the images of the basis vectors form a
27234:. Graduate Texts in Mathematics. New York, NY: Springer.
27099:, Society for Industrial and Applied Mathematics (SIAM),
24843:{\displaystyle \det(A)=\varepsilon \det(L)\cdot \det(U).}
21881:
12628:
This can be shown by writing out each term in components
7228:{\displaystyle \det \left(A^{\textsf {T}}\right)=\det(A)}
5361:{\displaystyle |a_{1},\dots ,v,\dots ,v,\dots ,a_{n}|=0.}
501:, and determinants can be used to solve these equations (
27635:
Calculator for matrix determinants, up to the 8th order.
27150:(1947) "Some identities in the theory of determinants",
25826:
A Brief History of Linear Algebra and Matrix Theory at:
25600:
http://www.ee.ic.ac.uk/hp/staff/dmb/matrix/proof003.html
23898:{\displaystyle \det(a+ib+jc+kd)=a^{2}+b^{2}+c^{2}+d^{2}}
22017:{\displaystyle A\in \operatorname {Mat} _{n\times n}(R)}
20617:
is the linear map given by multiplication with a matrix
19020:
Determinants can be used to describe the solutions of a
3681:{\displaystyle \sigma (1),\sigma (2),\ldots ,\sigma (n)}
478:, the determinant of the matrix that represents it on a
27422:
Introduction à l'analyse des lignes courbes algébriques
25718:
23443:. For this reason, the highest non-zero exterior power
21140:{\displaystyle {\frac {1}{6}}\cdot |\det(a-b,b-c,c-d)|}
20500:
of the lengths of the basis vectors. For instance, an
12272:{\displaystyle \det(A+B)\geq \det(A)+\det(B){\text{.}}}
449:
The exchange of two rows multiplies the determinant by
27639:
Matrices and Linear Algebra on the Earliest Uses Pages
26345:
26300:
26258:
26212:
26170:
23560:
20810:{\displaystyle f:\mathbf {R} ^{n}\to \mathbf {R} ^{m}}
20610:{\displaystyle f:\mathbf {R} ^{n}\to \mathbf {R} ^{n}}
20073:
19432:
19342:
18857:; he proved many special cases of general identities.
18549:
18465:. The above formula shows that its Lie algebra is the
16136:
14990:) is by definition the sum of the diagonal entries of
14689:
14075:
entries. Then, by the Fundamental Theorem of Algebra,
13600:
13558:
11558:
From this general result several consequences follow.
11290:
11077:
10824:
10668:
10625:
10567:
10267:
10111:
10068:
10010:
9917:
9848:
9302:
8911:
8682:
8640:
8598:
8529:
6714:
6619:
6524:
6427:
6296:
4664:
3989:
3843:
3303:
3025:
2734:
2292:
2223:
2161:
1889:
1857:
1456:
1348:
1256:
1217:
1104:
1021:
965:
816:
774:
691:
652:
589:
214:
130:
26836:
26572:
26495:
26451:
26141:
25829:"A Brief History of Linear Algebra and Matrix Theory"
25457:, §III.8, Proposition 1 proves this result using the
25199:
25156:
25105:
25058:
25020:
24994:
24952:
24923:
24903:
24864:
24791:
24768:
24745:
24722:
24702:
24682:
24662:
24642:
24622:
24602:
24582:
24562:
24521:
24498:
24460:
24421:
24376:
24315:
24286:
24263:
24240:
24184:
24008:
23918:
23803:
23783:{\displaystyle A=\operatorname {Mat} _{n\times n}(F)}
23745:
23703:
23680:
23656:
23618:
23575:
23537:
23507:
23479:
23449:
23422:
23402:
23270:
23234:
23214:
23186:
23156:
23126:
22942:
22919:
22899:
22869:
22846:
22826:
22806:
22779:
22759:
22727:
22662:
22618:
22591:
22559:
22539:
22516:{\displaystyle f(\det((a_{i,j})))=\det((f(a_{i,j})))}
22429:
22406:
22386:
22297:
22265:
22227:
22203:
22177:
22115:
22074:
22054:
22030:
21979:
21941:
21915:
21895:
21826:
21668:
21463:
21306:
21234:
21164:
21071:
21033:
21020:
bounded by four points, they can be used to identify
20925:
20893:
20869:
20849:
20823:
20773:
20753:
20714:
20685:
20643:
20623:
20573:
20414:
20394:
20017:
19980:
19916:
19886:
19852:
19797:
19777:
19754:
19691:
19601:
19567:
19311:
19288:
19268:
19248:
19221:
19115:
19079:
19059:
19030:
18628:
18589:
18537:
18511:
18474:
18442:
18415:
18312:
18283:
18088:
17949:
17926:
17810:
17787:
17767:
17731:
17625:
17592:
17557:
17374:
17250:
17036:
16853:
16702:
16505:
16414:
16115:
15932:
15345:
15305:{\displaystyle \det(A)=\exp(\operatorname {tr} (L)).}
15256:
15208:
15088:
15007:
14960:
14940:
14920:
14670:
14610:
14587:
14563:
14539:
14519:
14495:
14426:
14400:
14380:
14360:
14340:
14320:
14300:
14188:
14107:
14081:
14051:
14031:
13958:
13919:
13792:
13758:
13729:
13690:
13670:
13650:
13529:
13503:
13428:
13249:
13104:
13081:
13052:
13032:
12828:
12683:
12634:
12508:
12479:
12362:
12336:
12302:
12215:
12112:
12092:
12072:
12052:
12012:
11979:
11956:
11936:
11916:
11778:
11671:
11583:
11429:
11281:
11258:
11238:
11212:
11186:
11157:
11068:
11033:
11010:
10990:
10554:
10531:
10502:
10482:
10422:
9997:
9970:
9839:
9806:
9780:
9754:
9728:
9690:
9660:
9592:
9510:
9409:
9374:
9300:
9282:-term Laplace expansion along a row or column can be
8905:
8785:
8760:
8732:
8523:
8494:
8370:
8347:
8285:
8265:
8245:
8225:
8175:
8142:
8115:
8062:
7996:
7954:
7911:
7868:
7862:. This homomorphism is surjective and its kernel is
7848:
7821:
7782:
7753:
of matrices of determinant one. Examples include the
7691:
7645:
7622:
7602:
7502:
7470:
7447:
7424:
7404:
7384:
7322:
7295:
7275:
7260:
7186:
7159:
7057:
7005:
6953:
6904:
6856:
6818:
6702:
6607:
6512:
6415:
6389:
6284:
6261:
6227:
6115:
6089:
6063:
6027:
6003:
5717:
5540:
5517:
5491:
5434:
5380:
5288:
5008:
4942:
4919:
4880:
4843:
4803:
4784:{\displaystyle A={\big (}a_{1},\dots ,a_{n}{\big )},}
4728:
4705:
4675:
4622:
4451:
4402:
4341:
4208:
3968:
3831:
3803:
3777:
3757:
3725:
3698:
3629:
3609:
3562:
3528:
3297:
3019:
2968:
2722:
2599:
2467:
2399:
2364:
2152:
2065:
1751:
1342:
1319:
1098:
1015:
959:
930:
765:
643:
583:
380:
208:
124:
27573:
25355:
from the original on 2021-12-11 – via YouTube.
25270:
24098:
23975:, also arise as special cases of this construction.
21217:, much of the above carries over by considering the
16395:
where the sum is taken over the set of all integers
16106:
In the general case, this may also be obtained from
1313:
Finally, if any column is multiplied by some number
22103:The determinant being multiplicative, it defines a
19102:is nonzero. In this case, the solution is given by
19003:by Sylvester; and symmetric gauche determinants by
7901:(the matrices with determinant one). Hence, by the
7135:{\displaystyle |A|=-|E|=-(18\cdot 3\cdot (-1))=54.}
3507:matrix does not carry over into higher dimensions.
1516:If the matrix entries are real numbers, the matrix
26394:
25631:
25392:
25205:
25181:
25130:
25083:
25044:
25006:
24980:
24938:
24909:
24889:
24842:
24774:
24754:
24731:
24708:
24688:
24668:
24648:
24636:. The determinants of the two triangular matrices
24628:
24608:
24588:
24568:
24542:
24504:
24481:
24446:
24391:
24324:
24298:
24269:
24249:
24199:
24119:there is a notion of Fuglede−Kadison determinant.
24080:
23951:
23897:
23782:
23724:
23686:
23662:
23631:
23593:
23549:
23523:
23485:
23465:
23435:
23408:
23385:
23253:
23220:
23192:
23172:
23142:
23109:
22925:
22905:
22885:
22852:
22832:
22812:
22785:
22765:
22745:
22699:
22637:
22604:
22565:
22545:
22515:
22412:
22392:
22372:
22283:
22233:
22209:
22189:
22156:
22088:
22060:
22036:
22016:
21962:
21923:
21901:
21844:
21799:
21642:are similar, if there exists an invertible matrix
21595:
21394:
21273:
21200:
21139:
21057:
21005:
20908:
20875:
20855:
20835:
20809:
20759:
20739:
20700:
20664:
20629:
20609:
20554:
20426:
20400:
20377:
19992:
19966:
19898:
19865:
19838:
19783:
19763:
19716:
19674:
19580:
19550:
19294:
19274:
19254:
19234:
19204:
19094:
19065:
19045:
18782:
18607:
18575:
18523:
18494:
18457:
18428:
18390:
18295:
18261:
18068:
17932:
17909:
17793:
17773:
17749:
17714:
17600:
17578:
17512:
17333:
17182:
17010:
16821:
16617:
16460:
16384:
16095:
15878:
15304:
15232:
15148:
15064:
14966:
14946:
14926:
14903:
14638:
14593:
14569:
14545:
14525:
14501:
14478:
14406:
14386:
14366:
14346:
14326:
14306:
14276:
14152:
14087:
14063:
14037:
13996:
13944:
13902:
13775:
13744:
13715:
13676:
13656:
13636:
13515:
13482:
13408:
13229:
13087:
13067:
13038:
13015:
12808:
12666:
12613:
12491:
12449:
12348:
12314:
12271:
12201:
12098:
12078:
12058:
12024:
11988:
11965:
11942:
11922:
11899:
11760:
11638:
11506:
11376:
11365:
11264:
11244:
11224:
11198:
11169:
11140:
11051:
11016:
10996:
10962:
10537:
10514:
10488:
10468:
10405:
9976:
9953:
9818:
9792:
9766:
9740:
9714:
9672:
9638:
9568:
9490:
9392:
9327:
9270:
8884:
8766:
8744:
8715:
8506:
8469:
8353:
8329:
8271:
8251:
8231:
8211:
8161:
8121:
8075:
8048:
7979:
7936:
7893:
7854:
7834:
7807:
7741:
7670:
7628:
7608:
7584:
7476:
7453:
7433:
7410:
7390:
7367:
7301:
7281:
7227:
7165:
7134:
7037:
6988:
6936:
6888:
6834:
6780:
6685:
6590:
6496:
6395:
6371:
6267:
6233:
6209:
6101:
6075:
6049:
6009:
5963:
5703:
5523:
5503:
5477:
5392:
5360:
5260:
4973:
4925:
4886:
4866:
4828:) is composed of the entries of the matrix in the
4816:
4783:
4711:
4687:
4649:
4601:
4426:
4381:{\displaystyle \varepsilon _{i_{1},\ldots ,i_{n}}}
4380:
4320:
4184:
3955:the Leibniz formula for its determinant is, using
3944:
3812:
3789:
3763:
3743:
3711:
3680:
3615:
3592:
3540:
3441:
3242:
2987:
2951:
2681:produces a linear transformation which is neither
2637:
2576:
2453:
2370:
2347:
2135:
1933:
1492:
1325:
1302:
1078:
998:
942:
908:
745:
622:
434:matrices that has the four following properties:
389:
349:
184:
27592:
26415:Supersymmetry for mathematicians: An introduction
25432:, 2nd Edition, Addison-Wesley, 1971, pp 173, 191.
21625:
14160:. (Here it is understood that an eigenvalue with
4569:
1977:formed by imagining two vectors each with origin
28092:
27461:
27314:
26956:Mathematics for Physical Science and Engineering
26632:Elementary Linear Algebra (Applications Version)
26439:
25773:
25347:(video lecture). WildLinAlg. Sydney, Australia:
24825:
24810:
24792:
24377:
24150:as value on some pair of arguments implies that
24009:
23999:by an appropriate generalization of the formula
23804:
23704:
23493:and similarly for more involved objects such as
23336:
22663:
22473:
22436:
21942:
21782:
21767:
21745:
21733:
21718:
21706:
21684:
21669:
21547:
21090:
20955:
20720:
19755:
19646:
19536:
19424:
19334:
19312:
19153:
19132:
19080:
18741:
18693:
18645:
18502:consisting of those matrices having trace zero.
18443:
18313:
18188:
18113:
18089:
18020:
17956:
17843:
17817:
17632:
17414:
17295:
16854:
16506:
16116:
15611:
15455:
15350:
15257:
15116:
15008:
14581:of this polynomial, i.e., those complex numbers
14449:
14189:
13974:
13959:
13857:
13832:
13793:
13120:
13105:
13053:
12545:
12530:
12509:
12468:
12424:
12398:
12366:
12252:
12237:
12216:
12176:
12155:
12140:
12113:
11856:
11815:
11803:
11779:
11708:
11696:
11672:
11584:
11467:
11430:
11342:
11324:
11282:
11111:
11069:
10919:
10907:
10816:
10804:
10773:
10743:
10660:
10617:
10605:
10559:
10423:
10362:
10350:
10259:
10247:
10216:
10186:
10103:
10060:
10048:
10002:
9909:
9894:
9882:
9840:
9615:
9514:
8786:
8371:
8090:is a generalization of that product formula for
7557:
7536:
7503:
7425:
7353:
7341:
7323:
7213:
7187:
6116:
5463:
5435:
5000:is expressible as a similar linear combination:
4844:
4659:
4452:
4209:
3969:
2600:
766:
644:
27493:Journal für die reine und angewandte Mathematik
26972:Kleiner, Israel (2007), Kleiner, Israel (ed.),
26033:Linear Algebra and Its Applications 6th Edition
25228:allows rapid calculation of the determinant of
23995:defines the determinant for operators known as
21874:and therefore only depends on the endomorphism
21409:, appears in the higher-dimensional version of
20567:spanned by those vectors. As a consequence, if
19839:{\displaystyle v_{1},v_{2}\in \mathbf {R} ^{3}}
5982:columns (i.e. the columns of the matrix form a
5420:These rules have several further consequences:
3469:negative for an odd number. For the example of
27047:
26099:
24178:(i.e., matrices whose entries are elements of
14479:{\displaystyle \chi _{A}(t)=\det(t\cdot I-A).}
9684:, i.e., a matrix composed of four submatrices
1609:. (The parallelogram formed by the columns of
374:, which expresses the determinant as a sum of
27664:
27438:An introduction to the history of mathematics
27424:, Genève: Frères Cramer & Cl. Philibert,
26771:
26484:
25084:{\displaystyle \operatorname {O} (n^{2.376})}
23601:occurring in Leibniz's rule are omitted. The
22573:(the latter determinant being computed using
21620:
21209:parallelogram to that of the original square.
20442:. Another such use of the determinant is the
9318:
9305:
5035:
4773:
4737:
27615:Determinant Interactive Program and Tutorial
27540:: CS1 maint: multiple names: authors list (
26811:Dummit, David S.; Foote, Richard M. (2004),
23643:Determinants for finite-dimensional algebras
23594:{\displaystyle \operatorname {sgn}(\sigma )}
22252:The determinant is a natural transformation.
19592:. The rule is also implied by the identity
18831:were led to determinants by the question of
17027:is the identity matrix. More generally, if
14417:The characteristic polynomial is defined as
13997:{\displaystyle \det(I)=\det(\mathbf {i} )=1}
10984:size further formulas hold. For example, if
4641:
4623:
4421:
4403:
4199:for the product, this can be shortened into
3744:{\displaystyle \operatorname {sgn}(\sigma )}
3587:
3563:
2638:{\displaystyle \det(A)=\pm {\text{vol}}(P),}
16:In mathematics, invariant of square matrices
27440:(6 ed.), Saunders College Publishing,
27163:
27005:
26916:
26810:
26411:
26057:
26045:
25983:
25801:
25799:
25797:
25761:
25705:
25693:
25553:
25494:
25482:
25470:
24716:of the corresponding permutation (which is
23228:satisfying the following identity (for all
22721:The determinant of a linear transformation
21810:The determinant is therefore also called a
21024:. The volume of any tetrahedron, given its
18928:. About the time of Jacobi's last memoirs,
14533:is the identity matrix of the same size as
13945:{\displaystyle {\text{tr}}(\mathbf {i} )=0}
13716:{\displaystyle {\text{tr}}(\mathbf {i} )=0}
9338:, each the product of the determinant of a
6802:add 3 times the third column to the second
2593:-dimensional volume of this parallelotope,
1532:, and one that maps them to the columns of
27671:
27657:
27555:
27528:Histoire de l'Académie Royale des Sciences
27399:Théorie générale des equations algébriques
27096:Matrix Analysis and Applied Linear Algebra
25914:, Cahier 17, Tome 10, pages 29–112 (1815).
25717:A proof can be found in the Appendix B of
25340:
24739:for an even number of permutations and is
24454:, which is a significant improvement over
9349:and the determinant of the complementary (
7980:{\displaystyle \operatorname {GL} _{n}(K)}
7937:{\displaystyle \operatorname {SL} _{n}(K)}
7894:{\displaystyle \operatorname {SL} _{n}(K)}
7808:{\displaystyle \operatorname {GL} _{n}(K)}
7671:{\displaystyle \operatorname {GL} _{n}(K)}
7313:equals the product of their determinants:
5989:Adding a scalar multiple of one column to
2649:-dimensional volume scaling factor of the
2059:is written in terms of its column vectors
27315:Trefethen, Lloyd; Bau III, David (1997),
27274:
27050:Commutative Algebra: Constructive Methods
26937:
26884:
26870:
26862:
26852:
26801:
26791:
26605:
26518:
26469:
26115:
26075:Natural transformation § Determinant
25649:
25616:
25567:
25506:
25390:
25182:{\displaystyle \operatorname {O} (n^{3})}
25150:(or LU decomposition) method is of order
25131:{\displaystyle \operatorname {O} (n^{4})}
24890:{\displaystyle \operatorname {O} (n^{3})}
24782:, its determinant is readily computed as
24447:{\displaystyle \operatorname {O} (n^{3})}
24343:consists of left multiplying a matrix by
24187:
24123:Related notions for non-commutative rings
22684:
21581:
21497:
20969:
20665:{\displaystyle S\subset \mathbf {R} ^{n}}
20457:
19744:Determinants can be used to characterize
19717:{\displaystyle \operatorname {O} (n^{3})}
19658:
19639:
19605:
18805:The Nine Chapters on the Mathematical Art
17230:
17215:are zero and the remaining polynomial is
17176:
17004:
14013:Eigenvalues and characteristic polynomial
11890:
11874:
10980:If the blocks are square matrices of the
10769:
10765:
10742:
10212:
10208:
10185:
9559:
7200:
6383:Computation of the determinant of matrix
5415:
1840:
1836:
1822:
1795:
1791:
1775:
27339:
27333:
27123:A treatise on the theory of determinants
27072:Categories for the Working Mathematician
27065:
27048:Lombardi, Henri; Quitté, Claude (2015),
26753:
26463:
26103:
26087:
26069:
25865:
25794:
25454:
24492:For example, LU decomposition expresses
24365:
23201:
22247:
21634:have the same determinant: two matrices
21153:
19967:{\displaystyle f_{1}(x),\dots ,f_{n}(x)}
18904:also presented one on the subject. (See
18409:. For example, the special linear group
17612:. Its derivative can be expressed using
17579:{\displaystyle \mathbf {R} ^{n\times n}}
16461:{\displaystyle \sum _{l=1}^{n}lk_{l}=n.}
8169:is defined to be the determinant of the
3487:
1943:
1592:, as shown in the accompanying diagram.
1507:
509:of a square matrix, whose roots are the
413:of determinants of submatrices, or with
108:is the product of its diagonal entries.
27604:MacTutor History of Mathematics Archive
27518:
26971:
26707:
26664:
26576:
25973:http://jeff560.tripod.com/matrices.html
25897:
25853:
24335:
22649:, from the general linear group to the
22605:{\displaystyle \operatorname {GL} _{n}}
22044:) if and only if its determinant is an
20551:switches the orientation of the basis.
18983:, by Cayley; continuants by Sylvester;
18842:
18835:passing through a given set of points.
18429:{\displaystyle \operatorname {SL} _{n}}
17750:{\displaystyle \operatorname {adj} (A)}
11232:, the following formula holds (even if
10973:Both results can be combined to derive
9393:{\displaystyle \operatorname {adj} (A)}
7368:{\displaystyle \det(AB)=\det(A)\det(B)}
2645:and hence describes more generally the
1828:
1808:
1782:
1766:
409:, which expresses the determinant as a
97:and the corresponding linear map is an
28093:
28062:Comparison of linear algebra libraries
27483:
27419:
27406:
27395:
27369:
27342:Elements of the history of mathematics
26953:
26772:Bunch, J. R.; Hopcroft, J. E. (1974).
26587:
25959:
25935:History of matrices and determinants:
25610:
25413:
24482:{\displaystyle \operatorname {O} (n!)}
22717:Exterior algebra § Linear algebra
21882:Square matrices over commutative rings
21294:matrix whose entries are given by the
20519:More generally, if the determinant of
19739:
19242:is the matrix formed by replacing the
19053:. This equation has a unique solution
18937:
18911:
18878:The next contributor of importance is
18828:
18824:
18812:in 1545 by a determinant-like entity.
18079:Yet another equivalent formulation is
14639:{\displaystyle \chi _{A}(\lambda )=0.}
11771:For a column and row vector as above:
8109:expresses the determinant of a matrix
3464:from the second row first column, and
2713:columns, so that it can be written as
1742:according to the following equations:
568:
23:. For determinants in immunology, see
27652:
27574:
27253:(7th ed.), Pearson Prentice Hall
27132:Linear Algebra: A Modern Introduction
27129:
27092:
27006:Kung, Joseph P.S.; Rota, Gian-Carlo;
26638:
26629:
26624:Linear algebra § Further reading
26519:Fang, Xin Gui; Havas, George (1997).
25664:
23208:the determinant to be the element in
20527:represents an orientation-preserving
20440:the Wronskian and linear independence
17920:Expressed in terms of the entries of
14657:if all its eigenvalues are positive.
13240:We can then write the cross-terms as
9654:The formula for the determinant of a
7418:. Moreover, they both take the value
6842:times the second column to the first
5478:{\displaystyle \det(cA)=c^{n}\det(A)}
4867:{\displaystyle \det \left(I\right)=1}
3623:from this set to itself, with values
2005:. The bivector magnitude (denoted by
27435:
27319:(1st ed.), Philadelphia: SIAM,
27257:
27248:
27229:
27211:
27193:
27117:
27093:Meyer, Carl D. (February 15, 2001),
26815:(3rd ed.), Hoboken, NJ: Wiley,
26507:
26018:
26006:
25994:
25947:
25814:
25689:
25638:Electronic Journal of Linear Algebra
25441:
25328:
24166:-determinant on quantum groups, the
23978:
22529:For example, the determinant of the
20767:. More generally, if the linear map
19685:Cramer's rule can be implemented in
18890:rows, which for the special case of
18495:{\displaystyle {\mathfrak {sl}}_{n}}
18401:This identity is used in describing
8726:Unwinding the determinants of these
8101:
7309:of equal size, the determinant of a
1503:
470:. This implies that, given a linear
27489:"De Determinantibus functionalibus"
27032:Linear Algebra and Its Applications
27029:
26917:Habgood, Ken; Arel, Itamar (2012).
26634:(9th ed.), Wiley International
26594:Linear Algebra and Its Applications
26030:
25556:Linear Algebra and Its Applications
25052:. This means, for example, that an
23967:of a skew-symmetric matrix and the
23561:Generalizations and related notions
22710:
20446:, which gives a criterion when two
18979:, in connection with the theory of
18975:, and Scott; skew determinants and
18819:in 1683 in Japan and parallelly of
18481:
18478:
17535:, which is, in turn, less than the
13095:, so the expression can be written
11650:More generally, for any invertible
10469:{\displaystyle \det(A)(D-CA^{-1}B)}
6797:add the second column to the first
4665:Characterization of the determinant
919:
466:The determinant is invariant under
13:
27678:
27462:Grattan-Guinness, I., ed. (2003),
27267:Computational discrete mathematics
25708:, Observation 7.1.2, Theorem 7.2.5
25157:
25106:
25059:
24865:
24853:
24461:
24422:
23605:generalizes both by introducing a
22380:given by replacing all entries in
21553:
21345:
21330:
19692:
18730:
18682:
18634:
17970:
17953:
17608:. In particular, it is everywhere
17098:
17053:
16936:
16891:
16844:is less than 1 in absolute value,
16836:identity can be obtained from the
15179:corresponds to the eigenvalue exp(
11722:
11598:
11481:
11444:
9360:
9309:
7261:Multiplicativity and matrix groups
3548:matrix is an expression involving
3460:from the first row second column,
3264:
2560:
14:
28127:
27549:
27079:(2nd ed.), Springer-Verlag,
27030:Lay, David C. (August 22, 2005),
26793:10.1090/S0025-5718-1974-0331751-8
26573:Fisikopoulos & Peñaranda 2016
26496:Fisikopoulos & Peñaranda 2016
26452:Fisikopoulos & Peñaranda 2016
24099:Operators in von Neumann algebras
21016:By calculating the volume of the
16832:An important arbitrary dimension
13026:The terms which are quadratic in
11555:identity matrices, respectively.
9649:
8330:{\displaystyle (-1)^{i+j}M_{i,j}}
8212:{\displaystyle (n-1)\times (n-1)}
7051:Combining these equalities gives
3510:
3275:
1540:that represents the image of the
28075:
28074:
28052:Basic Linear Algebra Subprograms
27810:
27530:(seconde partie), Paris: 267–376
27251:Linear Algebra With Applications
27075:, Graduate Texts in Mathematics
27034:(3rd ed.), Addison Wesley,
26949:from the original on 2019-05-05.
25925:http://jeff560.tripod.com/d.html
25912:Journal de l'Ecole Polytechnique
25509:"Determinants of Block Matrices"
25273:
25256:Alice's Adventures in Wonderland
25224:have already been computed, the
24200:{\displaystyle \mathbb {Z} _{2}}
22082:
21917:
21606:The Jacobian also occurs in the
21586:
21569:
21532:
21502:
21490:
21258:
21243:
21188:
21173:
20797:
20782:
20652:
20597:
20582:
19826:
19015:
18769:
18761:
18735:
18721:
18713:
18687:
18673:
18665:
18639:
17594:
17560:
15919:the signed constant term of the
13981:
13929:
13864:
13812:
13769:
13700:
13543:
12461:th root of the determinant is a
12330:are Hermitian positive-definite
9328:{\displaystyle {\tbinom {n}{k}}}
6835:{\displaystyle -{\frac {13}{3}}}
4974:{\displaystyle a_{j}=r\cdot v+w}
4650:{\displaystyle \{1,\ldots ,n\}.}
4612:where the sum is taken over all
3593:{\displaystyle \{1,2,\dots ,n\}}
3271:Leibniz formula for determinants
2522:
2491:
2438:
2417:
2402:
2332:
2263:
2201:
2115:
2096:
2082:
2029:, which is also the determinant
27950:Seven-dimensional cross product
26749:from the original on 2006-09-01
26703:from the original on 2012-10-25
26581:
26566:
26512:
26501:
26489:
26478:
26457:
26445:
26432:
26405:
26120:
26109:
26093:
26081:
26063:
26051:
26039:
26024:
26012:
26000:
25988:
25977:
25965:
25953:
25941:
25929:
25923:Origins of mathematical terms:
25917:
25903:
25891:
25871:
25859:
25847:
25820:
25808:
25779:
25767:
25755:
25711:
25699:
25683:
25658:
25632:Paksoy; Turkmen; Zhang (2014).
25625:
25604:
25592:
25547:
25500:
25488:
25476:
25399:. Dover Publications. pp.
25395:Applications of Tensor Analysis
24616:and an upper triangular matrix
23674:as a vector space over a field
23524:{\displaystyle \bigwedge ^{k}V}
23466:{\displaystyle \bigwedge ^{n}V}
23173:{\displaystyle \bigwedge ^{n}T}
23143:{\displaystyle \bigwedge ^{n}V}
22886:{\displaystyle \bigwedge ^{n}V}
20555:Volume and Jacobian determinant
19168:
19010:
17235:For a positive definite matrix
13913:This result followed just from
13776:{\displaystyle B=b\mathbf {i} }
11950:of the same size, the matrices
11382:Sylvester's determinant theorem
11377:Sylvester's determinant theorem
10975:Sylvester's determinant theorem
8514:) gives the following formula:
4427:{\displaystyle \{1,\ldots ,n\}}
4034:
3888:
2283:
2276:
2214:
476:finite-dimensional vector space
27468:Johns Hopkins University Press
27409:Cambridge Mathematical Journal
27196:Introduction to Linear Algebra
27016:, Cambridge University Press,
26926:Journal of Discrete Algorithms
26418:, American Mathematical Soc.,
25464:
25447:
25435:
25419:
25407:
25384:
25359:
25341:Wildberger, Norman J. (2010).
25334:
25322:
25176:
25163:
25125:
25112:
25093:Coppersmith–Winograd algorithm
25078:
25065:
25039:
25036:
25030:
25024:
24981:{\displaystyle M(n)\geq n^{a}}
24962:
24956:
24933:
24927:
24884:
24871:
24834:
24828:
24819:
24813:
24801:
24795:
24476:
24467:
24441:
24428:
24415:). These methods are of order
24386:
24380:
24218:
24072:
24069:
24066:
24054:
24045:
24036:
24024:
24012:
23952:{\displaystyle N_{L/F}:L\to F}
23943:
23840:
23807:
23777:
23771:
23713:
23588:
23582:
23345:
23339:
23046:
22980:
22737:
22679:
22626:
22619:
22510:
22507:
22504:
22485:
22479:
22476:
22467:
22464:
22461:
22442:
22439:
22433:
22367:
22361:
22345:
22342:
22336:
22317:
22311:
22275:
22138:
22135:
22129:
22089:{\displaystyle R=\mathbf {Z} }
22011:
22005:
21951:
21945:
21836:
21791:
21785:
21776:
21770:
21755:
21748:
21742:
21736:
21727:
21721:
21715:
21709:
21694:
21687:
21678:
21672:
21626:Determinant of an endomorphism
21573:
21565:
21562:
21550:
21539:
21536:
21528:
21522:
21494:
21486:
21478:
21472:
21316:
21310:
21253:
21183:
21133:
21129:
21093:
21086:
20997:
20991:
20947:
20944:
20938:
20932:
20903:
20897:
20792:
20733:
20729:
20723:
20716:
20695:
20689:
20592:
20361:
20355:
20350:
20338:
20318:
20312:
20307:
20295:
20280:
20274:
20269:
20257:
20218:
20212:
20189:
20183:
20165:
20159:
20139:
20133:
20113:
20107:
20092:
20086:
20062:
20056:
20053:
20021:
19961:
19955:
19933:
19927:
19711:
19698:
19655:
19649:
19636:
19630:
19618:
19612:
19545:
19539:
19328:
19315:
19162:
19156:
19148:
19135:
19089:
19083:
18955:determinants by Sylvester and
18750:
18744:
18702:
18696:
18654:
18648:
18355:
18349:
18331:
18316:
18303:, the identity matrix, yields
18197:
18191:
18155:
18149:
18143:
18134:
18122:
18116:
18107:
18092:
18029:
18023:
18005:
17998:
17965:
17959:
17852:
17846:
17826:
17820:
17744:
17738:
17681:
17675:
17641:
17635:
17463:
17457:
17424:
17417:
17328:
17316:
17304:
17298:
17116:
17106:
16954:
16944:
16869:
16857:
16785:
16779:
16713:
16703:
16637:with multidimensional indices
16609:
16564:
16534:
16524:
16515:
16509:
16284:
16274:
16125:
16119:
16084:
16066:
15923:, determined recursively from
15831:
15825:
15725:
15719:
15663:
15657:
15620:
15614:
15539:
15533:
15507:
15501:
15464:
15458:
15402:
15396:
15359:
15353:
15296:
15293:
15287:
15278:
15266:
15260:
15221:
15215:
15140:
15137:
15134:
15128:
15119:
15113:
15101:
15095:
15059:
15056:
15050:
15041:
15029:
15026:
15020:
15011:
14627:
14621:
14470:
14452:
14443:
14437:
14198:
14192:
13985:
13977:
13968:
13962:
13933:
13925:
13868:
13860:
13841:
13835:
13816:
13796:
13704:
13696:
13474:
13465:
13454:
13448:
13440:
13434:
13403:
13311:
13305:
13279:
13276:
13250:
13129:
13123:
13114:
13108:
13062:
13056:
12800:
12774:
12771:
12745:
12739:
12713:
12710:
12684:
12605:
12596:
12585:
12579:
12571:
12565:
12554:
12548:
12539:
12533:
12524:
12512:
12433:
12427:
12407:
12401:
12381:
12369:
12261:
12255:
12246:
12240:
12231:
12219:
12191:
12179:
12170:
12158:
12149:
12143:
12134:
12116:
12045:positive semidefinite matrices
11887:
11881:
11865:
11859:
11812:
11806:
11797:
11782:
11705:
11699:
11690:
11675:
11563:For the case of column vector
11357:
11345:
11339:
11327:
11132:
11114:
10977:, which is also stated below.
10950:
10922:
10916:
10910:
10813:
10807:
10780:
10770:
10762:
10746:
10614:
10608:
10463:
10435:
10432:
10426:
10393:
10365:
10359:
10353:
10256:
10250:
10223:
10213:
10205:
10189:
10057:
10051:
9903:
9897:
9891:
9885:
9556:
9544:
9520:
9511:
9457:
9447:
9429:
9425:
9419:
9410:
9387:
9381:
8832:
8822:
8795:
8789:
8417:
8407:
8380:
8374:
8296:
8286:
8206:
8194:
8188:
8176:
8043:
8037:
8016:
8010:
7974:
7968:
7931:
7925:
7888:
7882:
7815:into the multiplicative group
7802:
7796:
7736:
7730:
7711:
7705:
7665:
7659:
7570:
7566:
7560:
7554:
7545:
7539:
7362:
7356:
7350:
7344:
7335:
7326:
7222:
7216:
7123:
7120:
7111:
7096:
7086:
7078:
7067:
7059:
7031:
7023:
7015:
7007:
6982:
6974:
6963:
6955:
6930:
6922:
6914:
6906:
6882:
6874:
6866:
6858:
6125:
6119:
5954:
5881:
5873:
5800:
5789:
5719:
5694:
5622:
5611:
5542:
5472:
5466:
5447:
5438:
5348:
5290:
5250:
5204:
5196:
5153:
5132:
5022:
5014:
4461:
4455:
4308:
4302:
4264:
4258:
4218:
4212:
4174:
4168:
4146:
4140:
4123:
4117:
3978:
3972:
3738:
3732:
3675:
3669:
3654:
3648:
3639:
3633:
3473:, the single transposition of
2629:
2623:
2609:
2603:
1787:
1777:
1771:
1761:
1439:
1421:
1203:
1186:
1180:
1163:
894:
882:
876:
873:
864:
855:
50:. The determinant of a matrix
1:
27620:Linear algebra: determinants.
27152:American Mathematical Monthly
27134:(2nd ed.), Brooks/Cole,
26974:A history of abstract algebra
26873:American Mathematical Monthly
26617:
25665:Serre, Denis (Oct 18, 2010).
25349:University of New South Wales
24170:on Capelli matrices, and the
23725:{\displaystyle \det :A\to F.}
23694:, there is a determinant map
22638:{\displaystyle (-)^{\times }}
21613:When applied to the field of
18405:associated to certain matrix
17542:
17344:with equality if and only if
14374:is invertible if and only if
12667:{\displaystyle A_{ij},B_{ij}}
12469:Sum identity for 2×2 matrices
7636:) forms a group known as the
4660:Properties of the determinant
2692:
1520:can be used to represent two
27792:Eigenvalues and eigenvectors
26864:10.1016/j.comgeo.2015.12.001
26588:Abeles, Francine F. (2008).
26440:Trefethen & Bau III 1997
25971:History of matrix notation:
24709:{\displaystyle \varepsilon }
24576:(which has exactly a single
23150:is one-dimensional, the map
21924:{\displaystyle \mathbf {Z} }
21855:for some finite-dimensional
20547:), while if it is negative,
19734:singular value decomposition
19024:, written in matrix form as
17601:{\displaystyle \mathbf {R} }
15187:). In particular, given any
13783:in the above identity gives
12288:th root of determinant is a
10525:A similar result holds when
8754:Laplace expansion along the
8482:Laplace expansion along the
7269:, i.e., for square matrices
7253:rows, the determinant is an
7249:matrix as being composed of
7144:
6083:or, alternatively, whenever
2689:, and so is not invertible.
1964:and orientation-preserving.
7:
27599:"Matrices and determinants"
27563:Encyclopedia of Mathematics
27013:Combinatorics: The Rota Way
25266:
24109:Fuglede−Kadison determinant
21621:Abstract algebraic aspects
21411:integration by substitution
21284:the Jacobian matrix is the
18436:is defined by the equation
17356:Kullback-Leibler divergence
15899:Faddeev–LeVerrier algorithm
15171:, because every eigenvalue
13497:This has an application to
13493:which completes the proof.
13419:which can be recognized as
12006:The determinant of the sum
8279:-th column. The expression
8076:{\displaystyle K^{\times }}
7835:{\displaystyle K^{\times }}
6807:swap the first two columns
3556:. A permutation of the set
551:exterior differential forms
417:, which allows computing a
10:
28132:
27556:Suprunenko, D.A. (2001) ,
27340:Bourbaki, Nicolas (1994),
27177:Cambridge University Press
26779:Mathematics of Computation
26754:Bourbaki, Nicolas (1998),
26674:Mathematics of Computation
26621:
26412:Varadarajan, V. S (2004),
26100:Lombardi & Quitté 2015
25667:"Concavity of det over HPD
25367:"Determinants and Volumes"
24917:can be multiplied in time
24413:positive definite matrices
24347:for getting a matrix in a
24103:For operators in a finite
23254:{\displaystyle v_{i}\in V}
22773:-dimensional vector space
22714:
22197:-matrices with entries in
20559:As pointed out above, the
20464:Orientation (vector space)
20461:
19022:linear system of equations
18800:system of linear equations
18793:
18467:special linear Lie algebra
9501:For every matrix, one has
8219:-matrix that results from
7761:is 2 or 3 consists of all
7173:equals the determinant of
6245:
4996:, then the determinant of
3268:
2586:The determinant gives the
499:system of linear equations
18:
28070:
28032:
27988:
27925:
27877:
27819:
27808:
27704:
27686:
27520:Laplace, Pierre-Simon, de
27485:Jacobi, Carl Gustav Jakob
27354:10.1007/978-3-642-61693-8
26982:10.1007/978-0-8176-4685-1
26954:Harris, Frank E. (2014),
26939:10.1016/j.jda.2011.06.007
26644:Linear Algebra Done Right
26607:10.1016/j.laa.2007.11.022
26485:Bunch & Hopcroft 1974
25962:, Vol. II, p. 92, no. 462
25578:10.1016/j.laa.2016.10.004
25507:Silvester, J. R. (2000).
25483:Kung, Rota & Yan 2009
25371:textbooks.math.gatech.edu
25295:Cayley–Menger determinant
24299:{\displaystyle n\times n}
22585:between the two functors
22190:{\displaystyle n\times n}
22171:(the group of invertible
21963:{\displaystyle \det(I)=1}
21440:) of some other function
21413:: for suitable functions
20836:{\displaystyle m\times n}
20740:{\displaystyle |\det(A)|}
20388:It is non-zero (for some
19899:{\displaystyle 3\times 3}
18981:orthogonal transformation
18524:{\displaystyle 3\times 3}
18277:. The special case where
17531:, which is less than the
17201:then all coefficients of
15921:characteristic polynomial
15233:{\displaystyle \exp(L)=A}
14577:: they are precisely the
14064:{\displaystyle n\times n}
14023:characteristic polynomial
13516:{\displaystyle 2\times 2}
12492:{\displaystyle 2\times 2}
12349:{\displaystyle n\times n}
12315:{\displaystyle n\times n}
11170:{\displaystyle 2\times 2}
10515:{\displaystyle 1\times 1}
9819:{\displaystyle n\times n}
9793:{\displaystyle n\times m}
9767:{\displaystyle m\times n}
9741:{\displaystyle m\times m}
9673:{\displaystyle 2\times 2}
8745:{\displaystyle 2\times 2}
7903:first isomorphism theorem
5504:{\displaystyle n\times n}
5393:{\displaystyle n\times n}
4699:as being composed of its
4688:{\displaystyle n\times n}
3541:{\displaystyle n\times n}
3288:matrix is the following:
3284:for the determinant of a
943:{\displaystyle 2\times 2}
507:characteristic polynomial
195:and the determinant of a
27609:University of St Andrews
27505:10.1515/crll.1841.22.319
27420:Cramer, Gabriel (1750),
27396:Bézout, Étienne (1779),
27370:Cajori, Florian (1993),
27317:Numerical Linear Algebra
27249:Leon, Steven J. (2006),
25788:A History of Mathematics
25721:Zeitschrift für Physik A
25316:
25310:Determinantal conjecture
25226:matrix determinant lemma
24225:numerical linear algebra
23200:. Some authors such as (
22820:over a commutative ring
22746:{\displaystyle T:V\to V}
22284:{\displaystyle f:R\to S}
21845:{\displaystyle T:V\to V}
21608:inverse function theorem
18458:{\displaystyle \det A=1}
17193:is expanded as a formal
16405:satisfying the equation
14977:
14914:being positive, for all
14594:{\displaystyle \lambda }
14394:is not an eigenvalue of
12674:. The left-hand side is
12473:For the special case of
12322:matrices. Therefore, if
9294:determinant as a sum of
7755:special orthogonal group
6989:{\displaystyle |D|=-|C|}
6050:{\displaystyle a_{ij}=0}
1967:The object known as the
85:represented, on a given
28116:Homogeneous polynomials
27276:10.1007/3-540-45506-X_9
26756:Algebra I, Chapters 1-3
26665:Bareiss, Erwin (1968),
26058:Dummit & Foote 2004
26046:Dummit & Foote 2004
26035:. Pearson. p. 172.
25984:Habgood & Arel 2012
25762:Horn & Johnson 2018
25706:Horn & Johnson 2018
25694:Horn & Johnson 2018
25651:10.13001/1081-3810.1622
25598:Proofs can be found in
25495:Horn & Johnson 2018
25471:Horn & Johnson 2018
25045:{\displaystyle O(M(n))}
24392:{\displaystyle \det(A)}
22581:, the determinant is a
21909:, such as the integers
21814:. The determinant of a
21215:differentiable function
21058:{\displaystyle a,b,c,d}
19095:{\displaystyle \det(A)}
17801:is invertible, we have
15901:. That is, for generic
15891:Cayley-Hamilton theorem
14314:is zero if and only if
13068:{\displaystyle \det(A)}
12282:Brunn–Minkowski theorem
9715:{\displaystyle A,B,C,D}
8361:, one has the equality
8162:{\displaystyle M_{i,j}}
7842:of nonzero elements of
7149:The determinant of the
7038:{\displaystyle |E|=|D|}
6937:{\displaystyle |B|=|C|}
6889:{\displaystyle |A|=|B|}
6234:{\displaystyle \sigma }
5273:
4902:
4837:
4719:columns, so denoted as
4616:-tuples of integers in
3764:{\displaystyle \sigma }
3616:{\displaystyle \sigma }
2988:{\displaystyle a_{1,1}}
2393:defined by the vectors
2136:{\displaystyle A=\left}
1528:vectors to the rows of
438:The determinant of the
27777:Row and column vectors
27633:Determinant Calculator
27169:Johnson, Charles Royal
26840:Computational Geometry
26630:Anton, Howard (2005),
26396:
26021:, §VII.6, Theorem 6.10
25461:of the exterior power.
25249:Charles Dodgson (i.e.
25216:If the determinant of
25207:
25183:
25132:
25085:
25046:
25008:
25007:{\displaystyle a>2}
24982:
24940:
24911:
24891:
24844:
24776:
24756:
24733:
24710:
24690:
24670:
24650:
24630:
24610:
24590:
24570:
24544:
24543:{\displaystyle A=PLU.}
24506:
24483:
24448:
24409:Cholesky decomposition
24393:
24326:
24300:
24271:
24251:
24229:Computational geometry
24201:
24093:functional determinant
24082:
23973:central simple algebra
23953:
23899:
23784:
23726:
23688:
23664:
23633:
23595:
23551:
23550:{\displaystyle k<n}
23525:
23487:
23467:
23437:
23416:-tuples of vectors in
23410:
23387:
23255:
23222:
23194:
23174:
23144:
23111:
22927:
22907:
22887:
22854:
22834:
22814:
22787:
22767:
22747:
22701:
22639:
22606:
22583:natural transformation
22577:). In the language of
22567:
22547:
22517:
22414:
22400:by their images under
22394:
22374:
22285:
22253:
22235:
22211:
22191:
22158:
22090:
22062:
22038:
22018:
21964:
21925:
21903:
21846:
21801:
21597:
21396:
21275:
21210:
21202:
21141:
21059:
21007:
20910:
20877:
20857:
20837:
20817:is represented by the
20811:
20761:
20741:
20702:
20666:
20631:
20611:
20458:Orientation of a basis
20428:
20402:
20379:
19994:
19968:
19908:differential equations
19900:
19867:
19840:
19785:
19765:
19764:{\displaystyle \det A}
19718:
19676:
19582:
19552:
19413:
19296:
19276:
19256:
19236:
19206:
19096:
19067:
19047:
18784:
18609:
18577:
18525:
18496:
18459:
18430:
18392:
18297:
18263:
18070:
17934:
17911:
17795:
17775:
17751:
17716:
17602:
17580:
17514:
17335:
17231:Upper and lower bounds
17184:
17102:
17057:
17012:
16940:
16895:
16823:
16619:
16462:
16435:
16386:
16270:
16097:
16013:
15880:
15306:
15234:
15195:, that is, any matrix
15150:
15075:or, for real matrices
15066:
14968:
14948:
14928:
14905:
14640:
14595:
14571:
14547:
14527:
14513:of the polynomial and
14503:
14480:
14408:
14388:
14368:
14348:
14328:
14308:
14278:
14224:
14179:of these eigenvalues,
14162:algebraic multiplicity
14154:
14089:
14065:
14039:
13998:
13946:
13904:
13777:
13746:
13717:
13678:
13658:
13638:
13517:
13484:
13410:
13231:
13089:
13069:
13040:
13017:
12810:
12668:
12615:
12493:
12451:
12350:
12316:
12273:
12203:
12100:
12080:
12060:
12026:
12001:
11990:
11967:
11944:
11924:
11901:
11762:
11640:
11508:
11367:
11266:
11246:
11226:
11200:
11171:
11142:
11053:
11018:
10998:
10964:
10545:is invertible, namely
10539:
10516:
10490:
10470:
10407:
9978:
9955:
9820:
9794:
9768:
9742:
9716:
9674:
9640:
9570:
9492:
9394:
9329:
9272:
8886:
8821:
8768:
8746:
8717:
8508:
8471:
8406:
8355:
8331:
8273:
8253:
8233:
8213:
8163:
8123:
8077:
8050:
7981:
7938:
7895:
7856:
7836:
7809:
7743:
7672:
7630:
7610:
7586:
7478:
7455:
7435:
7434:{\displaystyle \det B}
7412:
7392:
7369:
7303:
7283:
7229:
7167:
7136:
7039:
6990:
6938:
6890:
6836:
6782:
6687:
6592:
6498:
6397:
6373:
6269:
6235:
6211:
6190:
6103:
6102:{\displaystyle i<j}
6077:
6076:{\displaystyle i>j}
6051:
6011:
5965:
5705:
5525:
5505:
5479:
5416:Immediate consequences
5394:
5362:
5262:
4975:
4927:
4913:th column of a matrix
4888:
4868:
4818:
4785:
4713:
4689:
4651:
4603:
4428:
4382:
4322:
4287:
4186:
3946:
3814:
3791:
3765:
3745:
3713:
3692:, is commonly denoted
3682:
3617:
3594:
3542:
3495:
3443:
3244:
2989:
2953:
2639:
2578:
2455:
2372:
2349:
2137:
1975:oriented plane segment
1953:
1935:
1595:The absolute value of
1513:
1494:
1327:
1304:
1080:
1000:
944:
910:
747:
630:is denoted either by "
624:
391:
360:The determinant of an
351:
186:
27782:Row and column spaces
27727:Scalar multiplication
27436:Eves, Howard (1990),
27334:Historical references
27258:Rote, Günter (2001),
27130:Poole, David (2006),
26732:10.1145/122272.122273
26720:ACM SIGNUM Newsletter
26536:10.1145/258726.258740
26397:
25774:Grattan-Guinness 2003
25300:Dieudonné determinant
25208:
25184:
25133:
25086:
25047:
25009:
24983:
24941:
24912:
24892:
24845:
24777:
24757:
24734:
24711:
24691:
24671:
24651:
24631:
24611:
24591:
24571:
24545:
24507:
24484:
24449:
24394:
24370:Some methods compute
24366:Decomposition methods
24327:
24301:
24272:
24252:
24202:
24160:Dieudonné determinant
24083:
23997:trace class operators
23954:
23900:
23785:
23727:
23689:
23665:
23634:
23632:{\displaystyle S_{n}}
23596:
23552:
23526:
23488:
23468:
23438:
23436:{\displaystyle R^{n}}
23411:
23388:
23256:
23223:
23195:
23175:
23145:
23112:
22933:induces a linear map
22928:
22908:
22888:
22855:
22835:
22815:
22793:or, more generally a
22788:
22768:
22748:
22702:
22640:
22607:
22568:
22548:
22518:
22415:
22395:
22375:
22286:
22251:
22236:
22212:
22192:
22159:
22091:
22063:
22039:
22019:
21965:
21926:
21904:
21847:
21816:linear transformation
21802:
21598:
21432:), the integral over
21405:Its determinant, the
21397:
21276:
21203:
21157:
21142:
21060:
21008:
20911:
20878:
20858:
20838:
20812:
20762:
20742:
20703:
20679:, then the volume of
20667:
20632:
20612:
20529:linear transformation
20429:
20403:
20380:
19995:
19969:
19901:
19868:
19866:{\displaystyle v_{3}}
19841:
19786:
19766:
19719:
19677:
19583:
19581:{\displaystyle a_{j}}
19553:
19393:
19297:
19282:by the column vector
19277:
19257:
19237:
19235:{\displaystyle A_{i}}
19207:
19097:
19068:
19048:
18785:
18610:
18608:{\displaystyle a,b,c}
18578:
18526:
18497:
18460:
18431:
18393:
18298:
18264:
18071:
17935:
17912:
17796:
17776:
17752:
17717:
17603:
17581:
17515:
17336:
17185:
17082:
17037:
17013:
16920:
16875:
16824:
16620:
16463:
16415:
16387:
16250:
16098:
15993:
15881:
15307:
15235:
15151:
15067:
14969:
14949:
14929:
14906:
14659:Sylvester's criterion
14641:
14596:
14572:
14548:
14528:
14504:
14481:
14409:
14389:
14369:
14349:
14329:
14309:
14279:
14204:
14155:
14090:
14066:
14040:
13999:
13947:
13905:
13778:
13747:
13718:
13679:
13659:
13639:
13518:
13485:
13411:
13232:
13090:
13070:
13041:
13018:
12811:
12669:
12616:
12494:
12452:
12351:
12317:
12292:, when restricted to
12274:
12204:
12101:
12081:
12061:
12027:
11991:
11968:
11945:
11925:
11902:
11763:
11641:
11509:
11368:
11267:
11247:
11227:
11201:
11172:
11143:
11054:
11052:{\displaystyle CD=DC}
11019:
10999:
10965:
10540:
10517:
10491:
10471:
10408:
9979:
9956:
9821:
9795:
9769:
9743:
9717:
9675:
9641:
9571:
9493:
9395:
9330:
9273:
8887:
8801:
8769:
8747:
8718:
8509:
8472:
8386:
8356:
8332:
8274:
8254:
8234:
8214:
8164:
8124:
8078:
8051:
7982:
7939:
7896:
7857:
7837:
7810:
7767:special unitary group
7744:
7673:
7631:
7611:
7587:
7479:
7456:
7436:
7413:
7393:
7370:
7304:
7284:
7265:The determinant is a
7230:
7168:
7137:
7040:
6991:
6939:
6891:
6837:
6783:
6688:
6593:
6499:
6398:
6374:
6270:
6236:
6212:
6170:
6104:
6078:
6052:
6012:
5966:
5706:
5526:
5506:
5480:
5424:The determinant is a
5395:
5363:
5263:
4976:
4928:
4889:
4869:
4819:
4817:{\displaystyle a_{i}}
4786:
4714:
4690:
4652:
4604:
4429:
4383:
4323:
4267:
4187:
3947:
3815:
3792:
3766:
3746:
3714:
3712:{\displaystyle S_{n}}
3683:
3618:
3595:
3543:
3491:
3444:
3245:
2990:
2954:
2651:linear transformation
2640:
2579:
2456:
2373:
2350:
2138:
1947:
1936:
1511:
1495:
1328:
1305:
1081:
1001:
945:
911:
748:
625:
573:The determinant of a
392:
352:
187:
111:The determinant of a
27917:Gram–Schmidt process
27869:Gaussian elimination
27595:Robertson, Edmund F.
27230:Lang, Serge (2002).
27212:Lang, Serge (1987),
27194:Lang, Serge (1985),
26139:
25835:on 10 September 2012
25444:, §VI.7, Theorem 7.5
25261:Dodgson condensation
25246:are column vectors.
25197:
25154:
25148:Gaussian elimination
25140:closed ordered walks
25103:
25056:
25018:
24992:
24950:
24939:{\displaystyle M(n)}
24921:
24901:
24862:
24789:
24766:
24743:
24720:
24700:
24680:
24660:
24640:
24620:
24600:
24580:
24560:
24519:
24496:
24458:
24419:
24374:
24341:Gaussian elimination
24336:Gaussian elemination
24313:
24284:
24261:
24238:
24182:
24006:
23993:Fredholm determinant
23916:
23801:
23743:
23701:
23678:
23654:
23616:
23573:
23535:
23505:
23477:
23447:
23420:
23400:
23268:
23232:
23212:
23184:
23154:
23124:
22940:
22917:
22897:
22867:
22844:
22824:
22804:
22777:
22757:
22725:
22660:
22651:multiplicative group
22616:
22589:
22557:
22537:
22427:
22404:
22384:
22295:
22263:
22225:
22219:multiplicative group
22201:
22175:
22169:general linear group
22113:
22072:
22052:
22028:
21977:
21939:
21931:, as opposed to the
21913:
21893:
21824:
21812:similarity invariant
21666:
21461:
21407:Jacobian determinant
21304:
21232:
21162:
21069:
21031:
20923:
20909:{\displaystyle f(S)}
20891:
20867:
20847:
20821:
20771:
20751:
20747:times the volume of
20712:
20701:{\displaystyle f(S)}
20683:
20641:
20621:
20571:
20412:
20392:
20015:
19978:
19914:
19884:
19850:
19795:
19775:
19752:
19689:
19599:
19565:
19309:
19286:
19266:
19246:
19219:
19113:
19077:
19057:
19046:{\displaystyle Ax=b}
19028:
18918:. In his memoirs in
18906:Cauchy–Binet formula
18626:
18587:
18535:
18509:
18472:
18440:
18413:
18310:
18281:
18086:
17947:
17924:
17808:
17785:
17781:. In particular, if
17765:
17729:
17623:
17590:
17555:
17372:
17248:
17034:
16851:
16700:
16503:
16412:
16113:
15930:
15343:
15254:
15206:
15086:
15005:
14958:
14938:
14918:
14668:
14608:
14585:
14561:
14537:
14517:
14493:
14424:
14398:
14378:
14358:
14338:
14334:is an eigenvalue of
14318:
14298:
14186:
14105:
14079:
14049:
14029:
13956:
13917:
13790:
13756:
13745:{\displaystyle A=aI}
13727:
13688:
13668:
13648:
13527:
13501:
13426:
13247:
13102:
13079:
13075:, and similarly for
13050:
13030:
12826:
12681:
12632:
12506:
12477:
12463:homogeneous function
12360:
12334:
12300:
12213:
12110:
12090:
12070:
12050:
12010:
11977:
11954:
11934:
11914:
11910:For square matrices
11776:
11669:
11581:
11427:
11279:
11256:
11236:
11210:
11184:
11155:
11066:
11031:
11008:
10988:
10552:
10529:
10500:
10480:
10420:
10416:which simplifies to
9995:
9968:
9837:
9804:
9778:
9752:
9726:
9688:
9658:
9590:
9508:
9407:
9372:
9298:
8903:
8783:
8758:
8730:
8521:
8492:
8480:which is called the
8368:
8345:
8283:
8263:
8243:
8223:
8173:
8140:
8113:
8088:Cauchy–Binet formula
8060:
7994:
7952:
7909:
7866:
7846:
7819:
7780:
7689:
7684:special linear group
7643:
7638:general linear group
7620:
7600:
7500:
7468:
7445:
7422:
7402:
7382:
7320:
7293:
7273:
7184:
7157:
7055:
7003:
6951:
6902:
6854:
6816:
6700:
6605:
6510:
6413:
6387:
6282:
6259:
6252:Gaussian elimination
6225:
6113:
6087:
6061:
6025:
6001:
5715:
5538:
5515:
5489:
5432:
5426:homogeneous function
5378:
5286:
5006:
4940:
4917:
4878:
4841:
4801:
4726:
4703:
4673:
4620:
4449:
4400:
4339:
4206:
3966:
3829:
3801:
3775:
3755:
3723:
3696:
3627:
3607:
3560:
3526:
3295:
3017:
2966:
2720:
2597:
2465:
2397:
2362:
2150:
2063:
1749:
1524:: one that maps the
1340:
1317:
1096:
1013:
957:
928:
763:
641:
581:
559:changes of variables
557:, in particular for
555:Jacobian determinant
415:Gaussian elimination
378:
206:
122:
56:is commonly denoted
46:of the entries of a
28047:Numerical stability
27927:Multilinear algebra
27902:Inner product space
27752:Linear independence
27593:O'Connor, John J.;
27430:10.3931/e-rara-4048
26713:"An empty exercise"
26060:, §11.4, Theorem 30
26031:Lay, David (2021).
25733:1992ZPhyA.344...99K
25220:and the inverse of
24345:elementary matrices
24168:Capelli determinant
24117:von Neumann algebra
23986:Functional analysis
23649:associative algebra
23639:in Leibniz's rule.
23204:) use this fact to
21296:partial derivatives
21151:over the vertices.
20427:{\displaystyle n-1}
20354:
20311:
20273:
20211:
20182:
20158:
19993:{\displaystyle n-1}
19740:Linear independence
19588:are the columns of
18296:{\displaystyle A=I}
17360:multivariate normal
16815:
16769:
16754:
16726:
15895:Newton's identities
15243:the determinant of
12209:with the corollary
12025:{\displaystyle A+B}
11225:{\displaystyle B=C}
11199:{\displaystyle A=D}
9192:
9164:
9141:
9118:
9066:
9044:
9027:
9010:
8507:{\displaystyle i=1}
6403:
6275:using that method:
5975:of the permutation.
5274:The determinant is
4903:The determinant is
3813:{\displaystyle -1.}
3790:{\displaystyle +1,}
3002:The determinant of
1948:The volume of this
569:Two by two matrices
537:determines how the
27757:Linear combination
27625:2008-12-04 at the
27576:Weisstein, Eric W.
27067:Mac Lane, Saunders
26640:Axler, Sheldon Jay
26392:
26390:
26370:
26325:
26283:
26237:
26195:
25877:Muir, Sir Thomas,
25741:10.1007/BF01291027
25391:McConnell (1957).
25305:Slater determinant
25290:Cauchy determinant
25281:Mathematics portal
25203:
25179:
25128:
25099:having complexity
25081:
25042:
25004:
24978:
24936:
24907:
24887:
24840:
24772:
24755:{\displaystyle -1}
24752:
24732:{\displaystyle +1}
24729:
24706:
24686:
24666:
24646:
24626:
24606:
24586:
24566:
24555:permutation matrix
24540:
24502:
24479:
24444:
24389:
24325:{\displaystyle n!}
24322:
24296:
24280:) products for an
24267:
24250:{\displaystyle n!}
24247:
24197:
24078:
23949:
23895:
23780:
23722:
23684:
23672:finite-dimensional
23660:
23629:
23591:
23547:
23521:
23483:
23463:
23433:
23406:
23383:
23251:
23218:
23190:
23170:
23140:
23107:
23105:
22923:
22903:
22883:
22850:
22830:
22810:
22783:
22763:
22743:
22697:
22635:
22602:
22575:modular arithmetic
22563:
22543:
22513:
22410:
22390:
22370:
22281:
22254:
22243:group homomorphism
22231:
22207:
22187:
22154:
22105:group homomorphism
22086:
22058:
22046:invertible element
22034:
22014:
21960:
21921:
21899:
21842:
21797:
21593:
21392:
21271:
21211:
21198:
21137:
21055:
21003:
20906:
20873:
20853:
20833:
20807:
20757:
20737:
20698:
20662:
20627:
20607:
20543:matrix, this is a
20436:analytic functions
20424:
20398:
20375:
20366:
20328:
20285:
20247:
20199:
20170:
20146:
19990:
19964:
19910:: given functions
19896:
19863:
19836:
19781:
19761:
19746:linearly dependent
19714:
19672:
19578:
19561:where the vectors
19548:
19517:
19384:
19292:
19272:
19252:
19232:
19202:
19092:
19063:
19043:
18936:began their work.
18855:elimination theory
18780:
18778:
18619:of the other two:
18605:
18573:
18567:
18521:
18492:
18455:
18426:
18388:
18293:
18259:
18066:
17930:
17907:
17791:
17771:
17747:
17712:
17598:
17576:
17510:
17331:
17180:
17008:
16819:
16801:
16800:
16755:
16740:
16739:
16712:
16615:
16458:
16382:
16249:
16247:
16093:
15876:
15874:
15302:
15230:
15165:matrix exponential
15146:
15062:
14964:
14944:
14924:
14901:
14895:
14636:
14591:
14567:
14543:
14523:
14499:
14476:
14404:
14384:
14364:
14354:. In other words,
14344:
14324:
14304:
14289:pseudo-determinant
14274:
14150:
14095:must have exactly
14085:
14061:
14035:
13994:
13942:
13900:
13773:
13742:
13713:
13674:
13654:
13634:
13628:
13583:
13513:
13480:
13406:
13227:
13085:
13065:
13036:
13013:
12806:
12664:
12626:
12611:
12489:
12447:
12356:matrices, one has
12346:
12312:
12296:positive-definite
12269:
12199:
12096:
12076:
12056:
12022:
11989:{\displaystyle BA}
11986:
11966:{\displaystyle AB}
11963:
11940:
11920:
11897:
11758:
11636:
11504:
11363:
11315:
11262:
11242:
11222:
11196:
11167:
11138:
11102:
11049:
11014:
10994:
10960:
10958:
10891:
10793:
10736:
10726:
10650:
10592:
10535:
10512:
10486:
10466:
10403:
10401:
10334:
10236:
10179:
10169:
10093:
10035:
9974:
9951:
9942:
9873:
9816:
9790:
9764:
9738:
9712:
9670:
9636:
9581:nonsingular matrix
9566:
9488:
9390:
9325:
9323:
9268:
9231:
9195:
9172:
9144:
9121:
9098:
9052:
9030:
9013:
8996:
8898:Vandermonde matrix
8882:
8764:
8742:
8713:
8707:
8665:
8623:
8581:
8504:
8467:
8351:
8327:
8269:
8249:
8229:
8209:
8159:
8119:
8073:
8046:
7977:
7934:
7905:, this shows that
7891:
7852:
7832:
7805:
7774:group homomorphism
7739:
7668:
7626:
7606:
7582:
7484:with entries in a
7474:
7451:
7431:
7408:
7388:
7365:
7299:
7279:
7267:multiplicative map
7257:-linear function.
7225:
7163:
7132:
7035:
6986:
6934:
6886:
6832:
6778:
6772:
6683:
6677:
6588:
6582:
6494:
6488:
6393:
6382:
6369:
6360:
6265:
6231:
6207:
6099:
6073:
6047:
6007:
5984:linearly dependent
5961:
5701:
5521:
5501:
5475:
5390:
5358:
5258:
5256:
4971:
4935:linear combination
4923:
4884:
4864:
4814:
4781:
4709:
4685:
4647:
4599:
4515:
4424:
4388:is defined on the
4378:
4334:Levi-Civita symbol
4318:
4246:
4182:
4110:
4079:
3942:
3933:
3810:
3787:
3761:
3741:
3709:
3678:
3613:
3602:bijective function
3590:
3538:
3496:
3439:
3355:
3240:
3231:
3006:is denoted by det(
2985:
2949:
2940:
2635:
2574:
2451:
2368:
2345:
2321:
2252:
2190:
2133:
2127:
1981:, and coordinates
1954:
1931:
1904:
1875:
1514:
1490:
1481:
1385:
1323:
1300:
1291:
1242:
1151:
1076:
1046:
996:
990:
940:
906:
846:
802:
743:
716:
677:
620:
614:
563:multiple integrals
411:linear combination
390:{\displaystyle n!}
387:
347:
266:
182:
155:
28088:
28087:
27955:Geometric algebra
27912:Kronecker product
27747:Linear projection
27732:Vector projection
27326:978-0-89871-361-9
27286:978-3-540-42775-9
27241:978-0-387-95385-4
27186:978-0-521-54823-6
27106:978-0-89871-454-8
27041:978-0-321-28713-7
26991:978-0-8176-4684-4
26657:978-3-319-11079-0
26425:978-0-8218-3574-6
26072:, §I.4. See also
25206:{\displaystyle n}
25191:Bareiss Algorithm
24910:{\displaystyle n}
24775:{\displaystyle A}
24696:is just the sign
24689:{\displaystyle P}
24669:{\displaystyle U}
24649:{\displaystyle L}
24629:{\displaystyle U}
24609:{\displaystyle L}
24589:{\displaystyle 1}
24569:{\displaystyle P}
24505:{\displaystyle A}
24357:triangular matrix
24270:{\displaystyle n}
24156:quasideterminants
23979:Infinite matrices
23963:, as well as the
23687:{\displaystyle F}
23663:{\displaystyle A}
23517:
23486:{\displaystyle V}
23459:
23409:{\displaystyle n}
23285:
23221:{\displaystyle R}
23193:{\displaystyle R}
23166:
23136:
22992:
22972:
22956:
22926:{\displaystyle T}
22906:{\displaystyle V}
22879:
22853:{\displaystyle n}
22833:{\displaystyle R}
22813:{\displaystyle n}
22786:{\displaystyle V}
22766:{\displaystyle n}
22566:{\displaystyle m}
22546:{\displaystyle m}
22531:complex conjugate
22413:{\displaystyle f}
22393:{\displaystyle R}
22291:, there is a map
22258:ring homomorphism
22234:{\displaystyle R}
22210:{\displaystyle R}
22061:{\displaystyle R}
22037:{\displaystyle R}
21902:{\displaystyle R}
21359:
21080:
20983:
20971:
20876:{\displaystyle n}
20856:{\displaystyle A}
20760:{\displaystyle S}
20630:{\displaystyle A}
20535:is an orthogonal
20510:orthonormal basis
20502:orthogonal matrix
20401:{\displaystyle x}
20008:is defined to be
19846:, a third vector
19784:{\displaystyle A}
19295:{\displaystyle b}
19275:{\displaystyle A}
19255:{\displaystyle i}
19166:
19066:{\displaystyle x}
18951:, and Sylvester;
18865:theory of numbers
17987:
17933:{\displaystyle A}
17897:
17838:
17794:{\displaystyle A}
17774:{\displaystyle A}
17702:
17653:
17527:is less than the
17505:
17479:
17449:
17435:
17409:
17139:
17071:
16967:
16909:
16791:
16730:
16552:
16339:
16131:
16089:
16065:
16062:
15991:
15960:
15957:
15954:
15818:
15638:
15544:
15482:
15377:
15315:For example, for
14967:{\displaystyle n}
14947:{\displaystyle 1}
14927:{\displaystyle k}
14655:positive definite
14570:{\displaystyle A}
14546:{\displaystyle A}
14526:{\displaystyle I}
14502:{\displaystyle t}
14407:{\displaystyle A}
14387:{\displaystyle 0}
14367:{\displaystyle A}
14347:{\displaystyle A}
14327:{\displaystyle 0}
14307:{\displaystyle A}
14088:{\displaystyle A}
14038:{\displaystyle A}
14025:of a matrix. Let
13923:
13694:
13677:{\displaystyle b}
13657:{\displaystyle a}
13463:
13446:
13432:
13088:{\displaystyle B}
13039:{\displaystyle A}
12625:Proof of identity
12624:
12594:
12577:
12563:
12442:
12416:
12390:
12284:implies that the
12267:
12197:
12099:{\displaystyle C}
12079:{\displaystyle B}
12059:{\displaystyle A}
11943:{\displaystyle B}
11923:{\displaystyle A}
11265:{\displaystyle B}
11245:{\displaystyle A}
11017:{\displaystyle D}
10997:{\displaystyle C}
10658:
10656:
10538:{\displaystyle D}
10489:{\displaystyle D}
10101:
10099:
9977:{\displaystyle A}
9622:
9316:
9204:
8767:{\displaystyle j}
8354:{\displaystyle i}
8272:{\displaystyle j}
8252:{\displaystyle i}
8232:{\displaystyle A}
8122:{\displaystyle A}
8107:Laplace expansion
8102:Laplace expansion
8096:compound matrices
8056:is isomorphic to
7855:{\displaystyle K}
7763:rotation matrices
7678:(respectively, a
7629:{\displaystyle K}
7609:{\displaystyle n}
7549:
7477:{\displaystyle A}
7454:{\displaystyle A}
7411:{\displaystyle A}
7391:{\displaystyle B}
7302:{\displaystyle B}
7282:{\displaystyle A}
7202:
7166:{\displaystyle A}
7049:
7048:
6830:
6396:{\displaystyle A}
6268:{\displaystyle A}
6019:triangular matrix
6010:{\displaystyle A}
5524:{\displaystyle A}
4926:{\displaystyle A}
4887:{\displaystyle I}
4712:{\displaystyle n}
4467:
4224:
4088:
3751:of a permutation
3438:
2621:
2559:
2371:{\displaystyle A}
1755:
1504:Geometric meaning
1326:{\displaystyle r}
488:coordinate system
468:matrix similarity
407:Laplace expansion
106:triangular matrix
28123:
28078:
28077:
27960:Exterior algebra
27897:Hadamard product
27814:
27802:Linear equations
27673:
27666:
27659:
27650:
27649:
27611:
27589:
27588:
27570:
27545:
27539:
27531:
27515:
27480:
27458:
27432:
27416:
27403:
27392:
27366:
27344:, translated by
27329:
27311:
27310:
27309:
27303:
27297:, archived from
27278:
27264:
27254:
27245:
27226:
27208:
27190:
27175:(2nd ed.).
27165:Horn, Roger Alan
27144:
27126:
27114:
27109:, archived from
27089:
27062:
27044:
27026:
27002:
26968:
26950:
26948:
26941:
26923:
26913:
26888:
26867:
26866:
26856:
26833:
26813:Abstract algebra
26807:
26805:
26795:
26786:(125): 231–236.
26768:
26750:
26748:
26717:
26704:
26702:
26680:(102): 565–578,
26671:
26661:
26646:(3rd ed.).
26635:
26612:
26611:
26609:
26600:(2–3): 429–438.
26585:
26579:
26570:
26564:
26563:
26561:
26560:
26554:
26548:. Archived from
26525:
26516:
26510:
26505:
26499:
26493:
26487:
26482:
26476:
26475:
26473:
26461:
26455:
26449:
26443:
26436:
26430:
26429:
26409:
26403:
26401:
26399:
26398:
26393:
26391:
26375:
26374:
26330:
26329:
26288:
26287:
26246:
26242:
26241:
26200:
26199:
26124:
26118:
26113:
26107:
26097:
26091:
26085:
26079:
26067:
26061:
26055:
26049:
26043:
26037:
26036:
26028:
26022:
26016:
26010:
26004:
25998:
25992:
25986:
25981:
25975:
25969:
25963:
25957:
25951:
25945:
25939:
25933:
25927:
25921:
25915:
25907:
25901:
25895:
25889:
25875:
25869:
25863:
25857:
25851:
25845:
25844:
25842:
25840:
25831:. Archived from
25824:
25818:
25812:
25806:
25803:
25792:
25783:
25777:
25771:
25765:
25759:
25753:
25752:
25715:
25709:
25703:
25697:
25687:
25681:
25680:
25662:
25656:
25655:
25653:
25629:
25623:
25622:
25620:
25608:
25602:
25596:
25590:
25589:
25571:
25551:
25545:
25544:
25519:(501): 460–467.
25504:
25498:
25492:
25486:
25480:
25474:
25468:
25462:
25451:
25445:
25439:
25433:
25423:
25417:
25411:
25405:
25404:
25398:
25388:
25382:
25381:
25379:
25377:
25363:
25357:
25356:
25338:
25332:
25326:
25283:
25278:
25277:
25237:
25212:
25210:
25209:
25204:
25188:
25186:
25185:
25180:
25175:
25174:
25137:
25135:
25134:
25129:
25124:
25123:
25090:
25088:
25087:
25082:
25077:
25076:
25051:
25049:
25048:
25043:
25013:
25011:
25010:
25005:
24987:
24985:
24984:
24979:
24977:
24976:
24945:
24943:
24942:
24937:
24916:
24914:
24913:
24908:
24896:
24894:
24893:
24888:
24883:
24882:
24849:
24847:
24846:
24841:
24781:
24779:
24778:
24773:
24761:
24759:
24758:
24753:
24738:
24736:
24735:
24730:
24715:
24713:
24712:
24707:
24695:
24693:
24692:
24687:
24675:
24673:
24672:
24667:
24655:
24653:
24652:
24647:
24635:
24633:
24632:
24627:
24615:
24613:
24612:
24607:
24595:
24593:
24592:
24587:
24575:
24573:
24572:
24567:
24549:
24547:
24546:
24541:
24511:
24509:
24508:
24503:
24488:
24486:
24485:
24480:
24453:
24451:
24450:
24445:
24440:
24439:
24405:QR decomposition
24401:LU decomposition
24398:
24396:
24395:
24390:
24354:
24349:row echelon form
24331:
24329:
24328:
24323:
24305:
24303:
24302:
24297:
24276:
24274:
24273:
24268:
24256:
24254:
24253:
24248:
24206:
24204:
24203:
24198:
24196:
24195:
24190:
24141:
24133:
24087:
24085:
24084:
24079:
23958:
23956:
23955:
23950:
23936:
23935:
23931:
23904:
23902:
23901:
23896:
23894:
23893:
23881:
23880:
23868:
23867:
23855:
23854:
23789:
23787:
23786:
23781:
23767:
23766:
23731:
23729:
23728:
23723:
23693:
23691:
23690:
23685:
23669:
23667:
23666:
23661:
23638:
23636:
23635:
23630:
23628:
23627:
23600:
23598:
23597:
23592:
23556:
23554:
23553:
23548:
23530:
23528:
23527:
23522:
23516:
23508:
23492:
23490:
23489:
23484:
23472:
23470:
23469:
23464:
23458:
23450:
23442:
23440:
23439:
23434:
23432:
23431:
23415:
23413:
23412:
23407:
23392:
23390:
23389:
23384:
23379:
23378:
23360:
23359:
23332:
23328:
23327:
23326:
23308:
23307:
23293:
23289:
23284:
23276:
23260:
23258:
23257:
23252:
23244:
23243:
23227:
23225:
23224:
23219:
23199:
23197:
23196:
23191:
23179:
23177:
23176:
23171:
23165:
23157:
23149:
23147:
23146:
23141:
23135:
23127:
23116:
23114:
23113:
23108:
23106:
23099:
23098:
23077:
23076:
23061:
23060:
23041:
23040:
23022:
23021:
23009:
23008:
22991:
22983:
22971:
22963:
22955:
22947:
22932:
22930:
22929:
22924:
22912:
22910:
22909:
22904:
22892:
22890:
22889:
22884:
22878:
22870:
22859:
22857:
22856:
22851:
22839:
22837:
22836:
22831:
22819:
22817:
22816:
22811:
22792:
22790:
22789:
22784:
22772:
22770:
22769:
22764:
22752:
22750:
22749:
22744:
22711:Exterior algebra
22706:
22704:
22703:
22698:
22693:
22692:
22687:
22678:
22677:
22647:algebraic groups
22644:
22642:
22641:
22636:
22634:
22633:
22611:
22609:
22608:
22603:
22601:
22600:
22572:
22570:
22569:
22564:
22552:
22550:
22549:
22544:
22522:
22520:
22519:
22514:
22503:
22502:
22460:
22459:
22419:
22417:
22416:
22411:
22399:
22397:
22396:
22391:
22379:
22377:
22376:
22371:
22357:
22356:
22332:
22331:
22307:
22306:
22290:
22288:
22287:
22282:
22240:
22238:
22237:
22232:
22216:
22214:
22213:
22208:
22196:
22194:
22193:
22188:
22163:
22161:
22160:
22155:
22150:
22149:
22125:
22124:
22095:
22093:
22092:
22087:
22085:
22067:
22065:
22064:
22059:
22043:
22041:
22040:
22035:
22023:
22021:
22020:
22015:
22001:
22000:
21969:
21967:
21966:
21961:
21930:
21928:
21927:
21922:
21920:
21908:
21906:
21905:
21900:
21888:commutative ring
21851:
21849:
21848:
21843:
21806:
21804:
21803:
21798:
21766:
21765:
21705:
21704:
21658:
21632:similar matrices
21602:
21600:
21599:
21594:
21589:
21580:
21576:
21572:
21535:
21518:
21517:
21505:
21493:
21482:
21481:
21453:
21401:
21399:
21398:
21393:
21388:
21387:
21364:
21360:
21358:
21357:
21356:
21343:
21342:
21341:
21328:
21293:
21280:
21278:
21277:
21272:
21267:
21266:
21261:
21252:
21251:
21246:
21207:
21205:
21204:
21199:
21197:
21196:
21191:
21182:
21181:
21176:
21158:A nonlinear map
21146:
21144:
21143:
21138:
21136:
21089:
21081:
21073:
21064:
21062:
21061:
21056:
21012:
21010:
21009:
21004:
20984:
20982:
20978:
20974:
20973:
20972:
20954:
20915:
20913:
20912:
20907:
20882:
20880:
20879:
20874:
20862:
20860:
20859:
20854:
20842:
20840:
20839:
20834:
20816:
20814:
20813:
20808:
20806:
20805:
20800:
20791:
20790:
20785:
20766:
20764:
20763:
20758:
20746:
20744:
20743:
20738:
20736:
20719:
20707:
20705:
20704:
20699:
20671:
20669:
20668:
20663:
20661:
20660:
20655:
20636:
20634:
20633:
20628:
20616:
20614:
20613:
20608:
20606:
20605:
20600:
20591:
20590:
20585:
20542:
20538:
20504:with entries in
20433:
20431:
20430:
20425:
20407:
20405:
20404:
20399:
20384:
20382:
20381:
20376:
20371:
20370:
20353:
20336:
20310:
20293:
20272:
20255:
20207:
20178:
20154:
20132:
20131:
20106:
20105:
20085:
20084:
20052:
20051:
20033:
20032:
19999:
19997:
19996:
19991:
19974:(supposed to be
19973:
19971:
19970:
19965:
19954:
19953:
19926:
19925:
19905:
19903:
19902:
19897:
19872:
19870:
19869:
19864:
19862:
19861:
19845:
19843:
19842:
19837:
19835:
19834:
19829:
19820:
19819:
19807:
19806:
19790:
19788:
19787:
19782:
19770:
19768:
19767:
19762:
19723:
19721:
19720:
19715:
19710:
19709:
19681:
19679:
19678:
19673:
19668:
19667:
19587:
19585:
19584:
19579:
19577:
19576:
19557:
19555:
19554:
19549:
19535:
19534:
19522:
19521:
19514:
19513:
19497:
19496:
19479:
19478:
19467:
19466:
19444:
19443:
19423:
19422:
19412:
19407:
19389:
19388:
19381:
19380:
19354:
19353:
19327:
19326:
19301:
19299:
19298:
19293:
19281:
19279:
19278:
19273:
19261:
19259:
19258:
19253:
19241:
19239:
19238:
19233:
19231:
19230:
19211:
19209:
19208:
19203:
19167:
19165:
19151:
19147:
19146:
19130:
19125:
19124:
19101:
19099:
19098:
19093:
19072:
19070:
19069:
19064:
19052:
19050:
19049:
19044:
18921:Crelle's Journal
18899:
18789:
18787:
18786:
18781:
18779:
18772:
18764:
18740:
18739:
18738:
18724:
18716:
18692:
18691:
18690:
18676:
18668:
18644:
18643:
18642:
18614:
18612:
18611:
18606:
18582:
18580:
18579:
18574:
18572:
18571:
18530:
18528:
18527:
18522:
18501:
18499:
18498:
18493:
18491:
18490:
18485:
18484:
18464:
18462:
18461:
18456:
18435:
18433:
18432:
18427:
18425:
18424:
18397:
18395:
18394:
18389:
18384:
18380:
18379:
18302:
18300:
18299:
18294:
18268:
18266:
18265:
18260:
18258:
18254:
18253:
18231:
18227:
18223:
18222:
18184:
18180:
18179:
18075:
18073:
18072:
18067:
18062:
18061:
18053:
18049:
18048:
18016:
18015:
17988:
17986:
17985:
17984:
17968:
17951:
17939:
17937:
17936:
17931:
17916:
17914:
17913:
17908:
17903:
17899:
17898:
17896:
17888:
17880:
17878:
17877:
17839:
17837:
17829:
17812:
17800:
17798:
17797:
17792:
17780:
17778:
17777:
17772:
17756:
17754:
17753:
17748:
17721:
17719:
17718:
17713:
17708:
17704:
17703:
17701:
17693:
17685:
17654:
17652:
17644:
17627:
17614:Jacobi's formula
17607:
17605:
17604:
17599:
17597:
17585:
17583:
17582:
17577:
17575:
17574:
17563:
17537:root mean square
17519:
17517:
17516:
17511:
17506:
17504:
17500:
17499:
17480:
17472:
17470:
17450:
17442:
17437:
17436:
17428:
17410:
17408:
17407:
17403:
17402:
17376:
17353:
17340:
17338:
17337:
17332:
17285:
17281:
17280:
17279:
17240:
17226:
17214:
17204:
17200:
17189:
17187:
17186:
17181:
17175:
17174:
17169:
17165:
17164:
17160:
17159:
17140:
17135:
17134:
17133:
17124:
17123:
17104:
17101:
17096:
17072:
17070:
17059:
17056:
17051:
17026:
17017:
17015:
17014:
17009:
17003:
17002:
16997:
16993:
16992:
16988:
16987:
16968:
16963:
16962:
16961:
16942:
16939:
16934:
16910:
16908:
16897:
16894:
16889:
16835:
16828:
16826:
16825:
16820:
16814:
16809:
16799:
16768:
16763:
16753:
16748:
16738:
16725:
16720:
16692:
16664:
16636:
16624:
16622:
16621:
16616:
16608:
16607:
16589:
16588:
16576:
16575:
16563:
16562:
16553:
16551:
16543:
16542:
16541:
16522:
16467:
16465:
16464:
16459:
16448:
16447:
16434:
16429:
16404:
16391:
16389:
16388:
16383:
16378:
16377:
16376:
16375:
16365:
16361:
16360:
16340:
16338:
16334:
16333:
16324:
16323:
16322:
16321:
16306:
16305:
16304:
16297:
16296:
16272:
16269:
16264:
16248:
16238:
16237:
16216:
16215:
16200:
16199:
16180:
16179:
16161:
16160:
16148:
16147:
16102:
16100:
16099:
16094:
16087:
16063:
16060:
16059:
16055:
16054:
16035:
16034:
16012:
16007:
15992:
15984:
15976:
15975:
15958:
15955:
15952:
15942:
15941:
15918:
15904:
15885:
15883:
15882:
15877:
15875:
15868:
15864:
15863:
15859:
15858:
15816:
15815:
15811:
15810:
15785:
15784:
15779:
15775:
15774:
15770:
15769:
15738:
15737:
15732:
15728:
15706:
15702:
15701:
15676:
15675:
15670:
15666:
15639:
15631:
15603:
15599:
15598:
15594:
15593:
15568:
15564:
15563:
15542:
15520:
15519:
15514:
15510:
15483:
15475:
15447:
15443:
15442:
15438:
15437:
15415:
15414:
15409:
15405:
15378:
15370:
15336:, respectively,
15335:
15328:
15321:
15311:
15309:
15308:
15303:
15246:
15239:
15237:
15236:
15231:
15198:
15194:
15186:
15182:
15178:
15174:
15170:
15162:
15155:
15153:
15152:
15147:
15078:
15071:
15069:
15068:
15063:
14997:
14993:
14973:
14971:
14970:
14965:
14953:
14951:
14950:
14945:
14933:
14931:
14930:
14925:
14910:
14908:
14907:
14902:
14900:
14899:
14892:
14891:
14869:
14868:
14851:
14850:
14809:
14808:
14786:
14785:
14768:
14767:
14748:
14747:
14725:
14724:
14707:
14706:
14680:
14679:
14651:Hermitian matrix
14645:
14643:
14642:
14637:
14620:
14619:
14600:
14598:
14597:
14592:
14576:
14574:
14573:
14568:
14552:
14550:
14549:
14544:
14532:
14530:
14529:
14524:
14508:
14506:
14505:
14500:
14485:
14483:
14482:
14477:
14436:
14435:
14413:
14411:
14410:
14405:
14393:
14391:
14390:
14385:
14373:
14371:
14370:
14365:
14353:
14351:
14350:
14345:
14333:
14331:
14330:
14325:
14313:
14311:
14310:
14305:
14283:
14281:
14280:
14275:
14270:
14269:
14257:
14256:
14247:
14246:
14234:
14233:
14223:
14218:
14175:is equal to the
14174:
14170:
14166:
14159:
14157:
14156:
14151:
14149:
14148:
14130:
14129:
14117:
14116:
14094:
14092:
14091:
14086:
14070:
14068:
14067:
14062:
14044:
14042:
14041:
14036:
14003:
14001:
14000:
13995:
13984:
13951:
13949:
13948:
13943:
13932:
13924:
13921:
13909:
13907:
13906:
13901:
13896:
13895:
13883:
13882:
13867:
13856:
13855:
13831:
13830:
13815:
13782:
13780:
13779:
13774:
13772:
13751:
13749:
13748:
13743:
13722:
13720:
13719:
13714:
13703:
13695:
13692:
13683:
13681:
13680:
13675:
13663:
13661:
13660:
13655:
13643:
13641:
13640:
13635:
13633:
13632:
13588:
13587:
13546:
13522:
13520:
13519:
13514:
13489:
13487:
13486:
13481:
13464:
13461:
13447:
13444:
13433:
13430:
13415:
13413:
13412:
13407:
13402:
13401:
13392:
13391:
13379:
13378:
13369:
13368:
13356:
13355:
13346:
13345:
13333:
13332:
13323:
13322:
13304:
13303:
13291:
13290:
13275:
13274:
13262:
13261:
13236:
13234:
13233:
13228:
13223:
13222:
13213:
13212:
13200:
13199:
13190:
13189:
13177:
13176:
13167:
13166:
13154:
13153:
13144:
13143:
13094:
13092:
13091:
13086:
13074:
13072:
13071:
13066:
13045:
13043:
13042:
13037:
13022:
13020:
13019:
13014:
13009:
13008:
12999:
12998:
12986:
12985:
12976:
12975:
12963:
12962:
12953:
12952:
12940:
12939:
12930:
12929:
12917:
12916:
12907:
12906:
12894:
12893:
12884:
12883:
12871:
12870:
12861:
12860:
12848:
12847:
12838:
12837:
12819:Expanding gives
12815:
12813:
12812:
12807:
12799:
12798:
12786:
12785:
12770:
12769:
12757:
12756:
12738:
12737:
12725:
12724:
12709:
12708:
12696:
12695:
12673:
12671:
12670:
12665:
12663:
12662:
12647:
12646:
12620:
12618:
12617:
12612:
12595:
12592:
12578:
12575:
12564:
12561:
12498:
12496:
12495:
12490:
12460:
12456:
12454:
12453:
12448:
12443:
12441:
12436:
12422:
12417:
12415:
12410:
12396:
12391:
12389:
12384:
12364:
12355:
12353:
12352:
12347:
12329:
12325:
12321:
12319:
12318:
12313:
12290:concave function
12287:
12278:
12276:
12275:
12270:
12268:
12265:
12208:
12206:
12205:
12200:
12198:
12195:
12105:
12103:
12102:
12097:
12085:
12083:
12082:
12077:
12065:
12063:
12062:
12057:
12031:
12029:
12028:
12023:
11995:
11993:
11992:
11987:
11972:
11970:
11969:
11964:
11949:
11947:
11946:
11941:
11929:
11927:
11926:
11921:
11906:
11904:
11903:
11898:
11852:
11848:
11844:
11843:
11767:
11765:
11764:
11759:
11754:
11750:
11746:
11745:
11727:
11726:
11725:
11659:
11645:
11643:
11642:
11637:
11617:
11613:
11603:
11602:
11601:
11554:
11544:
11513:
11511:
11510:
11505:
11500:
11496:
11486:
11485:
11484:
11463:
11459:
11449:
11448:
11447:
11412:matrix (so that
11411:
11397:
11384:states that for
11372:
11370:
11369:
11364:
11320:
11319:
11272:do not commute)
11271:
11269:
11268:
11263:
11251:
11249:
11248:
11243:
11231:
11229:
11228:
11223:
11205:
11203:
11202:
11197:
11176:
11174:
11173:
11168:
11147:
11145:
11144:
11139:
11107:
11106:
11058:
11056:
11055:
11050:
11023:
11021:
11020:
11015:
11003:
11001:
11000:
10995:
10969:
10967:
10966:
10961:
10959:
10946:
10945:
10900:
10896:
10895:
10888:
10887:
10869:
10868:
10848:
10847:
10797:
10792:
10791:
10790:
10761:
10760:
10737:
10732:
10731:
10730:
10723:
10722:
10705:
10704:
10680:
10679:
10655:
10654:
10597:
10596:
10544:
10542:
10541:
10536:
10521:
10519:
10518:
10513:
10495:
10493:
10492:
10487:
10475:
10473:
10472:
10467:
10459:
10458:
10412:
10410:
10409:
10404:
10402:
10389:
10388:
10343:
10339:
10338:
10328:
10327:
10304:
10303:
10279:
10278:
10240:
10235:
10234:
10233:
10204:
10203:
10180:
10175:
10174:
10173:
10166:
10165:
10144:
10143:
10126:
10125:
10098:
10097:
10040:
10039:
9983:
9981:
9980:
9975:
9960:
9958:
9957:
9952:
9947:
9946:
9878:
9877:
9828:Schur complement
9825:
9823:
9822:
9817:
9799:
9797:
9796:
9791:
9773:
9771:
9770:
9765:
9747:
9745:
9744:
9739:
9721:
9719:
9718:
9713:
9679:
9677:
9676:
9671:
9645:
9643:
9642:
9637:
9623:
9621:
9610:
9605:
9604:
9575:
9573:
9572:
9567:
9497:
9495:
9494:
9489:
9484:
9483:
9471:
9470:
9443:
9442:
9399:
9397:
9396:
9391:
9334:
9332:
9331:
9326:
9324:
9322:
9321:
9308:
9277:
9275:
9274:
9269:
9264:
9260:
9259:
9258:
9246:
9245:
9230:
9200:
9199:
9191:
9180:
9163:
9152:
9140:
9129:
9117:
9106:
9065:
9060:
9043:
9038:
9026:
9021:
9009:
9004:
8991:
8990:
8974:
8973:
8962:
8961:
8950:
8949:
8891:
8889:
8888:
8883:
8878:
8877:
8862:
8861:
8846:
8845:
8820:
8815:
8776:is the equality
8773:
8771:
8770:
8765:
8751:
8749:
8748:
8743:
8722:
8720:
8719:
8714:
8712:
8711:
8670:
8669:
8628:
8627:
8586:
8585:
8513:
8511:
8510:
8505:
8485:
8476:
8474:
8473:
8468:
8463:
8462:
8447:
8446:
8431:
8430:
8405:
8400:
8360:
8358:
8357:
8352:
8336:
8334:
8333:
8328:
8326:
8325:
8310:
8309:
8278:
8276:
8275:
8270:
8259:-th row and the
8258:
8256:
8255:
8250:
8239:by removing the
8238:
8236:
8235:
8230:
8218:
8216:
8215:
8210:
8168:
8166:
8165:
8160:
8158:
8157:
8128:
8126:
8125:
8120:
8082:
8080:
8079:
8074:
8072:
8071:
8055:
8053:
8052:
8047:
8033:
8032:
8023:
8006:
8005:
7986:
7984:
7983:
7978:
7964:
7963:
7943:
7941:
7940:
7935:
7921:
7920:
7900:
7898:
7897:
7892:
7878:
7877:
7861:
7859:
7858:
7853:
7841:
7839:
7838:
7833:
7831:
7830:
7814:
7812:
7811:
7806:
7792:
7791:
7748:
7746:
7745:
7740:
7726:
7725:
7701:
7700:
7677:
7675:
7674:
7669:
7655:
7654:
7635:
7633:
7632:
7627:
7615:
7613:
7612:
7607:
7591:
7589:
7588:
7583:
7581:
7580:
7550:
7548:
7531:
7526:
7522:
7521:
7483:
7481:
7480:
7475:
7460:
7458:
7457:
7452:
7440:
7438:
7437:
7432:
7417:
7415:
7414:
7409:
7397:
7395:
7394:
7389:
7374:
7372:
7371:
7366:
7308:
7306:
7305:
7300:
7288:
7286:
7285:
7280:
7248:
7234:
7232:
7231:
7226:
7209:
7205:
7204:
7203:
7172:
7170:
7169:
7164:
7141:
7139:
7138:
7133:
7089:
7081:
7070:
7062:
7044:
7042:
7041:
7036:
7034:
7026:
7018:
7010:
6995:
6993:
6992:
6987:
6985:
6977:
6966:
6958:
6943:
6941:
6940:
6935:
6933:
6925:
6917:
6909:
6895:
6893:
6892:
6887:
6885:
6877:
6869:
6861:
6841:
6839:
6838:
6833:
6831:
6823:
6787:
6785:
6784:
6779:
6777:
6776:
6692:
6690:
6689:
6684:
6682:
6681:
6597:
6595:
6594:
6589:
6587:
6586:
6503:
6501:
6500:
6495:
6493:
6492:
6404:
6402:
6400:
6399:
6394:
6381:
6378:
6376:
6375:
6370:
6365:
6364:
6274:
6272:
6271:
6266:
6240:
6238:
6237:
6232:
6216:
6214:
6213:
6208:
6203:
6202:
6189:
6184:
6166:
6165:
6150:
6149:
6140:
6139:
6108:
6106:
6105:
6100:
6082:
6080:
6079:
6074:
6056:
6054:
6053:
6048:
6040:
6039:
6016:
6014:
6013:
6008:
5970:
5968:
5967:
5962:
5957:
5952:
5951:
5933:
5932:
5920:
5919:
5907:
5906:
5894:
5893:
5884:
5876:
5871:
5870:
5852:
5851:
5839:
5838:
5826:
5825:
5813:
5812:
5803:
5792:
5787:
5786:
5771:
5770:
5758:
5757:
5745:
5744:
5732:
5731:
5722:
5710:
5708:
5707:
5702:
5697:
5692:
5691:
5673:
5672:
5654:
5653:
5635:
5634:
5625:
5614:
5609:
5608:
5590:
5589:
5574:
5573:
5555:
5554:
5545:
5530:
5528:
5527:
5522:
5510:
5508:
5507:
5502:
5484:
5482:
5481:
5476:
5462:
5461:
5399:
5397:
5396:
5391:
5367:
5365:
5364:
5359:
5351:
5346:
5345:
5303:
5302:
5293:
5267:
5265:
5264:
5259:
5257:
5253:
5248:
5247:
5217:
5216:
5207:
5199:
5194:
5193:
5166:
5165:
5156:
5139:
5135:
5130:
5129:
5111:
5110:
5074:
5073:
5049:
5048:
5039:
5038:
5025:
5017:
4980:
4978:
4977:
4972:
4952:
4951:
4933:is written as a
4932:
4930:
4929:
4924:
4893:
4891:
4890:
4885:
4873:
4871:
4870:
4865:
4857:
4823:
4821:
4820:
4815:
4813:
4812:
4790:
4788:
4787:
4782:
4777:
4776:
4770:
4769:
4751:
4750:
4741:
4740:
4718:
4716:
4715:
4710:
4694:
4692:
4691:
4686:
4656:
4654:
4653:
4648:
4615:
4608:
4606:
4605:
4600:
4595:
4594:
4593:
4592:
4568:
4567:
4566:
4565:
4545:
4544:
4543:
4542:
4530:
4529:
4514:
4513:
4512:
4494:
4493:
4481:
4480:
4437:
4433:
4431:
4430:
4425:
4391:
4387:
4385:
4384:
4379:
4377:
4376:
4375:
4374:
4356:
4355:
4327:
4325:
4324:
4319:
4317:
4313:
4312:
4311:
4286:
4281:
4245:
4244:
4243:
4191:
4189:
4188:
4183:
4178:
4177:
4150:
4149:
4109:
4108:
4107:
4084:
4083:
4076:
4075:
4057:
4056:
4026:
4025:
4007:
4006:
3951:
3949:
3948:
3943:
3938:
3937:
3930:
3929:
3911:
3910:
3880:
3879:
3861:
3860:
3819:
3817:
3816:
3811:
3796:
3794:
3793:
3788:
3770:
3768:
3767:
3762:
3750:
3748:
3747:
3742:
3719:. The signature
3718:
3716:
3715:
3710:
3708:
3707:
3687:
3685:
3684:
3679:
3622:
3620:
3619:
3614:
3599:
3597:
3596:
3591:
3547:
3545:
3544:
3539:
3506:
3448:
3446:
3445:
3440:
3436:
3360:
3359:
3287:
3249:
3247:
3246:
3241:
3236:
3235:
3228:
3227:
3205:
3204:
3187:
3186:
3145:
3144:
3122:
3121:
3104:
3103:
3084:
3083:
3061:
3060:
3043:
3042:
2997:commutative ring
2994:
2992:
2991:
2986:
2984:
2983:
2958:
2956:
2955:
2950:
2945:
2944:
2937:
2936:
2914:
2913:
2896:
2895:
2854:
2853:
2831:
2830:
2813:
2812:
2793:
2792:
2770:
2769:
2752:
2751:
2644:
2642:
2641:
2636:
2622:
2619:
2583:
2581:
2580:
2575:
2570:
2566:
2557:
2550:
2549:
2531:
2530:
2525:
2519:
2518:
2500:
2499:
2494:
2488:
2487:
2460:
2458:
2457:
2452:
2447:
2446:
2441:
2426:
2425:
2420:
2411:
2410:
2405:
2377:
2375:
2374:
2369:
2358:This means that
2354:
2352:
2351:
2346:
2341:
2340:
2335:
2326:
2325:
2272:
2271:
2266:
2257:
2256:
2210:
2209:
2204:
2195:
2194:
2142:
2140:
2139:
2134:
2132:
2128:
2124:
2123:
2118:
2105:
2104:
2099:
2091:
2090:
2085:
2051:
2038:
2024:
2004:
1992:
1980:
1940:
1938:
1937:
1932:
1909:
1908:
1880:
1879:
1848:
1835:
1831:
1821:
1817:
1816:
1811:
1790:
1785:
1780:
1774:
1769:
1764:
1756:
1753:
1741:
1727:
1712:
1685:
1670:
1655:
1640:
1612:
1608:
1604:
1591:
1579:
1559:
1547:
1535:
1531:
1519:
1499:
1497:
1496:
1491:
1486:
1485:
1390:
1389:
1332:
1330:
1329:
1324:
1309:
1307:
1306:
1301:
1296:
1295:
1288:
1271:
1247:
1246:
1202:
1179:
1156:
1155:
1148:
1125:
1085:
1083:
1082:
1077:
1051:
1050:
1005:
1003:
1002:
997:
995:
994:
949:
947:
946:
941:
920:First properties
915:
913:
912:
907:
851:
850:
843:
807:
806:
752:
750:
749:
744:
721:
720:
682:
681:
633:
629:
627:
626:
621:
619:
618:
576:
544:
528:
520:
452:
445:
433:
419:row echelon form
404:
396:
394:
393:
388:
369:
356:
354:
353:
348:
271:
270:
198:
191:
189:
188:
183:
160:
159:
114:
80:
78:
70:
63:
55:
28131:
28130:
28126:
28125:
28124:
28122:
28121:
28120:
28091:
28090:
28089:
28084:
28066:
28028:
27984:
27921:
27873:
27815:
27806:
27772:Change of basis
27762:Multilinear map
27700:
27682:
27677:
27627:Wayback Machine
27552:
27533:
27532:
27499:(22): 320–359,
27478:
27466:, vol. 1,
27448:
27382:
27364:
27336:
27327:
27307:
27305:
27301:
27287:
27262:
27242:
27224:
27206:
27187:
27173:Matrix Analysis
27142:
27107:
27087:
27060:
27042:
27024:
26992:
26966:
26946:
26921:
26895:10.2307/4145188
26823:
26766:
26746:
26715:
26700:
26686:10.2307/2004533
26669:
26658:
26626:
26620:
26615:
26586:
26582:
26571:
26567:
26558:
26556:
26552:
26546:
26523:
26517:
26513:
26506:
26502:
26494:
26490:
26483:
26479:
26462:
26458:
26450:
26446:
26437:
26433:
26426:
26410:
26406:
26389:
26388:
26369:
26368:
26363:
26357:
26356:
26351:
26341:
26340:
26324:
26323:
26318:
26312:
26311:
26306:
26296:
26295:
26282:
26281:
26276:
26270:
26269:
26264:
26254:
26253:
26244:
26243:
26236:
26235:
26230:
26224:
26223:
26218:
26208:
26207:
26194:
26193:
26188:
26182:
26181:
26176:
26166:
26165:
26152:
26142:
26140:
26137:
26136:
26125:
26121:
26114:
26110:
26098:
26094:
26086:
26082:
26068:
26064:
26056:
26052:
26044:
26040:
26029:
26025:
26017:
26013:
26005:
26001:
25993:
25989:
25982:
25978:
25970:
25966:
25958:
25954:
25946:
25942:
25934:
25930:
25922:
25918:
25908:
25904:
25896:
25892:
25876:
25872:
25864:
25860:
25852:
25848:
25838:
25836:
25827:
25825:
25821:
25813:
25809:
25804:
25795:
25784:
25780:
25772:
25768:
25764:, § 0.8.10
25760:
25756:
25716:
25712:
25704:
25700:
25688:
25684:
25672:
25663:
25659:
25630:
25626:
25609:
25605:
25597:
25593:
25552:
25548:
25525:10.2307/3620776
25505:
25501:
25493:
25489:
25481:
25477:
25469:
25465:
25453:Alternatively,
25452:
25448:
25440:
25436:
25424:
25420:
25412:
25408:
25389:
25385:
25375:
25373:
25365:
25364:
25360:
25339:
25335:
25327:
25323:
25319:
25314:
25279:
25272:
25269:
25229:
25198:
25195:
25194:
25170:
25166:
25155:
25152:
25151:
25119:
25115:
25104:
25101:
25100:
25072:
25068:
25057:
25054:
25053:
25019:
25016:
25015:
24993:
24990:
24989:
24972:
24968:
24951:
24948:
24947:
24922:
24919:
24918:
24902:
24899:
24898:
24878:
24874:
24863:
24860:
24859:
24856:
24854:Further methods
24790:
24787:
24786:
24767:
24764:
24763:
24744:
24741:
24740:
24721:
24718:
24717:
24701:
24698:
24697:
24681:
24678:
24677:
24661:
24658:
24657:
24641:
24638:
24637:
24621:
24618:
24617:
24601:
24598:
24597:
24581:
24578:
24577:
24561:
24558:
24557:
24520:
24517:
24516:
24497:
24494:
24493:
24459:
24456:
24455:
24435:
24431:
24420:
24417:
24416:
24375:
24372:
24371:
24368:
24352:
24338:
24314:
24311:
24310:
24285:
24282:
24281:
24262:
24259:
24258:
24239:
24236:
24235:
24221:
24191:
24186:
24185:
24183:
24180:
24179:
24144:regular element
24139:
24128:
24125:
24101:
24007:
24004:
24003:
23981:
23961:field extension
23927:
23923:
23919:
23917:
23914:
23913:
23889:
23885:
23876:
23872:
23863:
23859:
23850:
23846:
23802:
23799:
23798:
23756:
23752:
23744:
23741:
23740:
23702:
23699:
23698:
23679:
23676:
23675:
23655:
23652:
23651:
23645:
23623:
23619:
23617:
23614:
23613:
23611:symmetric group
23574:
23571:
23570:
23563:
23536:
23533:
23532:
23512:
23506:
23503:
23502:
23499:chain complexes
23478:
23475:
23474:
23454:
23448:
23445:
23444:
23427:
23423:
23421:
23418:
23417:
23401:
23398:
23397:
23374:
23370:
23355:
23351:
23322:
23318:
23303:
23299:
23298:
23294:
23280:
23275:
23271:
23269:
23266:
23265:
23239:
23235:
23233:
23230:
23229:
23213:
23210:
23209:
23185:
23182:
23181:
23161:
23155:
23152:
23151:
23131:
23125:
23122:
23121:
23104:
23103:
23094:
23090:
23072:
23068:
23056:
23052:
23042:
23036:
23032:
23017:
23013:
23004:
23000:
22997:
22996:
22987:
22976:
22967:
22951:
22943:
22941:
22938:
22937:
22918:
22915:
22914:
22898:
22895:
22894:
22874:
22868:
22865:
22864:
22845:
22842:
22841:
22825:
22822:
22821:
22805:
22802:
22801:
22778:
22775:
22774:
22758:
22755:
22754:
22726:
22723:
22722:
22719:
22713:
22688:
22683:
22682:
22673:
22669:
22661:
22658:
22657:
22629:
22625:
22617:
22614:
22613:
22596:
22592:
22590:
22587:
22586:
22579:category theory
22558:
22555:
22554:
22538:
22535:
22534:
22492:
22488:
22449:
22445:
22428:
22425:
22424:
22405:
22402:
22401:
22385:
22382:
22381:
22352:
22348:
22327:
22323:
22302:
22298:
22296:
22293:
22292:
22264:
22261:
22260:
22226:
22223:
22222:
22202:
22199:
22198:
22176:
22173:
22172:
22145:
22141:
22120:
22116:
22114:
22111:
22110:
22081:
22073:
22070:
22069:
22053:
22050:
22049:
22029:
22026:
22025:
21990:
21986:
21978:
21975:
21974:
21940:
21937:
21936:
21916:
21914:
21911:
21910:
21894:
21891:
21890:
21884:
21825:
21822:
21821:
21758:
21754:
21697:
21693:
21667:
21664:
21663:
21647:
21628:
21623:
21585:
21568:
21546:
21542:
21531:
21513:
21509:
21501:
21489:
21468:
21464:
21462:
21459:
21458:
21441:
21428:(the domain of
21365:
21352:
21348:
21344:
21337:
21333:
21329:
21327:
21323:
21322:
21305:
21302:
21301:
21285:
21262:
21257:
21256:
21247:
21242:
21241:
21233:
21230:
21229:
21219:Jacobian matrix
21192:
21187:
21186:
21177:
21172:
21171:
21163:
21160:
21159:
21132:
21085:
21072:
21070:
21067:
21066:
21032:
21029:
21028:
20968:
20967:
20963:
20962:
20958:
20953:
20924:
20921:
20920:
20892:
20889:
20888:
20868:
20865:
20864:
20848:
20845:
20844:
20822:
20819:
20818:
20801:
20796:
20795:
20786:
20781:
20780:
20772:
20769:
20768:
20752:
20749:
20748:
20732:
20715:
20713:
20710:
20709:
20684:
20681:
20680:
20656:
20651:
20650:
20642:
20639:
20638:
20622:
20619:
20618:
20601:
20596:
20595:
20586:
20581:
20580:
20572:
20569:
20568:
20557:
20540:
20536:
20514:Euclidean space
20466:
20460:
20413:
20410:
20409:
20393:
20390:
20389:
20365:
20364:
20337:
20332:
20326:
20321:
20294:
20289:
20283:
20256:
20251:
20244:
20243:
20238:
20233:
20228:
20222:
20221:
20203:
20197:
20192:
20174:
20168:
20150:
20143:
20142:
20127:
20123:
20121:
20116:
20101:
20097:
20095:
20080:
20076:
20069:
20068:
20047:
20043:
20028:
20024:
20016:
20013:
20012:
19979:
19976:
19975:
19949:
19945:
19921:
19917:
19915:
19912:
19911:
19885:
19882:
19881:
19857:
19853:
19851:
19848:
19847:
19830:
19825:
19824:
19815:
19811:
19802:
19798:
19796:
19793:
19792:
19776:
19773:
19772:
19753:
19750:
19749:
19742:
19705:
19701:
19690:
19687:
19686:
19663:
19659:
19600:
19597:
19596:
19572:
19568:
19566:
19563:
19562:
19530:
19526:
19516:
19515:
19509:
19505:
19503:
19498:
19486:
19482:
19480:
19474:
19470:
19468:
19456:
19452:
19450:
19445:
19439:
19435:
19428:
19427:
19418:
19414:
19408:
19397:
19383:
19382:
19376:
19372:
19370:
19365:
19360:
19355:
19349:
19345:
19338:
19337:
19322:
19318:
19310:
19307:
19306:
19287:
19284:
19283:
19267:
19264:
19263:
19247:
19244:
19243:
19226:
19222:
19220:
19217:
19216:
19152:
19142:
19138:
19131:
19129:
19120:
19116:
19114:
19111:
19110:
19078:
19075:
19074:
19073:if and only if
19058:
19055:
19054:
19029:
19026:
19025:
19018:
19013:
18891:
18796:
18777:
18776:
18768:
18760:
18753:
18734:
18733:
18729:
18726:
18725:
18720:
18712:
18705:
18686:
18685:
18681:
18678:
18677:
18672:
18664:
18657:
18638:
18637:
18633:
18629:
18627:
18624:
18623:
18588:
18585:
18584:
18566:
18565:
18560:
18555:
18545:
18544:
18536:
18533:
18532:
18510:
18507:
18506:
18486:
18477:
18476:
18475:
18473:
18470:
18469:
18441:
18438:
18437:
18420:
18416:
18414:
18411:
18410:
18375:
18371:
18367:
18311:
18308:
18307:
18282:
18279:
18278:
18249:
18245:
18241:
18215:
18211:
18210:
18206:
18175:
18171:
18167:
18087:
18084:
18083:
18054:
18041:
18037:
18033:
18032:
18008:
18004:
17977:
17973:
17969:
17952:
17950:
17948:
17945:
17944:
17925:
17922:
17921:
17889:
17881:
17879:
17870:
17866:
17865:
17861:
17830:
17813:
17811:
17809:
17806:
17805:
17786:
17783:
17782:
17766:
17763:
17762:
17730:
17727:
17726:
17694:
17686:
17684:
17668:
17664:
17645:
17628:
17626:
17624:
17621:
17620:
17593:
17591:
17588:
17587:
17564:
17559:
17558:
17556:
17553:
17552:
17545:
17533:arithmetic mean
17495:
17491:
17487:
17471:
17469:
17441:
17427:
17423:
17395:
17391:
17387:
17380:
17375:
17373:
17370:
17369:
17362:distributions.
17345:
17272:
17268:
17261:
17257:
17249:
17246:
17245:
17236:
17233:
17216:
17206:
17202:
17198:
17170:
17155:
17151:
17147:
17129:
17125:
17119:
17115:
17105:
17103:
17097:
17086:
17078:
17074:
17073:
17063:
17058:
17052:
17041:
17035:
17032:
17031:
17022:
16998:
16983:
16979:
16975:
16957:
16953:
16943:
16941:
16935:
16924:
16916:
16912:
16911:
16901:
16896:
16890:
16879:
16852:
16849:
16848:
16838:Mercator series
16833:
16810:
16805:
16795:
16764:
16759:
16749:
16744:
16734:
16721:
16716:
16701:
16698:
16697:
16689:
16683:
16676:
16666:
16661:
16655:
16648:
16638:
16634:
16629:
16603:
16599:
16584:
16580:
16571:
16567:
16558:
16554:
16544:
16537:
16533:
16523:
16521:
16504:
16501:
16500:
16487:
16473:Bell polynomial
16443:
16439:
16430:
16419:
16413:
16410:
16409:
16401:
16396:
16371:
16367:
16366:
16356:
16352:
16348:
16347:
16329:
16325:
16317:
16313:
16312:
16308:
16307:
16292:
16288:
16287:
16283:
16273:
16271:
16265:
16254:
16246:
16245:
16233:
16229:
16211:
16207:
16195:
16191:
16188:
16187:
16175:
16171:
16156:
16152:
16143:
16139:
16135:
16114:
16111:
16110:
16050:
16046:
16042:
16018:
16014:
16008:
15997:
15983:
15965:
15961:
15937:
15933:
15931:
15928:
15927:
15917:
15906:
15902:
15873:
15872:
15854:
15850:
15846:
15806:
15802:
15798:
15780:
15765:
15761:
15757:
15750:
15746:
15745:
15733:
15712:
15708:
15707:
15697:
15693:
15689:
15671:
15650:
15646:
15645:
15644:
15640:
15630:
15623:
15608:
15607:
15589:
15585:
15581:
15559:
15555:
15551:
15515:
15494:
15490:
15489:
15488:
15484:
15474:
15467:
15452:
15451:
15433:
15429:
15425:
15410:
15389:
15385:
15384:
15383:
15379:
15369:
15362:
15346:
15344:
15341:
15340:
15330:
15323:
15316:
15255:
15252:
15251:
15244:
15207:
15204:
15203:
15196:
15192:
15184:
15180:
15176:
15172:
15168:
15160:
15087:
15084:
15083:
15076:
15006:
15003:
15002:
14995:
14991:
14980:
14959:
14956:
14955:
14939:
14936:
14935:
14919:
14916:
14915:
14894:
14893:
14881:
14877:
14875:
14870:
14858:
14854:
14852:
14840:
14836:
14833:
14832:
14827:
14822:
14817:
14811:
14810:
14798:
14794:
14792:
14787:
14775:
14771:
14769:
14757:
14753:
14750:
14749:
14737:
14733:
14731:
14726:
14714:
14710:
14708:
14696:
14692:
14685:
14684:
14675:
14671:
14669:
14666:
14665:
14615:
14611:
14609:
14606:
14605:
14586:
14583:
14582:
14562:
14559:
14558:
14538:
14535:
14534:
14518:
14515:
14514:
14494:
14491:
14490:
14431:
14427:
14425:
14422:
14421:
14399:
14396:
14395:
14379:
14376:
14375:
14359:
14356:
14355:
14339:
14336:
14335:
14319:
14316:
14315:
14299:
14296:
14295:
14265:
14261:
14252:
14248:
14242:
14238:
14229:
14225:
14219:
14208:
14187:
14184:
14183:
14172:
14168:
14164:
14144:
14140:
14125:
14121:
14112:
14108:
14106:
14103:
14102:
14080:
14077:
14076:
14050:
14047:
14046:
14030:
14027:
14026:
14015:
14010:
13980:
13957:
13954:
13953:
13928:
13920:
13918:
13915:
13914:
13891:
13887:
13878:
13874:
13863:
13851:
13847:
13826:
13822:
13811:
13791:
13788:
13787:
13768:
13757:
13754:
13753:
13728:
13725:
13724:
13699:
13691:
13689:
13686:
13685:
13669:
13666:
13665:
13649:
13646:
13645:
13627:
13626:
13621:
13615:
13614:
13606:
13596:
13595:
13582:
13581:
13576:
13570:
13569:
13564:
13554:
13553:
13542:
13528:
13525:
13524:
13502:
13499:
13498:
13495:
13460:
13443:
13429:
13427:
13424:
13423:
13397:
13393:
13387:
13383:
13374:
13370:
13364:
13360:
13351:
13347:
13341:
13337:
13328:
13324:
13318:
13314:
13299:
13295:
13286:
13282:
13270:
13266:
13257:
13253:
13248:
13245:
13244:
13218:
13214:
13208:
13204:
13195:
13191:
13185:
13181:
13172:
13168:
13162:
13158:
13149:
13145:
13139:
13135:
13103:
13100:
13099:
13080:
13077:
13076:
13051:
13048:
13047:
13046:are seen to be
13031:
13028:
13027:
13004:
13000:
12994:
12990:
12981:
12977:
12971:
12967:
12958:
12954:
12948:
12944:
12935:
12931:
12925:
12921:
12912:
12908:
12902:
12898:
12889:
12885:
12879:
12875:
12866:
12862:
12856:
12852:
12843:
12839:
12833:
12829:
12827:
12824:
12823:
12794:
12790:
12781:
12777:
12765:
12761:
12752:
12748:
12733:
12729:
12720:
12716:
12704:
12700:
12691:
12687:
12682:
12679:
12678:
12655:
12651:
12639:
12635:
12633:
12630:
12629:
12591:
12574:
12560:
12507:
12504:
12503:
12478:
12475:
12474:
12471:
12458:
12437:
12423:
12421:
12411:
12397:
12395:
12385:
12365:
12363:
12361:
12358:
12357:
12335:
12332:
12331:
12327:
12323:
12301:
12298:
12297:
12285:
12264:
12214:
12211:
12210:
12194:
12111:
12108:
12107:
12106:of equal size,
12091:
12088:
12087:
12071:
12068:
12067:
12051:
12048:
12047:
12011:
12008:
12007:
12004:
11999:
11978:
11975:
11974:
11955:
11952:
11951:
11935:
11932:
11931:
11915:
11912:
11911:
11836:
11832:
11822:
11818:
11777:
11774:
11773:
11738:
11734:
11721:
11720:
11716:
11715:
11711:
11670:
11667:
11666:
11651:
11597:
11596:
11592:
11591:
11587:
11582:
11579:
11578:
11567:and row vector
11546:
11536:
11534:
11525:
11480:
11479:
11475:
11474:
11470:
11443:
11442:
11438:
11437:
11433:
11428:
11425:
11424:
11403:
11389:
11379:
11314:
11313:
11308:
11302:
11301:
11296:
11286:
11285:
11280:
11277:
11276:
11257:
11254:
11253:
11237:
11234:
11233:
11211:
11208:
11207:
11185:
11182:
11181:
11156:
11153:
11152:
11101:
11100:
11095:
11089:
11088:
11083:
11073:
11072:
11067:
11064:
11063:
11032:
11029:
11028:
11009:
11006:
11005:
10989:
10986:
10985:
10957:
10956:
10938:
10934:
10898:
10897:
10890:
10889:
10883:
10879:
10877:
10871:
10870:
10861:
10857:
10852:
10840:
10836:
10820:
10819:
10795:
10794:
10783:
10779:
10753:
10749:
10738:
10725:
10724:
10715:
10711:
10709:
10697:
10693:
10687:
10686:
10681:
10675:
10671:
10664:
10663:
10659:
10657:
10649:
10648:
10643:
10637:
10636:
10631:
10621:
10620:
10598:
10591:
10590:
10585:
10579:
10578:
10573:
10563:
10562:
10555:
10553:
10550:
10549:
10530:
10527:
10526:
10501:
10498:
10497:
10481:
10478:
10477:
10451:
10447:
10421:
10418:
10417:
10400:
10399:
10381:
10377:
10341:
10340:
10333:
10332:
10320:
10316:
10305:
10296:
10292:
10286:
10285:
10280:
10274:
10270:
10263:
10262:
10238:
10237:
10226:
10222:
10196:
10192:
10181:
10168:
10167:
10161:
10157:
10155:
10149:
10148:
10136:
10132:
10127:
10118:
10114:
10107:
10106:
10102:
10100:
10092:
10091:
10086:
10080:
10079:
10074:
10064:
10063:
10041:
10034:
10033:
10028:
10022:
10021:
10016:
10006:
10005:
9998:
9996:
9993:
9992:
9969:
9966:
9965:
9941:
9940:
9935:
9929:
9928:
9923:
9913:
9912:
9872:
9871:
9866:
9860:
9859:
9854:
9844:
9843:
9838:
9835:
9834:
9805:
9802:
9801:
9779:
9776:
9775:
9753:
9750:
9749:
9727:
9724:
9723:
9689:
9686:
9685:
9659:
9656:
9655:
9652:
9614:
9609:
9597:
9593:
9591:
9588:
9587:
9509:
9506:
9505:
9476:
9472:
9460:
9456:
9432:
9428:
9408:
9405:
9404:
9373:
9370:
9369:
9367:adjugate matrix
9363:
9361:Adjugate matrix
9317:
9304:
9303:
9301:
9299:
9296:
9295:
9254:
9250:
9241:
9237:
9236:
9232:
9208:
9194:
9193:
9181:
9176:
9170:
9165:
9153:
9148:
9142:
9130:
9125:
9119:
9107:
9102:
9095:
9094:
9089:
9084:
9079:
9074:
9068:
9067:
9061:
9056:
9050:
9045:
9039:
9034:
9028:
9022:
9017:
9011:
9005:
9000:
8993:
8992:
8986:
8982:
8980:
8975:
8969:
8965:
8963:
8957:
8953:
8951:
8945:
8941:
8938:
8937:
8932:
8927:
8922:
8917:
8907:
8906:
8904:
8901:
8900:
8867:
8863:
8851:
8847:
8835:
8831:
8816:
8805:
8784:
8781:
8780:
8759:
8756:
8755:
8731:
8728:
8727:
8706:
8705:
8700:
8694:
8693:
8688:
8678:
8677:
8664:
8663:
8658:
8652:
8651:
8646:
8636:
8635:
8622:
8621:
8616:
8610:
8609:
8604:
8594:
8593:
8580:
8579:
8574:
8569:
8563:
8562:
8557:
8552:
8546:
8545:
8540:
8535:
8525:
8524:
8522:
8519:
8518:
8493:
8490:
8489:
8483:
8452:
8448:
8436:
8432:
8420:
8416:
8401:
8390:
8369:
8366:
8365:
8346:
8343:
8342:
8315:
8311:
8299:
8295:
8284:
8281:
8280:
8264:
8261:
8260:
8244:
8241:
8240:
8224:
8221:
8220:
8174:
8171:
8170:
8147:
8143:
8141:
8138:
8137:
8114:
8111:
8110:
8104:
8067:
8063:
8061:
8058:
8057:
8028:
8024:
8019:
8001:
7997:
7995:
7992:
7991:
7987:, and that the
7959:
7955:
7953:
7950:
7949:
7946:normal subgroup
7916:
7912:
7910:
7907:
7906:
7873:
7869:
7867:
7864:
7863:
7847:
7844:
7843:
7826:
7822:
7820:
7817:
7816:
7787:
7783:
7781:
7778:
7777:
7721:
7717:
7696:
7692:
7690:
7687:
7686:
7650:
7646:
7644:
7641:
7640:
7621:
7618:
7617:
7601:
7598:
7597:
7573:
7569:
7535:
7530:
7514:
7510:
7506:
7501:
7498:
7497:
7469:
7466:
7465:
7446:
7443:
7442:
7423:
7420:
7419:
7403:
7400:
7399:
7383:
7380:
7379:
7321:
7318:
7317:
7294:
7291:
7290:
7274:
7271:
7270:
7263:
7240:
7199:
7198:
7194:
7190:
7185:
7182:
7181:
7158:
7155:
7154:
7147:
7085:
7077:
7066:
7058:
7056:
7053:
7052:
7030:
7022:
7014:
7006:
7004:
7001:
7000:
6981:
6973:
6962:
6954:
6952:
6949:
6948:
6929:
6921:
6913:
6905:
6903:
6900:
6899:
6881:
6873:
6865:
6857:
6855:
6852:
6851:
6822:
6817:
6814:
6813:
6771:
6770:
6762:
6757:
6751:
6750:
6745:
6740:
6734:
6733:
6728:
6720:
6710:
6709:
6701:
6698:
6697:
6676:
6675:
6667:
6662:
6656:
6655:
6650:
6645:
6639:
6638:
6633:
6625:
6615:
6614:
6606:
6603:
6602:
6581:
6580:
6572:
6567:
6561:
6560:
6555:
6550:
6544:
6543:
6538:
6533:
6520:
6519:
6511:
6508:
6507:
6487:
6486:
6478:
6473:
6467:
6466:
6461:
6456:
6450:
6449:
6444:
6436:
6423:
6422:
6414:
6411:
6410:
6388:
6385:
6384:
6359:
6358:
6350:
6345:
6336:
6335:
6330:
6325:
6319:
6318:
6313:
6305:
6292:
6291:
6283:
6280:
6279:
6260:
6257:
6256:
6248:
6226:
6223:
6222:
6219:diagonal matrix
6195:
6191:
6185:
6174:
6158:
6154:
6145:
6141:
6135:
6131:
6114:
6111:
6110:
6088:
6085:
6084:
6062:
6059:
6058:
6032:
6028:
6026:
6023:
6022:
6002:
5999:
5998:
5953:
5947:
5943:
5928:
5924:
5915:
5911:
5902:
5898:
5889:
5885:
5880:
5872:
5866:
5862:
5847:
5843:
5834:
5830:
5821:
5817:
5808:
5804:
5799:
5788:
5782:
5778:
5766:
5762:
5753:
5749:
5740:
5736:
5727:
5723:
5718:
5716:
5713:
5712:
5693:
5687:
5683:
5668:
5664:
5649:
5645:
5630:
5626:
5621:
5610:
5604:
5600:
5585:
5581:
5569:
5565:
5550:
5546:
5541:
5539:
5536:
5535:
5516:
5513:
5512:
5490:
5487:
5486:
5457:
5453:
5433:
5430:
5429:
5418:
5379:
5376:
5375:
5347:
5341:
5337:
5298:
5294:
5289:
5287:
5284:
5283:
5255:
5254:
5249:
5243:
5239:
5212:
5208:
5203:
5195:
5189:
5185:
5161:
5157:
5152:
5137:
5136:
5131:
5125:
5121:
5100:
5096:
5063:
5059:
5044:
5040:
5034:
5033:
5026:
5021:
5013:
5009:
5007:
5004:
5003:
4947:
4943:
4941:
4938:
4937:
4918:
4915:
4914:
4896:identity matrix
4879:
4876:
4875:
4847:
4842:
4839:
4838:
4808:
4804:
4802:
4799:
4798:
4772:
4771:
4765:
4761:
4746:
4742:
4736:
4735:
4727:
4724:
4723:
4704:
4701:
4700:
4674:
4671:
4670:
4667:
4662:
4621:
4618:
4617:
4613:
4588:
4584:
4577:
4573:
4561:
4557:
4550:
4546:
4538:
4534:
4525:
4521:
4520:
4516:
4508:
4504:
4489:
4485:
4476:
4472:
4471:
4450:
4447:
4446:
4435:
4401:
4398:
4397:
4396:of integers in
4389:
4370:
4366:
4351:
4347:
4346:
4342:
4340:
4337:
4336:
4292:
4288:
4282:
4271:
4251:
4247:
4239:
4235:
4228:
4207:
4204:
4203:
4158:
4154:
4130:
4126:
4103:
4099:
4092:
4078:
4077:
4065:
4061:
4046:
4042:
4039:
4038:
4028:
4027:
4015:
4011:
3996:
3992:
3985:
3984:
3967:
3964:
3963:
3932:
3931:
3919:
3915:
3900:
3896:
3893:
3892:
3882:
3881:
3869:
3865:
3850:
3846:
3839:
3838:
3830:
3827:
3826:
3822:Given a matrix
3802:
3799:
3798:
3776:
3773:
3772:
3756:
3753:
3752:
3724:
3721:
3720:
3703:
3699:
3697:
3694:
3693:
3690:symmetric group
3628:
3625:
3624:
3608:
3605:
3604:
3561:
3558:
3557:
3527:
3524:
3523:
3520:
3504:
3354:
3353:
3348:
3343:
3337:
3336:
3331:
3326:
3320:
3319:
3314:
3309:
3299:
3298:
3296:
3293:
3292:
3285:
3282:Leibniz formula
3278:
3273:
3267:
3265:Leibniz formula
3259:Leibniz formula
3230:
3229:
3217:
3213:
3211:
3206:
3194:
3190:
3188:
3176:
3172:
3169:
3168:
3163:
3158:
3153:
3147:
3146:
3134:
3130:
3128:
3123:
3111:
3107:
3105:
3093:
3089:
3086:
3085:
3073:
3069:
3067:
3062:
3050:
3046:
3044:
3032:
3028:
3021:
3020:
3018:
3015:
3014:
2973:
2969:
2967:
2964:
2963:
2939:
2938:
2926:
2922:
2920:
2915:
2903:
2899:
2897:
2885:
2881:
2878:
2877:
2872:
2867:
2862:
2856:
2855:
2843:
2839:
2837:
2832:
2820:
2816:
2814:
2802:
2798:
2795:
2794:
2782:
2778:
2776:
2771:
2759:
2755:
2753:
2741:
2737:
2730:
2729:
2721:
2718:
2717:
2695:
2618:
2598:
2595:
2594:
2545:
2541:
2526:
2521:
2520:
2514:
2510:
2495:
2490:
2489:
2483:
2479:
2478:
2474:
2466:
2463:
2462:
2442:
2437:
2436:
2421:
2416:
2415:
2406:
2401:
2400:
2398:
2395:
2394:
2363:
2360:
2359:
2336:
2331:
2330:
2320:
2319:
2313:
2312:
2306:
2305:
2299:
2298:
2288:
2287:
2267:
2262:
2261:
2251:
2250:
2244:
2243:
2237:
2236:
2230:
2229:
2219:
2218:
2205:
2200:
2199:
2189:
2188:
2182:
2181:
2175:
2174:
2168:
2167:
2157:
2156:
2151:
2148:
2147:
2126:
2125:
2119:
2114:
2113:
2111:
2106:
2100:
2095:
2094:
2092:
2086:
2081:
2080:
2076:
2072:
2064:
2061:
2060:
2043:
2030:
2006:
1994:
1982:
1978:
1903:
1902:
1896:
1895:
1885:
1884:
1874:
1873:
1867:
1866:
1853:
1852:
1841:
1827:
1823:
1812:
1807:
1806:
1802:
1786:
1781:
1776:
1770:
1765:
1760:
1752:
1750:
1747:
1746:
1733:
1732:to be equal to
1714:
1699:
1672:
1657:
1642:
1632:
1626:identity matrix
1610:
1606:
1596:
1581:
1561:
1549:
1545:
1533:
1529:
1517:
1506:
1480:
1479:
1474:
1468:
1467:
1462:
1452:
1451:
1384:
1383:
1378:
1366:
1365:
1360:
1344:
1343:
1341:
1338:
1337:
1318:
1315:
1314:
1290:
1289:
1281:
1279:
1273:
1272:
1264:
1262:
1252:
1251:
1241:
1240:
1235:
1229:
1228:
1223:
1213:
1212:
1195:
1172:
1150:
1149:
1141:
1133:
1127:
1126:
1118:
1110:
1100:
1099:
1097:
1094:
1093:
1045:
1044:
1039:
1033:
1032:
1027:
1017:
1016:
1014:
1011:
1010:
989:
988:
983:
977:
976:
971:
961:
960:
958:
955:
954:
952:identity matrix
929:
926:
925:
922:
845:
844:
836:
834:
828:
827:
822:
812:
811:
801:
800:
792:
786:
785:
780:
770:
769:
764:
761:
760:
715:
714:
709:
703:
702:
697:
687:
686:
676:
675:
670:
664:
663:
658:
648:
647:
642:
639:
638:
631:
613:
612:
607:
601:
600:
595:
585:
584:
582:
579:
578:
574:
571:
542:
526:
518:
450:
443:
440:identity matrix
425:
402:
379:
376:
375:
372:Leibniz formula
361:
265:
264:
259:
254:
248:
247:
242:
237:
231:
230:
225:
220:
210:
209:
207:
204:
203:
196:
154:
153:
148:
142:
141:
136:
126:
125:
123:
120:
119:
112:
74:
72:
65:
57:
51:
28:
17:
12:
11:
5:
28129:
28119:
28118:
28113:
28111:Linear algebra
28108:
28103:
28086:
28085:
28083:
28082:
28071:
28068:
28067:
28065:
28064:
28059:
28054:
28049:
28044:
28042:Floating-point
28038:
28036:
28030:
28029:
28027:
28026:
28024:Tensor product
28021:
28016:
28011:
28009:Function space
28006:
28001:
27995:
27993:
27986:
27985:
27983:
27982:
27977:
27972:
27967:
27962:
27957:
27952:
27947:
27945:Triple product
27942:
27937:
27931:
27929:
27923:
27922:
27920:
27919:
27914:
27909:
27904:
27899:
27894:
27889:
27883:
27881:
27875:
27874:
27872:
27871:
27866:
27861:
27859:Transformation
27856:
27851:
27849:Multiplication
27846:
27841:
27836:
27831:
27825:
27823:
27817:
27816:
27809:
27807:
27805:
27804:
27799:
27794:
27789:
27784:
27779:
27774:
27769:
27764:
27759:
27754:
27749:
27744:
27739:
27734:
27729:
27724:
27719:
27714:
27708:
27706:
27705:Basic concepts
27702:
27701:
27699:
27698:
27693:
27687:
27684:
27683:
27680:Linear algebra
27676:
27675:
27668:
27661:
27653:
27647:
27646:
27641:
27636:
27630:
27617:
27612:
27590:
27571:
27551:
27550:External links
27548:
27547:
27546:
27516:
27481:
27476:
27459:
27446:
27433:
27417:
27404:
27393:
27380:
27367:
27362:
27335:
27332:
27331:
27330:
27325:
27312:
27285:
27255:
27246:
27240:
27227:
27222:
27214:Linear Algebra
27209:
27204:
27191:
27185:
27161:
27148:G. Baley Price
27145:
27140:
27127:
27115:
27105:
27090:
27085:
27063:
27058:
27045:
27040:
27027:
27022:
27008:Yan, Catherine
27003:
26990:
26976:, Birkhäuser,
26969:
26964:
26951:
26914:
26879:(9): 761–778,
26868:
26834:
26821:
26808:
26769:
26764:
26751:
26705:
26662:
26656:
26636:
26619:
26616:
26614:
26613:
26580:
26565:
26544:
26511:
26500:
26488:
26477:
26456:
26444:
26431:
26424:
26404:
26387:
26384:
26381:
26378:
26373:
26367:
26364:
26362:
26359:
26358:
26355:
26352:
26350:
26347:
26346:
26344:
26339:
26336:
26333:
26328:
26322:
26319:
26317:
26314:
26313:
26310:
26307:
26305:
26302:
26301:
26299:
26294:
26291:
26286:
26280:
26277:
26275:
26272:
26271:
26268:
26265:
26263:
26260:
26259:
26257:
26252:
26249:
26247:
26245:
26240:
26234:
26231:
26229:
26226:
26225:
26222:
26219:
26217:
26214:
26213:
26211:
26206:
26203:
26198:
26192:
26189:
26187:
26184:
26183:
26180:
26177:
26175:
26172:
26171:
26169:
26164:
26161:
26158:
26155:
26153:
26151:
26148:
26145:
26144:
26119:
26116:Garibaldi 2004
26108:
26092:
26080:
26062:
26050:
26038:
26023:
26011:
25999:
25987:
25976:
25964:
25952:
25940:
25928:
25916:
25902:
25890:
25870:
25866:Bourbaki (1994
25858:
25846:
25819:
25807:
25793:
25778:
25766:
25754:
25710:
25698:
25682:
25668:
25657:
25624:
25603:
25591:
25546:
25499:
25487:
25475:
25463:
25446:
25434:
25430:Linear Algebra
25418:
25406:
25383:
25358:
25344:Episode 4
25333:
25320:
25318:
25315:
25313:
25312:
25307:
25302:
25297:
25292:
25286:
25285:
25284:
25268:
25265:
25202:
25178:
25173:
25169:
25165:
25162:
25159:
25144:bit complexity
25127:
25122:
25118:
25114:
25111:
25108:
25080:
25075:
25071:
25067:
25064:
25061:
25041:
25038:
25035:
25032:
25029:
25026:
25023:
25003:
25000:
24997:
24975:
24971:
24967:
24964:
24961:
24958:
24955:
24935:
24932:
24929:
24926:
24906:
24886:
24881:
24877:
24873:
24870:
24867:
24855:
24852:
24851:
24850:
24839:
24836:
24833:
24830:
24827:
24824:
24821:
24818:
24815:
24812:
24809:
24806:
24803:
24800:
24797:
24794:
24771:
24751:
24748:
24728:
24725:
24705:
24685:
24665:
24645:
24625:
24605:
24585:
24565:
24551:
24550:
24539:
24536:
24533:
24530:
24527:
24524:
24501:
24478:
24475:
24472:
24469:
24466:
24463:
24443:
24438:
24434:
24430:
24427:
24424:
24388:
24385:
24382:
24379:
24367:
24364:
24337:
24334:
24321:
24318:
24295:
24292:
24289:
24266:
24246:
24243:
24220:
24217:
24213:Manin matrices
24194:
24189:
24124:
24121:
24100:
24097:
24089:
24088:
24077:
24074:
24071:
24068:
24065:
24062:
24059:
24056:
24053:
24050:
24047:
24044:
24041:
24038:
24035:
24032:
24029:
24026:
24023:
24020:
24017:
24014:
24011:
23980:
23977:
23948:
23945:
23942:
23939:
23934:
23930:
23926:
23922:
23907:
23906:
23892:
23888:
23884:
23879:
23875:
23871:
23866:
23862:
23858:
23853:
23849:
23845:
23842:
23839:
23836:
23833:
23830:
23827:
23824:
23821:
23818:
23815:
23812:
23809:
23806:
23779:
23776:
23773:
23770:
23765:
23762:
23759:
23755:
23751:
23748:
23738:matrix algebra
23733:
23732:
23721:
23718:
23715:
23712:
23709:
23706:
23683:
23659:
23644:
23641:
23626:
23622:
23590:
23587:
23584:
23581:
23578:
23562:
23559:
23546:
23543:
23540:
23520:
23515:
23511:
23495:vector bundles
23482:
23462:
23457:
23453:
23430:
23426:
23405:
23394:
23393:
23382:
23377:
23373:
23369:
23366:
23363:
23358:
23354:
23350:
23347:
23344:
23341:
23338:
23335:
23331:
23325:
23321:
23317:
23314:
23311:
23306:
23302:
23297:
23292:
23288:
23283:
23279:
23274:
23250:
23247:
23242:
23238:
23217:
23189:
23169:
23164:
23160:
23139:
23134:
23130:
23118:
23117:
23102:
23097:
23093:
23089:
23086:
23083:
23080:
23075:
23071:
23067:
23064:
23059:
23055:
23051:
23048:
23045:
23043:
23039:
23035:
23031:
23028:
23025:
23020:
23016:
23012:
23007:
23003:
22999:
22998:
22995:
22990:
22986:
22982:
22979:
22977:
22975:
22970:
22966:
22962:
22959:
22954:
22950:
22946:
22945:
22922:
22902:
22882:
22877:
22873:
22862:exterior power
22849:
22829:
22809:
22782:
22762:
22742:
22739:
22736:
22733:
22730:
22712:
22709:
22708:
22707:
22696:
22691:
22686:
22681:
22676:
22672:
22668:
22665:
22632:
22628:
22624:
22621:
22599:
22595:
22562:
22542:
22524:
22523:
22512:
22509:
22506:
22501:
22498:
22495:
22491:
22487:
22484:
22481:
22478:
22475:
22472:
22469:
22466:
22463:
22458:
22455:
22452:
22448:
22444:
22441:
22438:
22435:
22432:
22409:
22389:
22369:
22366:
22363:
22360:
22355:
22351:
22347:
22344:
22341:
22338:
22335:
22330:
22326:
22322:
22319:
22316:
22313:
22310:
22305:
22301:
22280:
22277:
22274:
22271:
22268:
22230:
22206:
22186:
22183:
22180:
22165:
22164:
22153:
22148:
22144:
22140:
22137:
22134:
22131:
22128:
22123:
22119:
22084:
22080:
22077:
22057:
22033:
22013:
22010:
22007:
22004:
21999:
21996:
21993:
21989:
21985:
21982:
21959:
21956:
21953:
21950:
21947:
21944:
21919:
21898:
21883:
21880:
21853:
21852:
21841:
21838:
21835:
21832:
21829:
21808:
21807:
21796:
21793:
21790:
21787:
21784:
21781:
21778:
21775:
21772:
21769:
21764:
21761:
21757:
21753:
21750:
21747:
21744:
21741:
21738:
21735:
21732:
21729:
21726:
21723:
21720:
21717:
21714:
21711:
21708:
21703:
21700:
21696:
21692:
21689:
21686:
21683:
21680:
21677:
21674:
21671:
21627:
21624:
21622:
21619:
21604:
21603:
21592:
21588:
21584:
21579:
21575:
21571:
21567:
21564:
21561:
21558:
21555:
21552:
21549:
21545:
21541:
21538:
21534:
21530:
21527:
21524:
21521:
21516:
21512:
21508:
21504:
21500:
21496:
21492:
21488:
21485:
21480:
21477:
21474:
21471:
21467:
21403:
21402:
21391:
21386:
21383:
21380:
21377:
21374:
21371:
21368:
21363:
21355:
21351:
21347:
21340:
21336:
21332:
21326:
21321:
21318:
21315:
21312:
21309:
21282:
21281:
21270:
21265:
21260:
21255:
21250:
21245:
21240:
21237:
21213:For a general
21195:
21190:
21185:
21180:
21175:
21170:
21167:
21135:
21131:
21128:
21125:
21122:
21119:
21116:
21113:
21110:
21107:
21104:
21101:
21098:
21095:
21092:
21088:
21084:
21079:
21076:
21054:
21051:
21048:
21045:
21042:
21039:
21036:
21014:
21013:
21002:
20999:
20996:
20993:
20990:
20987:
20981:
20977:
20966:
20961:
20957:
20952:
20949:
20946:
20943:
20940:
20937:
20934:
20931:
20928:
20905:
20902:
20899:
20896:
20872:
20852:
20832:
20829:
20826:
20804:
20799:
20794:
20789:
20784:
20779:
20776:
20756:
20735:
20731:
20728:
20725:
20722:
20718:
20697:
20694:
20691:
20688:
20659:
20654:
20649:
20646:
20626:
20604:
20599:
20594:
20589:
20584:
20579:
20576:
20565:parallelepiped
20561:absolute value
20556:
20553:
20508:represents an
20494:standard basis
20462:Main article:
20459:
20456:
20450:have a common
20423:
20420:
20417:
20397:
20386:
20385:
20374:
20369:
20363:
20360:
20357:
20352:
20349:
20346:
20343:
20340:
20335:
20331:
20327:
20325:
20322:
20320:
20317:
20314:
20309:
20306:
20303:
20300:
20297:
20292:
20288:
20284:
20282:
20279:
20276:
20271:
20268:
20265:
20262:
20259:
20254:
20250:
20246:
20245:
20242:
20239:
20237:
20234:
20232:
20229:
20227:
20224:
20223:
20220:
20217:
20214:
20210:
20206:
20202:
20198:
20196:
20193:
20191:
20188:
20185:
20181:
20177:
20173:
20169:
20167:
20164:
20161:
20157:
20153:
20149:
20145:
20144:
20141:
20138:
20135:
20130:
20126:
20122:
20120:
20117:
20115:
20112:
20109:
20104:
20100:
20096:
20094:
20091:
20088:
20083:
20079:
20075:
20074:
20072:
20067:
20064:
20061:
20058:
20055:
20050:
20046:
20042:
20039:
20036:
20031:
20027:
20023:
20020:
20002:differentiable
19989:
19986:
19983:
19963:
19960:
19957:
19952:
19948:
19944:
19941:
19938:
19935:
19932:
19929:
19924:
19920:
19895:
19892:
19889:
19860:
19856:
19833:
19828:
19823:
19818:
19814:
19810:
19805:
19801:
19780:
19760:
19757:
19741:
19738:
19713:
19708:
19704:
19700:
19697:
19694:
19683:
19682:
19671:
19666:
19662:
19657:
19654:
19651:
19648:
19645:
19642:
19638:
19635:
19632:
19629:
19626:
19623:
19620:
19617:
19614:
19611:
19608:
19604:
19575:
19571:
19559:
19558:
19547:
19544:
19541:
19538:
19533:
19529:
19525:
19520:
19512:
19508:
19504:
19502:
19499:
19495:
19492:
19489:
19485:
19481:
19477:
19473:
19469:
19465:
19462:
19459:
19455:
19451:
19449:
19446:
19442:
19438:
19434:
19433:
19431:
19426:
19421:
19417:
19411:
19406:
19403:
19400:
19396:
19392:
19387:
19379:
19375:
19371:
19369:
19366:
19364:
19361:
19359:
19356:
19352:
19348:
19344:
19343:
19341:
19336:
19333:
19330:
19325:
19321:
19317:
19314:
19291:
19271:
19262:-th column of
19251:
19229:
19225:
19213:
19212:
19201:
19198:
19195:
19192:
19189:
19186:
19183:
19180:
19177:
19174:
19171:
19164:
19161:
19158:
19155:
19150:
19145:
19141:
19137:
19134:
19128:
19123:
19119:
19091:
19088:
19085:
19082:
19062:
19042:
19039:
19036:
19033:
19017:
19014:
19012:
19009:
18987:(so called by
18843:Laplace (1772)
18795:
18792:
18791:
18790:
18775:
18771:
18767:
18763:
18759:
18756:
18754:
18752:
18749:
18746:
18743:
18737:
18732:
18728:
18727:
18723:
18719:
18715:
18711:
18708:
18706:
18704:
18701:
18698:
18695:
18689:
18684:
18680:
18679:
18675:
18671:
18667:
18663:
18660:
18658:
18656:
18653:
18650:
18647:
18641:
18636:
18632:
18631:
18604:
18601:
18598:
18595:
18592:
18570:
18564:
18561:
18559:
18556:
18554:
18551:
18550:
18548:
18543:
18540:
18520:
18517:
18514:
18489:
18483:
18480:
18454:
18451:
18448:
18445:
18423:
18419:
18399:
18398:
18387:
18383:
18378:
18374:
18370:
18366:
18363:
18360:
18357:
18354:
18351:
18348:
18345:
18342:
18339:
18336:
18333:
18330:
18327:
18324:
18321:
18318:
18315:
18292:
18289:
18286:
18275:big O notation
18271:
18270:
18257:
18252:
18248:
18244:
18240:
18237:
18234:
18230:
18226:
18221:
18218:
18214:
18209:
18205:
18202:
18199:
18196:
18193:
18190:
18187:
18183:
18178:
18174:
18170:
18166:
18163:
18160:
18157:
18154:
18151:
18148:
18145:
18142:
18139:
18136:
18133:
18130:
18127:
18124:
18121:
18118:
18115:
18112:
18109:
18106:
18103:
18100:
18097:
18094:
18091:
18077:
18076:
18065:
18060:
18057:
18052:
18047:
18044:
18040:
18036:
18031:
18028:
18025:
18022:
18019:
18014:
18011:
18007:
18003:
18000:
17997:
17994:
17991:
17983:
17980:
17976:
17972:
17967:
17964:
17961:
17958:
17955:
17929:
17918:
17917:
17906:
17902:
17895:
17892:
17887:
17884:
17876:
17873:
17869:
17864:
17860:
17857:
17854:
17851:
17848:
17845:
17842:
17836:
17833:
17828:
17825:
17822:
17819:
17816:
17790:
17770:
17746:
17743:
17740:
17737:
17734:
17723:
17722:
17711:
17707:
17700:
17697:
17692:
17689:
17683:
17680:
17677:
17674:
17671:
17667:
17663:
17660:
17657:
17651:
17648:
17643:
17640:
17637:
17634:
17631:
17610:differentiable
17596:
17573:
17570:
17567:
17562:
17551:function from
17544:
17541:
17529:geometric mean
17521:
17520:
17509:
17503:
17498:
17494:
17490:
17486:
17483:
17478:
17475:
17468:
17465:
17462:
17459:
17456:
17453:
17448:
17445:
17440:
17434:
17431:
17426:
17422:
17419:
17416:
17413:
17406:
17401:
17398:
17394:
17390:
17386:
17383:
17379:
17342:
17341:
17330:
17327:
17324:
17321:
17318:
17315:
17312:
17309:
17306:
17303:
17300:
17297:
17294:
17291:
17288:
17284:
17278:
17275:
17271:
17267:
17264:
17260:
17256:
17253:
17232:
17229:
17191:
17190:
17179:
17173:
17168:
17163:
17158:
17154:
17150:
17146:
17143:
17138:
17132:
17128:
17122:
17118:
17114:
17111:
17108:
17100:
17095:
17092:
17089:
17085:
17081:
17077:
17069:
17066:
17062:
17055:
17050:
17047:
17044:
17040:
17019:
17018:
17007:
17001:
16996:
16991:
16986:
16982:
16978:
16974:
16971:
16966:
16960:
16956:
16952:
16949:
16946:
16938:
16933:
16930:
16927:
16923:
16919:
16915:
16907:
16904:
16900:
16893:
16888:
16885:
16882:
16878:
16874:
16871:
16868:
16865:
16862:
16859:
16856:
16830:
16829:
16818:
16813:
16808:
16804:
16798:
16794:
16790:
16787:
16784:
16781:
16778:
16775:
16772:
16767:
16762:
16758:
16752:
16747:
16743:
16737:
16733:
16729:
16724:
16719:
16715:
16711:
16708:
16705:
16687:
16681:
16674:
16659:
16653:
16646:
16632:
16626:
16625:
16614:
16611:
16606:
16602:
16598:
16595:
16592:
16587:
16583:
16579:
16574:
16570:
16566:
16561:
16557:
16550:
16547:
16540:
16536:
16532:
16529:
16526:
16520:
16517:
16514:
16511:
16508:
16483:
16469:
16468:
16457:
16454:
16451:
16446:
16442:
16438:
16433:
16428:
16425:
16422:
16418:
16399:
16393:
16392:
16381:
16374:
16370:
16364:
16359:
16355:
16351:
16346:
16343:
16337:
16332:
16328:
16320:
16316:
16311:
16303:
16300:
16295:
16291:
16286:
16282:
16279:
16276:
16268:
16263:
16260:
16257:
16253:
16244:
16241:
16236:
16232:
16228:
16225:
16222:
16219:
16214:
16210:
16206:
16203:
16198:
16194:
16190:
16189:
16186:
16183:
16178:
16174:
16170:
16167:
16164:
16159:
16155:
16151:
16146:
16142:
16138:
16137:
16134:
16130:
16127:
16124:
16121:
16118:
16104:
16103:
16092:
16086:
16083:
16080:
16077:
16074:
16071:
16068:
16058:
16053:
16049:
16045:
16041:
16038:
16033:
16030:
16027:
16024:
16021:
16017:
16011:
16006:
16003:
16000:
15996:
15990:
15987:
15982:
15979:
15974:
15971:
15968:
15964:
15951:
15948:
15945:
15940:
15936:
15915:
15887:
15886:
15871:
15867:
15862:
15857:
15853:
15849:
15845:
15842:
15839:
15836:
15833:
15830:
15827:
15824:
15821:
15814:
15809:
15805:
15801:
15797:
15794:
15791:
15788:
15783:
15778:
15773:
15768:
15764:
15760:
15756:
15753:
15749:
15744:
15741:
15736:
15731:
15727:
15724:
15721:
15718:
15715:
15711:
15705:
15700:
15696:
15692:
15688:
15685:
15682:
15679:
15674:
15669:
15665:
15662:
15659:
15656:
15653:
15649:
15643:
15637:
15634:
15629:
15626:
15624:
15622:
15619:
15616:
15613:
15610:
15609:
15606:
15602:
15597:
15592:
15588:
15584:
15580:
15577:
15574:
15571:
15567:
15562:
15558:
15554:
15550:
15547:
15541:
15538:
15535:
15532:
15529:
15526:
15523:
15518:
15513:
15509:
15506:
15503:
15500:
15497:
15493:
15487:
15481:
15478:
15473:
15470:
15468:
15466:
15463:
15460:
15457:
15454:
15453:
15450:
15446:
15441:
15436:
15432:
15428:
15424:
15421:
15418:
15413:
15408:
15404:
15401:
15398:
15395:
15392:
15388:
15382:
15376:
15373:
15368:
15365:
15363:
15361:
15358:
15355:
15352:
15349:
15348:
15313:
15312:
15301:
15298:
15295:
15292:
15289:
15286:
15283:
15280:
15277:
15274:
15271:
15268:
15265:
15262:
15259:
15241:
15240:
15229:
15226:
15223:
15220:
15217:
15214:
15211:
15163:) denotes the
15157:
15156:
15145:
15142:
15139:
15136:
15133:
15130:
15127:
15124:
15121:
15118:
15115:
15112:
15109:
15106:
15103:
15100:
15097:
15094:
15091:
15073:
15072:
15061:
15058:
15055:
15052:
15049:
15046:
15043:
15040:
15037:
15034:
15031:
15028:
15025:
15022:
15019:
15016:
15013:
15010:
14979:
14976:
14963:
14943:
14923:
14912:
14911:
14898:
14890:
14887:
14884:
14880:
14876:
14874:
14871:
14867:
14864:
14861:
14857:
14853:
14849:
14846:
14843:
14839:
14835:
14834:
14831:
14828:
14826:
14823:
14821:
14818:
14816:
14813:
14812:
14807:
14804:
14801:
14797:
14793:
14791:
14788:
14784:
14781:
14778:
14774:
14770:
14766:
14763:
14760:
14756:
14752:
14751:
14746:
14743:
14740:
14736:
14732:
14730:
14727:
14723:
14720:
14717:
14713:
14709:
14705:
14702:
14699:
14695:
14691:
14690:
14688:
14683:
14678:
14674:
14647:
14646:
14635:
14632:
14629:
14626:
14623:
14618:
14614:
14590:
14566:
14557:of the matrix
14542:
14522:
14498:
14487:
14486:
14475:
14472:
14469:
14466:
14463:
14460:
14457:
14454:
14451:
14448:
14445:
14442:
14439:
14434:
14430:
14403:
14383:
14363:
14343:
14323:
14303:
14285:
14284:
14273:
14268:
14264:
14260:
14255:
14251:
14245:
14241:
14237:
14232:
14228:
14222:
14217:
14214:
14211:
14207:
14203:
14200:
14197:
14194:
14191:
14147:
14143:
14139:
14136:
14133:
14128:
14124:
14120:
14115:
14111:
14084:
14060:
14057:
14054:
14034:
14014:
14011:
14009:
14006:
13993:
13990:
13987:
13983:
13979:
13976:
13973:
13970:
13967:
13964:
13961:
13941:
13938:
13935:
13931:
13927:
13911:
13910:
13899:
13894:
13890:
13886:
13881:
13877:
13873:
13870:
13866:
13862:
13859:
13854:
13850:
13846:
13843:
13840:
13837:
13834:
13829:
13825:
13821:
13818:
13814:
13810:
13807:
13804:
13801:
13798:
13795:
13771:
13767:
13764:
13761:
13741:
13738:
13735:
13732:
13712:
13709:
13706:
13702:
13698:
13673:
13653:
13631:
13625:
13622:
13620:
13617:
13616:
13613:
13610:
13607:
13605:
13602:
13601:
13599:
13594:
13591:
13586:
13580:
13577:
13575:
13572:
13571:
13568:
13565:
13563:
13560:
13559:
13557:
13552:
13549:
13545:
13541:
13538:
13535:
13532:
13512:
13509:
13506:
13491:
13490:
13479:
13476:
13473:
13470:
13467:
13459:
13456:
13453:
13450:
13442:
13439:
13436:
13417:
13416:
13405:
13400:
13396:
13390:
13386:
13382:
13377:
13373:
13367:
13363:
13359:
13354:
13350:
13344:
13340:
13336:
13331:
13327:
13321:
13317:
13313:
13310:
13307:
13302:
13298:
13294:
13289:
13285:
13281:
13278:
13273:
13269:
13265:
13260:
13256:
13252:
13238:
13237:
13226:
13221:
13217:
13211:
13207:
13203:
13198:
13194:
13188:
13184:
13180:
13175:
13171:
13165:
13161:
13157:
13152:
13148:
13142:
13138:
13134:
13131:
13128:
13125:
13122:
13119:
13116:
13113:
13110:
13107:
13084:
13064:
13061:
13058:
13055:
13035:
13024:
13023:
13012:
13007:
13003:
12997:
12993:
12989:
12984:
12980:
12974:
12970:
12966:
12961:
12957:
12951:
12947:
12943:
12938:
12934:
12928:
12924:
12920:
12915:
12911:
12905:
12901:
12897:
12892:
12888:
12882:
12878:
12874:
12869:
12865:
12859:
12855:
12851:
12846:
12842:
12836:
12832:
12817:
12816:
12805:
12802:
12797:
12793:
12789:
12784:
12780:
12776:
12773:
12768:
12764:
12760:
12755:
12751:
12747:
12744:
12741:
12736:
12732:
12728:
12723:
12719:
12715:
12712:
12707:
12703:
12699:
12694:
12690:
12686:
12661:
12658:
12654:
12650:
12645:
12642:
12638:
12623:
12622:
12621:
12610:
12607:
12604:
12601:
12598:
12590:
12587:
12584:
12581:
12573:
12570:
12567:
12559:
12556:
12553:
12550:
12547:
12544:
12541:
12538:
12535:
12532:
12529:
12526:
12523:
12520:
12517:
12514:
12511:
12488:
12485:
12482:
12470:
12467:
12446:
12440:
12435:
12432:
12429:
12426:
12420:
12414:
12409:
12406:
12403:
12400:
12394:
12388:
12383:
12380:
12377:
12374:
12371:
12368:
12345:
12342:
12339:
12311:
12308:
12305:
12263:
12260:
12257:
12254:
12251:
12248:
12245:
12242:
12239:
12236:
12233:
12230:
12227:
12224:
12221:
12218:
12193:
12190:
12187:
12184:
12181:
12178:
12175:
12172:
12169:
12166:
12163:
12160:
12157:
12154:
12151:
12148:
12145:
12142:
12139:
12136:
12133:
12130:
12127:
12124:
12121:
12118:
12115:
12095:
12075:
12055:
12021:
12018:
12015:
12003:
12000:
11998:
11997:
11985:
11982:
11962:
11959:
11939:
11919:
11908:
11907:
11896:
11893:
11889:
11886:
11883:
11880:
11877:
11873:
11870:
11867:
11864:
11861:
11858:
11855:
11851:
11847:
11842:
11839:
11835:
11831:
11828:
11825:
11821:
11817:
11814:
11811:
11808:
11805:
11802:
11799:
11796:
11793:
11790:
11787:
11784:
11781:
11769:
11768:
11757:
11753:
11749:
11744:
11741:
11737:
11733:
11730:
11724:
11719:
11714:
11710:
11707:
11704:
11701:
11698:
11695:
11692:
11689:
11686:
11683:
11680:
11677:
11674:
11648:
11647:
11646:
11635:
11632:
11629:
11626:
11623:
11620:
11616:
11612:
11609:
11606:
11600:
11595:
11590:
11586:
11560:
11530:
11521:
11515:
11514:
11503:
11499:
11495:
11492:
11489:
11483:
11478:
11473:
11469:
11466:
11462:
11458:
11455:
11452:
11446:
11441:
11436:
11432:
11378:
11375:
11374:
11373:
11362:
11359:
11356:
11353:
11350:
11347:
11344:
11341:
11338:
11335:
11332:
11329:
11326:
11323:
11318:
11312:
11309:
11307:
11304:
11303:
11300:
11297:
11295:
11292:
11291:
11289:
11284:
11261:
11241:
11221:
11218:
11215:
11195:
11192:
11189:
11166:
11163:
11160:
11149:
11148:
11137:
11134:
11131:
11128:
11125:
11122:
11119:
11116:
11113:
11110:
11105:
11099:
11096:
11094:
11091:
11090:
11087:
11084:
11082:
11079:
11078:
11076:
11071:
11048:
11045:
11042:
11039:
11036:
11013:
10993:
10971:
10970:
10955:
10952:
10949:
10944:
10941:
10937:
10933:
10930:
10927:
10924:
10921:
10918:
10915:
10912:
10909:
10906:
10903:
10901:
10899:
10894:
10886:
10882:
10878:
10876:
10873:
10872:
10867:
10864:
10860:
10856:
10853:
10851:
10846:
10843:
10839:
10835:
10832:
10829:
10826:
10825:
10823:
10818:
10815:
10812:
10809:
10806:
10803:
10800:
10798:
10796:
10789:
10786:
10782:
10778:
10775:
10772:
10768:
10764:
10759:
10756:
10752:
10748:
10745:
10741:
10735:
10729:
10721:
10718:
10714:
10710:
10708:
10703:
10700:
10696:
10692:
10689:
10688:
10685:
10682:
10678:
10674:
10670:
10669:
10667:
10662:
10653:
10647:
10644:
10642:
10639:
10638:
10635:
10632:
10630:
10627:
10626:
10624:
10619:
10616:
10613:
10610:
10607:
10604:
10601:
10599:
10595:
10589:
10586:
10584:
10581:
10580:
10577:
10574:
10572:
10569:
10568:
10566:
10561:
10558:
10557:
10534:
10511:
10508:
10505:
10485:
10465:
10462:
10457:
10454:
10450:
10446:
10443:
10440:
10437:
10434:
10431:
10428:
10425:
10414:
10413:
10398:
10395:
10392:
10387:
10384:
10380:
10376:
10373:
10370:
10367:
10364:
10361:
10358:
10355:
10352:
10349:
10346:
10344:
10342:
10337:
10331:
10326:
10323:
10319:
10315:
10312:
10309:
10306:
10302:
10299:
10295:
10291:
10288:
10287:
10284:
10281:
10277:
10273:
10269:
10268:
10266:
10261:
10258:
10255:
10252:
10249:
10246:
10243:
10241:
10239:
10232:
10229:
10225:
10221:
10218:
10215:
10211:
10207:
10202:
10199:
10195:
10191:
10188:
10184:
10178:
10172:
10164:
10160:
10156:
10154:
10151:
10150:
10147:
10142:
10139:
10135:
10131:
10128:
10124:
10121:
10117:
10113:
10112:
10110:
10105:
10096:
10090:
10087:
10085:
10082:
10081:
10078:
10075:
10073:
10070:
10069:
10067:
10062:
10059:
10056:
10053:
10050:
10047:
10044:
10042:
10038:
10032:
10029:
10027:
10024:
10023:
10020:
10017:
10015:
10012:
10011:
10009:
10004:
10001:
10000:
9973:
9962:
9961:
9950:
9945:
9939:
9936:
9934:
9931:
9930:
9927:
9924:
9922:
9919:
9918:
9916:
9911:
9908:
9905:
9902:
9899:
9896:
9893:
9890:
9887:
9884:
9881:
9876:
9870:
9867:
9865:
9862:
9861:
9858:
9855:
9853:
9850:
9849:
9847:
9842:
9815:
9812:
9809:
9789:
9786:
9783:
9763:
9760:
9757:
9737:
9734:
9731:
9711:
9708:
9705:
9702:
9699:
9696:
9693:
9669:
9666:
9663:
9651:
9650:Block matrices
9648:
9647:
9646:
9635:
9632:
9629:
9626:
9620:
9617:
9613:
9608:
9603:
9600:
9596:
9577:
9576:
9565:
9562:
9558:
9555:
9552:
9549:
9546:
9543:
9540:
9537:
9534:
9531:
9528:
9525:
9522:
9519:
9516:
9513:
9499:
9498:
9487:
9482:
9479:
9475:
9469:
9466:
9463:
9459:
9455:
9452:
9449:
9446:
9441:
9438:
9435:
9431:
9427:
9424:
9421:
9418:
9415:
9412:
9389:
9386:
9383:
9380:
9377:
9362:
9359:
9320:
9315:
9312:
9307:
9267:
9263:
9257:
9253:
9249:
9244:
9240:
9235:
9229:
9226:
9223:
9220:
9217:
9214:
9211:
9207:
9203:
9198:
9190:
9187:
9184:
9179:
9175:
9171:
9169:
9166:
9162:
9159:
9156:
9151:
9147:
9143:
9139:
9136:
9133:
9128:
9124:
9120:
9116:
9113:
9110:
9105:
9101:
9097:
9096:
9093:
9090:
9088:
9085:
9083:
9080:
9078:
9075:
9073:
9070:
9069:
9064:
9059:
9055:
9051:
9049:
9046:
9042:
9037:
9033:
9029:
9025:
9020:
9016:
9012:
9008:
9003:
8999:
8995:
8994:
8989:
8985:
8981:
8979:
8976:
8972:
8968:
8964:
8960:
8956:
8952:
8948:
8944:
8940:
8939:
8936:
8933:
8931:
8928:
8926:
8923:
8921:
8918:
8916:
8913:
8912:
8910:
8893:
8892:
8881:
8876:
8873:
8870:
8866:
8860:
8857:
8854:
8850:
8844:
8841:
8838:
8834:
8830:
8827:
8824:
8819:
8814:
8811:
8808:
8804:
8800:
8797:
8794:
8791:
8788:
8763:
8741:
8738:
8735:
8724:
8723:
8710:
8704:
8701:
8699:
8696:
8695:
8692:
8689:
8687:
8684:
8683:
8681:
8676:
8673:
8668:
8662:
8659:
8657:
8654:
8653:
8650:
8647:
8645:
8642:
8641:
8639:
8634:
8631:
8626:
8620:
8617:
8615:
8612:
8611:
8608:
8605:
8603:
8600:
8599:
8597:
8592:
8589:
8584:
8578:
8575:
8573:
8570:
8568:
8565:
8564:
8561:
8558:
8556:
8553:
8551:
8548:
8547:
8544:
8541:
8539:
8536:
8534:
8531:
8530:
8528:
8503:
8500:
8497:
8478:
8477:
8466:
8461:
8458:
8455:
8451:
8445:
8442:
8439:
8435:
8429:
8426:
8423:
8419:
8415:
8412:
8409:
8404:
8399:
8396:
8393:
8389:
8385:
8382:
8379:
8376:
8373:
8350:
8337:is known as a
8324:
8321:
8318:
8314:
8308:
8305:
8302:
8298:
8294:
8291:
8288:
8268:
8248:
8228:
8208:
8205:
8202:
8199:
8196:
8193:
8190:
8187:
8184:
8181:
8178:
8156:
8153:
8150:
8146:
8118:
8103:
8100:
8070:
8066:
8045:
8042:
8039:
8036:
8031:
8027:
8022:
8018:
8015:
8012:
8009:
8004:
8000:
7989:quotient group
7976:
7973:
7970:
7967:
7962:
7958:
7933:
7930:
7927:
7924:
7919:
7915:
7890:
7887:
7884:
7881:
7876:
7872:
7851:
7829:
7825:
7804:
7801:
7798:
7795:
7790:
7786:
7738:
7735:
7732:
7729:
7724:
7720:
7716:
7713:
7710:
7707:
7704:
7699:
7695:
7667:
7664:
7661:
7658:
7653:
7649:
7625:
7605:
7594:
7593:
7579:
7576:
7572:
7568:
7565:
7562:
7559:
7556:
7553:
7547:
7544:
7541:
7538:
7534:
7529:
7525:
7520:
7517:
7513:
7509:
7505:
7473:
7450:
7430:
7427:
7407:
7387:
7376:
7375:
7364:
7361:
7358:
7355:
7352:
7349:
7346:
7343:
7340:
7337:
7334:
7331:
7328:
7325:
7311:matrix product
7298:
7278:
7262:
7259:
7237:
7236:
7224:
7221:
7218:
7215:
7212:
7208:
7197:
7193:
7189:
7162:
7146:
7143:
7131:
7128:
7125:
7122:
7119:
7116:
7113:
7110:
7107:
7104:
7101:
7098:
7095:
7092:
7088:
7084:
7080:
7076:
7073:
7069:
7065:
7061:
7047:
7046:
7033:
7029:
7025:
7021:
7017:
7013:
7009:
6997:
6984:
6980:
6976:
6972:
6969:
6965:
6961:
6957:
6945:
6932:
6928:
6924:
6920:
6916:
6912:
6908:
6896:
6884:
6880:
6876:
6872:
6868:
6864:
6860:
6849:
6845:
6844:
6829:
6826:
6821:
6809:
6804:
6799:
6794:
6790:
6789:
6775:
6769:
6766:
6763:
6761:
6758:
6756:
6753:
6752:
6749:
6746:
6744:
6741:
6739:
6736:
6735:
6732:
6729:
6727:
6724:
6721:
6719:
6716:
6715:
6713:
6708:
6705:
6694:
6680:
6674:
6671:
6668:
6666:
6663:
6661:
6658:
6657:
6654:
6651:
6649:
6646:
6644:
6641:
6640:
6637:
6634:
6632:
6629:
6626:
6624:
6621:
6620:
6618:
6613:
6610:
6599:
6585:
6579:
6576:
6573:
6571:
6568:
6566:
6563:
6562:
6559:
6556:
6554:
6551:
6549:
6546:
6545:
6542:
6539:
6537:
6534:
6532:
6529:
6526:
6525:
6523:
6518:
6515:
6504:
6491:
6485:
6482:
6479:
6477:
6474:
6472:
6469:
6468:
6465:
6462:
6460:
6457:
6455:
6452:
6451:
6448:
6445:
6443:
6440:
6437:
6435:
6432:
6429:
6428:
6426:
6421:
6418:
6408:
6392:
6380:
6379:
6368:
6363:
6357:
6354:
6351:
6349:
6346:
6344:
6341:
6338:
6337:
6334:
6331:
6329:
6326:
6324:
6321:
6320:
6317:
6314:
6312:
6309:
6306:
6304:
6301:
6298:
6297:
6295:
6290:
6287:
6264:
6247:
6244:
6243:
6242:
6230:
6206:
6201:
6198:
6194:
6188:
6183:
6180:
6177:
6173:
6169:
6164:
6161:
6157:
6153:
6148:
6144:
6138:
6134:
6130:
6127:
6124:
6121:
6118:
6098:
6095:
6092:
6072:
6069:
6066:
6046:
6043:
6038:
6035:
6031:
6006:
5995:
5987:
5976:
5960:
5956:
5950:
5946:
5942:
5939:
5936:
5931:
5927:
5923:
5918:
5914:
5910:
5905:
5901:
5897:
5892:
5888:
5883:
5879:
5875:
5869:
5865:
5861:
5858:
5855:
5850:
5846:
5842:
5837:
5833:
5829:
5824:
5820:
5816:
5811:
5807:
5802:
5798:
5795:
5791:
5785:
5781:
5777:
5774:
5769:
5765:
5761:
5756:
5752:
5748:
5743:
5739:
5735:
5730:
5726:
5721:
5700:
5696:
5690:
5686:
5682:
5679:
5676:
5671:
5667:
5663:
5660:
5657:
5652:
5648:
5644:
5641:
5638:
5633:
5629:
5624:
5620:
5617:
5613:
5607:
5603:
5599:
5596:
5593:
5588:
5584:
5580:
5577:
5572:
5568:
5564:
5561:
5558:
5553:
5549:
5544:
5532:
5520:
5500:
5497:
5494:
5474:
5471:
5468:
5465:
5460:
5456:
5452:
5449:
5446:
5443:
5440:
5437:
5417:
5414:
5409:standard basis
5389:
5386:
5383:
5371:
5370:
5369:
5368:
5357:
5354:
5350:
5344:
5340:
5336:
5333:
5330:
5327:
5324:
5321:
5318:
5315:
5312:
5309:
5306:
5301:
5297:
5292:
5272:
5270:
5269:
5268:
5252:
5246:
5242:
5238:
5235:
5232:
5229:
5226:
5223:
5220:
5215:
5211:
5206:
5202:
5198:
5192:
5188:
5184:
5181:
5178:
5175:
5172:
5169:
5164:
5160:
5155:
5151:
5148:
5145:
5142:
5140:
5138:
5134:
5128:
5124:
5120:
5117:
5114:
5109:
5106:
5103:
5099:
5095:
5092:
5089:
5086:
5083:
5080:
5077:
5072:
5069:
5066:
5062:
5058:
5055:
5052:
5047:
5043:
5037:
5032:
5029:
5027:
5024:
5020:
5016:
5012:
5011:
4983:column vectors
4970:
4967:
4964:
4961:
4958:
4955:
4950:
4946:
4922:
4901:
4899:
4883:
4863:
4860:
4856:
4853:
4850:
4846:
4836:
4811:
4807:
4792:
4791:
4780:
4775:
4768:
4764:
4760:
4757:
4754:
4749:
4745:
4739:
4734:
4731:
4708:
4684:
4681:
4678:
4666:
4663:
4661:
4658:
4646:
4643:
4640:
4637:
4634:
4631:
4628:
4625:
4610:
4609:
4598:
4591:
4587:
4583:
4580:
4576:
4572:
4564:
4560:
4556:
4553:
4549:
4541:
4537:
4533:
4528:
4524:
4519:
4511:
4507:
4503:
4500:
4497:
4492:
4488:
4484:
4479:
4475:
4470:
4466:
4463:
4460:
4457:
4454:
4423:
4420:
4417:
4414:
4411:
4408:
4405:
4373:
4369:
4365:
4362:
4359:
4354:
4350:
4345:
4330:
4329:
4316:
4310:
4307:
4304:
4301:
4298:
4295:
4291:
4285:
4280:
4277:
4274:
4270:
4266:
4263:
4260:
4257:
4254:
4250:
4242:
4238:
4234:
4231:
4227:
4223:
4220:
4217:
4214:
4211:
4193:
4192:
4181:
4176:
4173:
4170:
4167:
4164:
4161:
4157:
4153:
4148:
4145:
4142:
4139:
4136:
4133:
4129:
4125:
4122:
4119:
4116:
4113:
4106:
4102:
4098:
4095:
4091:
4087:
4082:
4074:
4071:
4068:
4064:
4060:
4055:
4052:
4049:
4045:
4041:
4040:
4037:
4033:
4030:
4029:
4024:
4021:
4018:
4014:
4010:
4005:
4002:
3999:
3995:
3991:
3990:
3988:
3983:
3980:
3977:
3974:
3971:
3957:sigma notation
3953:
3952:
3941:
3936:
3928:
3925:
3922:
3918:
3914:
3909:
3906:
3903:
3899:
3895:
3894:
3891:
3887:
3884:
3883:
3878:
3875:
3872:
3868:
3864:
3859:
3856:
3853:
3849:
3845:
3844:
3842:
3837:
3834:
3809:
3806:
3786:
3783:
3780:
3760:
3740:
3737:
3734:
3731:
3728:
3706:
3702:
3677:
3674:
3671:
3668:
3665:
3662:
3659:
3656:
3653:
3650:
3647:
3644:
3641:
3638:
3635:
3632:
3612:
3589:
3586:
3583:
3580:
3577:
3574:
3571:
3568:
3565:
3537:
3534:
3531:
3519:
3509:
3500:rule of Sarrus
3493:Rule of Sarrus
3450:
3449:
3435:
3432:
3429:
3426:
3423:
3420:
3417:
3414:
3411:
3408:
3405:
3402:
3399:
3396:
3393:
3390:
3387:
3384:
3381:
3378:
3375:
3372:
3369:
3366:
3363:
3358:
3352:
3349:
3347:
3344:
3342:
3339:
3338:
3335:
3332:
3330:
3327:
3325:
3322:
3321:
3318:
3315:
3313:
3310:
3308:
3305:
3304:
3302:
3277:
3276:3 × 3 matrices
3274:
3269:Main article:
3266:
3263:
3251:
3250:
3239:
3234:
3226:
3223:
3220:
3216:
3212:
3210:
3207:
3203:
3200:
3197:
3193:
3189:
3185:
3182:
3179:
3175:
3171:
3170:
3167:
3164:
3162:
3159:
3157:
3154:
3152:
3149:
3148:
3143:
3140:
3137:
3133:
3129:
3127:
3124:
3120:
3117:
3114:
3110:
3106:
3102:
3099:
3096:
3092:
3088:
3087:
3082:
3079:
3076:
3072:
3068:
3066:
3063:
3059:
3056:
3053:
3049:
3045:
3041:
3038:
3035:
3031:
3027:
3026:
3024:
2982:
2979:
2976:
2972:
2960:
2959:
2948:
2943:
2935:
2932:
2929:
2925:
2921:
2919:
2916:
2912:
2909:
2906:
2902:
2898:
2894:
2891:
2888:
2884:
2880:
2879:
2876:
2873:
2871:
2868:
2866:
2863:
2861:
2858:
2857:
2852:
2849:
2846:
2842:
2838:
2836:
2833:
2829:
2826:
2823:
2819:
2815:
2811:
2808:
2805:
2801:
2797:
2796:
2791:
2788:
2785:
2781:
2777:
2775:
2772:
2768:
2765:
2762:
2758:
2754:
2750:
2747:
2744:
2740:
2736:
2735:
2733:
2728:
2725:
2694:
2691:
2634:
2631:
2628:
2625:
2617:
2614:
2611:
2608:
2605:
2602:
2573:
2569:
2565:
2562:
2556:
2553:
2548:
2544:
2540:
2537:
2534:
2529:
2524:
2517:
2513:
2509:
2506:
2503:
2498:
2493:
2486:
2482:
2477:
2473:
2470:
2450:
2445:
2440:
2435:
2432:
2429:
2424:
2419:
2414:
2409:
2404:
2378:maps the unit
2367:
2356:
2355:
2344:
2339:
2334:
2329:
2324:
2318:
2315:
2314:
2311:
2308:
2307:
2304:
2301:
2300:
2297:
2294:
2293:
2291:
2286:
2282:
2279:
2275:
2270:
2265:
2260:
2255:
2249:
2246:
2245:
2242:
2239:
2238:
2235:
2232:
2231:
2228:
2225:
2224:
2222:
2217:
2213:
2208:
2203:
2198:
2193:
2187:
2184:
2183:
2180:
2177:
2176:
2173:
2170:
2169:
2166:
2163:
2162:
2160:
2155:
2131:
2122:
2117:
2112:
2110:
2107:
2103:
2098:
2093:
2089:
2084:
2079:
2078:
2075:
2071:
2068:
1950:parallelepiped
1942:
1941:
1930:
1927:
1924:
1921:
1918:
1915:
1912:
1907:
1901:
1898:
1897:
1894:
1891:
1890:
1888:
1883:
1878:
1872:
1869:
1868:
1865:
1862:
1859:
1858:
1856:
1851:
1847:
1844:
1839:
1834:
1830:
1826:
1820:
1815:
1810:
1805:
1801:
1798:
1794:
1789:
1784:
1779:
1773:
1768:
1763:
1759:
1730:scalar product
1686:for the angle
1526:standard basis
1505:
1502:
1501:
1500:
1489:
1484:
1478:
1475:
1473:
1470:
1469:
1466:
1463:
1461:
1458:
1457:
1455:
1450:
1447:
1444:
1441:
1438:
1435:
1432:
1429:
1426:
1423:
1420:
1417:
1414:
1411:
1408:
1405:
1402:
1399:
1396:
1393:
1388:
1382:
1379:
1377:
1374:
1371:
1368:
1367:
1364:
1361:
1359:
1356:
1353:
1350:
1349:
1347:
1322:
1311:
1310:
1299:
1294:
1287:
1284:
1280:
1278:
1275:
1274:
1270:
1267:
1263:
1261:
1258:
1257:
1255:
1250:
1245:
1239:
1236:
1234:
1231:
1230:
1227:
1224:
1222:
1219:
1218:
1216:
1211:
1208:
1205:
1201:
1198:
1194:
1191:
1188:
1185:
1182:
1178:
1175:
1171:
1168:
1165:
1162:
1159:
1154:
1147:
1144:
1140:
1137:
1134:
1132:
1129:
1128:
1124:
1121:
1117:
1114:
1111:
1109:
1106:
1105:
1103:
1087:
1086:
1075:
1072:
1069:
1066:
1063:
1060:
1057:
1054:
1049:
1043:
1040:
1038:
1035:
1034:
1031:
1028:
1026:
1023:
1022:
1020:
993:
987:
984:
982:
979:
978:
975:
972:
970:
967:
966:
964:
939:
936:
933:
921:
918:
917:
916:
905:
902:
899:
896:
893:
890:
887:
884:
881:
878:
875:
872:
869:
866:
863:
860:
857:
854:
849:
842:
839:
835:
833:
830:
829:
826:
823:
821:
818:
817:
815:
810:
805:
799:
796:
793:
791:
788:
787:
784:
781:
779:
776:
775:
773:
768:
754:
753:
742:
739:
736:
733:
730:
727:
724:
719:
713:
710:
708:
705:
704:
701:
698:
696:
693:
692:
690:
685:
680:
674:
671:
669:
666:
665:
662:
659:
657:
654:
653:
651:
646:
617:
611:
608:
606:
603:
602:
599:
596:
594:
591:
590:
588:
570:
567:
531:parallelepiped
461:
460:
457:
454:
447:
386:
383:
358:
357:
346:
343:
340:
337:
334:
331:
328:
325:
322:
319:
316:
313:
310:
307:
304:
301:
298:
295:
292:
289:
286:
283:
280:
277:
274:
269:
263:
260:
258:
255:
253:
250:
249:
246:
243:
241:
238:
236:
233:
232:
229:
226:
224:
221:
219:
216:
215:
213:
193:
192:
181:
178:
175:
172:
169:
166:
163:
158:
152:
149:
147:
144:
143:
140:
137:
135:
132:
131:
129:
93:the matrix is
91:if and only if
15:
9:
6:
4:
3:
2:
28128:
28117:
28114:
28112:
28109:
28107:
28106:Matrix theory
28104:
28102:
28099:
28098:
28096:
28081:
28073:
28072:
28069:
28063:
28060:
28058:
28057:Sparse matrix
28055:
28053:
28050:
28048:
28045:
28043:
28040:
28039:
28037:
28035:
28031:
28025:
28022:
28020:
28017:
28015:
28012:
28010:
28007:
28005:
28002:
28000:
27997:
27996:
27994:
27992:constructions
27991:
27987:
27981:
27980:Outermorphism
27978:
27976:
27973:
27971:
27968:
27966:
27963:
27961:
27958:
27956:
27953:
27951:
27948:
27946:
27943:
27941:
27940:Cross product
27938:
27936:
27933:
27932:
27930:
27928:
27924:
27918:
27915:
27913:
27910:
27908:
27907:Outer product
27905:
27903:
27900:
27898:
27895:
27893:
27890:
27888:
27887:Orthogonality
27885:
27884:
27882:
27880:
27876:
27870:
27867:
27865:
27864:Cramer's rule
27862:
27860:
27857:
27855:
27852:
27850:
27847:
27845:
27842:
27840:
27837:
27835:
27834:Decomposition
27832:
27830:
27827:
27826:
27824:
27822:
27818:
27813:
27803:
27800:
27798:
27795:
27793:
27790:
27788:
27785:
27783:
27780:
27778:
27775:
27773:
27770:
27768:
27765:
27763:
27760:
27758:
27755:
27753:
27750:
27748:
27745:
27743:
27740:
27738:
27735:
27733:
27730:
27728:
27725:
27723:
27720:
27718:
27715:
27713:
27710:
27709:
27707:
27703:
27697:
27694:
27692:
27689:
27688:
27685:
27681:
27674:
27669:
27667:
27662:
27660:
27655:
27654:
27651:
27645:
27642:
27640:
27637:
27634:
27631:
27628:
27624:
27621:
27618:
27616:
27613:
27610:
27606:
27605:
27600:
27596:
27591:
27586:
27585:
27580:
27579:"Determinant"
27577:
27572:
27569:
27565:
27564:
27559:
27558:"Determinant"
27554:
27553:
27543:
27537:
27529:
27525:
27521:
27517:
27514:
27510:
27506:
27502:
27498:
27494:
27490:
27486:
27482:
27479:
27477:9780801873966
27473:
27469:
27465:
27460:
27457:
27453:
27449:
27447:0-03-029558-0
27443:
27439:
27434:
27431:
27427:
27423:
27418:
27414:
27410:
27405:
27401:
27400:
27394:
27391:
27387:
27383:
27381:0-486-67766-4
27377:
27373:
27368:
27365:
27363:3-540-19376-6
27359:
27355:
27351:
27347:
27346:Meldrum, John
27343:
27338:
27337:
27328:
27322:
27318:
27313:
27304:on 2007-02-01
27300:
27296:
27292:
27288:
27282:
27277:
27272:
27268:
27261:
27256:
27252:
27247:
27243:
27237:
27233:
27228:
27225:
27223:9780387964126
27219:
27215:
27210:
27207:
27205:9780387962054
27201:
27197:
27192:
27188:
27182:
27178:
27174:
27170:
27166:
27162:
27160:
27157:
27153:
27149:
27146:
27143:
27141:0-534-99845-3
27137:
27133:
27128:
27124:
27120:
27116:
27113:on 2009-10-31
27112:
27108:
27102:
27098:
27097:
27091:
27088:
27086:0-387-98403-8
27082:
27078:
27074:
27073:
27068:
27064:
27061:
27059:9789401799447
27055:
27051:
27046:
27043:
27037:
27033:
27028:
27025:
27023:9780521883894
27019:
27015:
27014:
27009:
27004:
27001:
26997:
26993:
26987:
26983:
26979:
26975:
26970:
26967:
26965:9780128010495
26961:
26957:
26952:
26945:
26940:
26935:
26931:
26927:
26920:
26915:
26912:
26908:
26904:
26900:
26896:
26892:
26887:
26882:
26878:
26874:
26869:
26865:
26860:
26855:
26850:
26846:
26842:
26841:
26835:
26832:
26828:
26824:
26822:9780471452348
26818:
26814:
26809:
26804:
26799:
26794:
26789:
26785:
26781:
26780:
26775:
26770:
26767:
26765:9783540642435
26761:
26757:
26752:
26745:
26741:
26737:
26733:
26729:
26725:
26721:
26714:
26710:
26709:de Boor, Carl
26706:
26699:
26695:
26691:
26687:
26683:
26679:
26675:
26668:
26663:
26659:
26653:
26649:
26645:
26641:
26637:
26633:
26628:
26627:
26625:
26608:
26603:
26599:
26595:
26591:
26584:
26578:
26574:
26569:
26555:on 2011-08-07
26551:
26547:
26545:0-89791-875-4
26541:
26537:
26533:
26529:
26522:
26515:
26509:
26504:
26497:
26492:
26486:
26481:
26472:
26467:
26460:
26453:
26448:
26441:
26435:
26427:
26421:
26417:
26416:
26408:
26385:
26382:
26379:
26376:
26371:
26365:
26360:
26353:
26348:
26342:
26337:
26334:
26331:
26326:
26320:
26315:
26308:
26303:
26297:
26292:
26289:
26284:
26278:
26273:
26266:
26261:
26255:
26250:
26248:
26238:
26232:
26227:
26220:
26215:
26209:
26204:
26201:
26196:
26190:
26185:
26178:
26173:
26167:
26162:
26159:
26156:
26154:
26149:
26146:
26134:
26130:
26123:
26117:
26112:
26105:
26104:Bourbaki 1998
26101:
26096:
26089:
26088:Bourbaki 1998
26084:
26077:
26076:
26071:
26070:Mac Lane 1998
26066:
26059:
26054:
26047:
26042:
26034:
26027:
26020:
26015:
26008:
26003:
25996:
25991:
25985:
25980:
25974:
25968:
25961:
25956:
25950:, p. 494
25949:
25944:
25938:
25932:
25926:
25920:
25913:
25906:
25899:
25894:
25888:
25884:
25880:
25874:
25868:, p. 59)
25867:
25862:
25855:
25850:
25834:
25830:
25823:
25817:, p. 405
25816:
25811:
25802:
25800:
25798:
25791:
25789:
25782:
25775:
25770:
25763:
25758:
25750:
25746:
25742:
25738:
25734:
25730:
25727:(1): 99–115.
25726:
25722:
25714:
25707:
25702:
25695:
25691:
25686:
25678:
25674:
25671:
25661:
25652:
25647:
25643:
25639:
25635:
25628:
25619:
25614:
25607:
25601:
25595:
25587:
25583:
25579:
25575:
25570:
25565:
25561:
25557:
25550:
25542:
25538:
25534:
25530:
25526:
25522:
25518:
25514:
25510:
25503:
25496:
25491:
25485:, p. 306
25484:
25479:
25472:
25467:
25460:
25459:functoriality
25456:
25455:Bourbaki 1998
25450:
25443:
25438:
25431:
25427:
25422:
25415:
25410:
25402:
25397:
25396:
25387:
25372:
25368:
25362:
25354:
25350:
25346:
25345:
25337:
25330:
25325:
25321:
25311:
25308:
25306:
25303:
25301:
25298:
25296:
25293:
25291:
25288:
25287:
25282:
25276:
25271:
25264:
25262:
25258:
25257:
25252:
25251:Lewis Carroll
25247:
25245:
25241:
25236:
25232:
25227:
25223:
25219:
25214:
25200:
25192:
25171:
25167:
25160:
25149:
25145:
25141:
25120:
25116:
25109:
25096:
25094:
25073:
25069:
25062:
25033:
25027:
25021:
25001:
24998:
24995:
24973:
24969:
24965:
24959:
24953:
24930:
24924:
24904:
24879:
24875:
24868:
24837:
24831:
24822:
24816:
24807:
24804:
24798:
24785:
24784:
24783:
24769:
24749:
24746:
24726:
24723:
24703:
24683:
24663:
24643:
24623:
24603:
24583:
24563:
24556:
24537:
24534:
24531:
24528:
24525:
24522:
24515:
24514:
24513:
24512:as a product
24499:
24490:
24473:
24470:
24464:
24436:
24432:
24425:
24414:
24410:
24406:
24402:
24383:
24363:
24360:
24358:
24350:
24346:
24342:
24333:
24319:
24316:
24309:
24293:
24290:
24287:
24279:
24264:
24244:
24241:
24232:
24230:
24226:
24216:
24214:
24210:
24192:
24177:
24176:supermatrices
24173:
24169:
24165:
24161:
24157:
24153:
24149:
24145:
24140:bilinear form
24135:
24131:
24120:
24118:
24114:
24113:tracial state
24110:
24106:
24096:
24094:
24075:
24063:
24060:
24057:
24051:
24048:
24042:
24039:
24033:
24030:
24027:
24021:
24018:
24015:
24002:
24001:
24000:
23998:
23994:
23989:
23987:
23976:
23974:
23970:
23966:
23962:
23946:
23940:
23937:
23932:
23928:
23924:
23920:
23912:
23890:
23886:
23882:
23877:
23873:
23869:
23864:
23860:
23856:
23851:
23847:
23843:
23837:
23834:
23831:
23828:
23825:
23822:
23819:
23816:
23813:
23810:
23797:
23796:
23795:
23793:
23774:
23768:
23763:
23760:
23757:
23753:
23749:
23746:
23739:
23719:
23716:
23710:
23707:
23697:
23696:
23695:
23681:
23673:
23657:
23650:
23640:
23624:
23620:
23612:
23608:
23604:
23585:
23579:
23576:
23568:
23558:
23544:
23541:
23538:
23518:
23513:
23509:
23500:
23496:
23480:
23460:
23455:
23451:
23428:
23424:
23403:
23380:
23375:
23371:
23367:
23364:
23361:
23356:
23352:
23348:
23342:
23333:
23329:
23323:
23319:
23315:
23312:
23309:
23304:
23300:
23295:
23290:
23286:
23281:
23277:
23272:
23264:
23263:
23262:
23248:
23245:
23240:
23236:
23215:
23207:
23203:
23202:Bourbaki 1998
23187:
23167:
23162:
23158:
23137:
23132:
23128:
23100:
23095:
23091:
23087:
23084:
23081:
23078:
23073:
23069:
23065:
23062:
23057:
23053:
23049:
23044:
23037:
23033:
23029:
23026:
23023:
23018:
23014:
23010:
23005:
23001:
22993:
22988:
22984:
22978:
22973:
22968:
22964:
22960:
22957:
22952:
22948:
22936:
22935:
22934:
22920:
22900:
22880:
22875:
22871:
22863:
22847:
22827:
22807:
22800:
22796:
22780:
22760:
22740:
22734:
22731:
22728:
22718:
22694:
22689:
22674:
22670:
22666:
22656:
22655:
22654:
22652:
22648:
22630:
22622:
22597:
22593:
22584:
22580:
22576:
22560:
22540:
22532:
22527:
22499:
22496:
22493:
22489:
22482:
22470:
22456:
22453:
22450:
22446:
22430:
22423:
22422:
22421:
22407:
22387:
22364:
22358:
22353:
22349:
22339:
22333:
22328:
22324:
22320:
22314:
22308:
22303:
22299:
22278:
22272:
22269:
22266:
22259:
22250:
22246:
22244:
22228:
22220:
22204:
22184:
22181:
22178:
22170:
22151:
22146:
22142:
22132:
22126:
22121:
22117:
22109:
22108:
22107:
22106:
22101:
22099:
22078:
22075:
22055:
22047:
22031:
22008:
22002:
21997:
21994:
21991:
21987:
21983:
21980:
21971:
21957:
21954:
21948:
21934:
21896:
21889:
21879:
21877:
21873:
21869:
21865:
21861:
21858:
21839:
21833:
21830:
21827:
21820:
21819:
21818:
21817:
21813:
21794:
21788:
21779:
21773:
21762:
21759:
21751:
21739:
21730:
21724:
21712:
21701:
21698:
21690:
21681:
21675:
21662:
21661:
21660:
21657:
21654:
21650:
21645:
21641:
21637:
21633:
21618:
21616:
21611:
21609:
21590:
21582:
21577:
21559:
21556:
21543:
21525:
21519:
21514:
21510:
21506:
21498:
21483:
21475:
21469:
21465:
21457:
21456:
21455:
21452:
21448:
21444:
21439:
21435:
21431:
21427:
21423:
21420:
21416:
21412:
21408:
21389:
21384:
21381:
21378:
21375:
21372:
21369:
21366:
21361:
21353:
21349:
21338:
21334:
21324:
21319:
21313:
21307:
21300:
21299:
21298:
21297:
21292:
21288:
21268:
21263:
21248:
21238:
21235:
21228:
21227:
21226:
21224:
21220:
21216:
21193:
21178:
21168:
21165:
21156:
21152:
21150:
21149:spanning tree
21126:
21123:
21120:
21117:
21114:
21111:
21108:
21105:
21102:
21099:
21096:
21082:
21077:
21074:
21052:
21049:
21046:
21043:
21040:
21037:
21034:
21027:
21023:
21019:
21000:
20994:
20988:
20985:
20979:
20975:
20964:
20959:
20950:
20941:
20935:
20929:
20926:
20919:
20918:
20917:
20916:is given by:
20900:
20894:
20886:
20870:
20850:
20830:
20827:
20824:
20802:
20787:
20777:
20774:
20754:
20726:
20692:
20686:
20678:
20675:
20657:
20647:
20644:
20624:
20602:
20587:
20577:
20574:
20566:
20562:
20552:
20550:
20546:
20534:
20530:
20526:
20523:is positive,
20522:
20517:
20515:
20511:
20507:
20503:
20499:
20495:
20491:
20487:
20483:
20479:
20475:
20471:
20465:
20455:
20453:
20449:
20445:
20441:
20437:
20421:
20418:
20415:
20395:
20372:
20367:
20358:
20347:
20344:
20341:
20333:
20329:
20323:
20315:
20304:
20301:
20298:
20290:
20286:
20277:
20266:
20263:
20260:
20252:
20248:
20240:
20235:
20230:
20225:
20215:
20208:
20204:
20200:
20194:
20186:
20179:
20175:
20171:
20162:
20155:
20151:
20147:
20136:
20128:
20124:
20118:
20110:
20102:
20098:
20089:
20081:
20077:
20070:
20065:
20059:
20048:
20044:
20040:
20037:
20034:
20029:
20025:
20018:
20011:
20010:
20009:
20007:
20003:
19987:
19984:
19981:
19958:
19950:
19946:
19942:
19939:
19936:
19930:
19922:
19918:
19909:
19893:
19890:
19887:
19879:
19876:
19858:
19854:
19831:
19821:
19816:
19812:
19808:
19803:
19799:
19778:
19758:
19747:
19737:
19735:
19731:
19727:
19706:
19702:
19695:
19669:
19664:
19660:
19652:
19643:
19640:
19633:
19627:
19624:
19621:
19615:
19609:
19606:
19602:
19595:
19594:
19593:
19591:
19573:
19569:
19542:
19531:
19527:
19523:
19518:
19510:
19506:
19500:
19493:
19490:
19487:
19483:
19475:
19471:
19463:
19460:
19457:
19453:
19447:
19440:
19436:
19429:
19419:
19415:
19409:
19404:
19401:
19398:
19394:
19390:
19385:
19377:
19373:
19367:
19362:
19357:
19350:
19346:
19339:
19331:
19323:
19319:
19305:
19304:
19303:
19289:
19269:
19249:
19227:
19223:
19199:
19196:
19193:
19190:
19187:
19184:
19181:
19178:
19175:
19172:
19169:
19159:
19143:
19139:
19126:
19121:
19117:
19109:
19108:
19107:
19105:
19104:Cramer's rule
19086:
19060:
19040:
19037:
19034:
19031:
19023:
19016:Cramer's rule
19008:
19006:
19002:
18998:
18994:
18990:
18986:
18982:
18978:
18974:
18970:
18966:
18962:
18958:
18954:
18950:
18946:
18941:
18939:
18935:
18931:
18927:
18923:
18922:
18917:
18913:
18912:Jacobi (1841)
18909:
18907:
18903:
18898:
18894:
18889:
18885:
18881:
18876:
18874:
18870:
18866:
18862:
18858:
18856:
18852:
18848:
18844:
18840:
18836:
18834:
18830:
18829:Bézout (1779)
18826:
18825:Cramer (1750)
18822:
18818:
18817:Seki Takakazu
18813:
18811:
18807:
18806:
18801:
18773:
18765:
18757:
18755:
18747:
18717:
18709:
18707:
18699:
18669:
18661:
18659:
18651:
18622:
18621:
18620:
18618:
18617:cross product
18602:
18599:
18596:
18593:
18590:
18568:
18562:
18557:
18552:
18546:
18541:
18538:
18518:
18515:
18512:
18503:
18487:
18468:
18452:
18449:
18446:
18421:
18417:
18408:
18404:
18385:
18381:
18376:
18372:
18368:
18364:
18361:
18358:
18352:
18346:
18343:
18340:
18337:
18334:
18328:
18325:
18322:
18319:
18306:
18305:
18304:
18290:
18287:
18284:
18276:
18255:
18250:
18246:
18242:
18238:
18235:
18232:
18228:
18224:
18219:
18216:
18212:
18207:
18203:
18200:
18194:
18185:
18181:
18176:
18172:
18168:
18164:
18161:
18158:
18152:
18146:
18140:
18137:
18131:
18128:
18125:
18119:
18110:
18104:
18101:
18098:
18095:
18082:
18081:
18080:
18063:
18058:
18055:
18050:
18045:
18042:
18038:
18034:
18026:
18017:
18012:
18009:
18001:
17995:
17992:
17989:
17981:
17978:
17974:
17962:
17943:
17942:
17941:
17927:
17904:
17900:
17893:
17890:
17885:
17882:
17874:
17871:
17867:
17862:
17858:
17855:
17849:
17840:
17834:
17831:
17823:
17814:
17804:
17803:
17802:
17788:
17768:
17760:
17741:
17735:
17732:
17709:
17705:
17698:
17695:
17690:
17687:
17678:
17672:
17669:
17665:
17661:
17658:
17655:
17649:
17646:
17638:
17629:
17619:
17618:
17617:
17615:
17611:
17571:
17568:
17565:
17550:
17540:
17538:
17534:
17530:
17526:
17525:harmonic mean
17507:
17501:
17496:
17492:
17488:
17484:
17481:
17476:
17473:
17466:
17460:
17454:
17451:
17446:
17443:
17438:
17432:
17429:
17420:
17411:
17404:
17399:
17396:
17392:
17388:
17384:
17381:
17377:
17368:
17367:
17366:
17363:
17361:
17357:
17352:
17348:
17325:
17322:
17319:
17313:
17310:
17307:
17301:
17292:
17289:
17286:
17282:
17276:
17273:
17269:
17265:
17262:
17258:
17254:
17251:
17244:
17243:
17242:
17239:
17228:
17224:
17220:
17213:
17209:
17196:
17177:
17171:
17166:
17161:
17156:
17152:
17148:
17144:
17141:
17136:
17130:
17126:
17120:
17112:
17109:
17093:
17090:
17087:
17083:
17079:
17075:
17067:
17064:
17060:
17048:
17045:
17042:
17038:
17030:
17029:
17028:
17025:
17005:
16999:
16994:
16989:
16984:
16980:
16976:
16972:
16969:
16964:
16958:
16950:
16947:
16931:
16928:
16925:
16921:
16917:
16913:
16905:
16902:
16898:
16886:
16883:
16880:
16876:
16872:
16866:
16863:
16860:
16847:
16846:
16845:
16843:
16839:
16816:
16811:
16806:
16802:
16796:
16792:
16788:
16782:
16776:
16773:
16770:
16765:
16760:
16756:
16750:
16745:
16741:
16735:
16731:
16727:
16722:
16717:
16709:
16706:
16696:
16695:
16694:
16690:
16680:
16673:
16669:
16662:
16652:
16645:
16641:
16635:
16612:
16604:
16600:
16596:
16593:
16590:
16585:
16581:
16577:
16572:
16568:
16559:
16555:
16548:
16545:
16538:
16530:
16527:
16518:
16512:
16499:
16498:
16497:
16495:
16491:
16486:
16482:
16478:
16474:
16455:
16452:
16449:
16444:
16440:
16436:
16431:
16426:
16423:
16420:
16416:
16408:
16407:
16406:
16402:
16379:
16372:
16368:
16362:
16357:
16353:
16349:
16344:
16341:
16335:
16330:
16326:
16318:
16314:
16309:
16301:
16298:
16293:
16289:
16280:
16277:
16266:
16261:
16258:
16255:
16251:
16242:
16239:
16234:
16230:
16226:
16223:
16220:
16217:
16212:
16208:
16204:
16201:
16196:
16192:
16184:
16181:
16176:
16172:
16168:
16165:
16162:
16157:
16153:
16149:
16144:
16140:
16132:
16128:
16122:
16109:
16108:
16107:
16090:
16081:
16078:
16075:
16072:
16069:
16056:
16051:
16047:
16043:
16039:
16036:
16031:
16028:
16025:
16022:
16019:
16015:
16009:
16004:
16001:
15998:
15994:
15988:
15985:
15980:
15977:
15972:
15969:
15966:
15962:
15949:
15946:
15943:
15938:
15934:
15926:
15925:
15924:
15922:
15914:
15910:
15900:
15896:
15892:
15869:
15865:
15860:
15855:
15851:
15847:
15843:
15840:
15837:
15834:
15828:
15822:
15819:
15812:
15807:
15803:
15799:
15795:
15792:
15789:
15786:
15781:
15776:
15771:
15766:
15762:
15758:
15754:
15751:
15747:
15742:
15739:
15734:
15729:
15722:
15716:
15713:
15709:
15703:
15698:
15694:
15690:
15686:
15683:
15680:
15677:
15672:
15667:
15660:
15654:
15651:
15647:
15641:
15635:
15632:
15627:
15625:
15617:
15604:
15600:
15595:
15590:
15586:
15582:
15578:
15575:
15572:
15569:
15565:
15560:
15556:
15552:
15548:
15545:
15536:
15530:
15527:
15524:
15521:
15516:
15511:
15504:
15498:
15495:
15491:
15485:
15479:
15476:
15471:
15469:
15461:
15448:
15444:
15439:
15434:
15430:
15426:
15422:
15419:
15416:
15411:
15406:
15399:
15393:
15390:
15386:
15380:
15374:
15371:
15366:
15364:
15356:
15339:
15338:
15337:
15333:
15326:
15319:
15299:
15290:
15284:
15281:
15275:
15272:
15269:
15263:
15250:
15249:
15248:
15227:
15224:
15218:
15212:
15209:
15202:
15201:
15200:
15190:
15166:
15143:
15131:
15125:
15122:
15110:
15107:
15104:
15098:
15092:
15089:
15082:
15081:
15080:
15053:
15047:
15044:
15038:
15035:
15032:
15023:
15017:
15014:
15001:
15000:
14999:
14989:
14985:
14975:
14961:
14941:
14921:
14896:
14888:
14885:
14882:
14878:
14872:
14865:
14862:
14859:
14855:
14847:
14844:
14841:
14837:
14829:
14824:
14819:
14814:
14805:
14802:
14799:
14795:
14789:
14782:
14779:
14776:
14772:
14764:
14761:
14758:
14754:
14744:
14741:
14738:
14734:
14728:
14721:
14718:
14715:
14711:
14703:
14700:
14697:
14693:
14686:
14681:
14676:
14672:
14664:
14663:
14662:
14660:
14656:
14652:
14633:
14630:
14624:
14616:
14612:
14604:
14603:
14602:
14588:
14580:
14564:
14556:
14540:
14520:
14512:
14511:indeterminate
14496:
14473:
14467:
14464:
14461:
14458:
14455:
14446:
14440:
14432:
14428:
14420:
14419:
14418:
14415:
14401:
14381:
14361:
14341:
14321:
14301:
14292:
14290:
14271:
14266:
14262:
14258:
14253:
14249:
14243:
14239:
14235:
14230:
14226:
14220:
14215:
14212:
14209:
14205:
14201:
14195:
14182:
14181:
14180:
14178:
14163:
14145:
14141:
14137:
14134:
14131:
14126:
14122:
14118:
14113:
14109:
14101:
14098:
14082:
14074:
14071:-matrix with
14058:
14055:
14052:
14032:
14024:
14020:
14005:
13991:
13988:
13971:
13965:
13939:
13936:
13897:
13892:
13888:
13884:
13879:
13875:
13871:
13852:
13848:
13844:
13838:
13827:
13823:
13819:
13808:
13805:
13802:
13799:
13786:
13785:
13784:
13765:
13762:
13759:
13739:
13736:
13733:
13730:
13710:
13707:
13671:
13651:
13629:
13623:
13618:
13611:
13608:
13603:
13597:
13592:
13589:
13584:
13578:
13573:
13566:
13561:
13555:
13550:
13547:
13539:
13536:
13533:
13530:
13510:
13507:
13504:
13494:
13477:
13471:
13468:
13457:
13451:
13437:
13422:
13421:
13420:
13398:
13394:
13388:
13384:
13380:
13375:
13371:
13365:
13361:
13357:
13352:
13348:
13342:
13338:
13334:
13329:
13325:
13319:
13315:
13308:
13300:
13296:
13292:
13287:
13283:
13271:
13267:
13263:
13258:
13254:
13243:
13242:
13241:
13224:
13219:
13215:
13209:
13205:
13201:
13196:
13192:
13186:
13182:
13178:
13173:
13169:
13163:
13159:
13155:
13150:
13146:
13140:
13136:
13132:
13126:
13117:
13111:
13098:
13097:
13096:
13082:
13059:
13033:
13010:
13005:
13001:
12995:
12991:
12987:
12982:
12978:
12972:
12968:
12964:
12959:
12955:
12949:
12945:
12941:
12936:
12932:
12926:
12922:
12918:
12913:
12909:
12903:
12899:
12895:
12890:
12886:
12880:
12876:
12872:
12867:
12863:
12857:
12853:
12849:
12844:
12840:
12834:
12830:
12822:
12821:
12820:
12803:
12795:
12791:
12787:
12782:
12778:
12766:
12762:
12758:
12753:
12749:
12742:
12734:
12730:
12726:
12721:
12717:
12705:
12701:
12697:
12692:
12688:
12677:
12676:
12675:
12659:
12656:
12652:
12648:
12643:
12640:
12636:
12608:
12602:
12599:
12588:
12582:
12568:
12557:
12551:
12542:
12536:
12527:
12521:
12518:
12515:
12502:
12501:
12500:
12486:
12483:
12480:
12466:
12464:
12444:
12438:
12430:
12418:
12412:
12404:
12392:
12386:
12378:
12375:
12372:
12343:
12340:
12337:
12309:
12306:
12303:
12295:
12291:
12283:
12279:
12258:
12249:
12243:
12234:
12228:
12225:
12222:
12188:
12185:
12182:
12173:
12167:
12164:
12161:
12152:
12146:
12137:
12131:
12128:
12125:
12122:
12119:
12093:
12073:
12053:
12046:
12043:However, for
12041:
12039:
12035:
12019:
12016:
12013:
11983:
11980:
11960:
11957:
11937:
11917:
11909:
11894:
11891:
11884:
11878:
11875:
11871:
11868:
11862:
11853:
11849:
11845:
11840:
11837:
11833:
11829:
11826:
11823:
11819:
11809:
11800:
11794:
11791:
11788:
11785:
11772:
11770:
11755:
11751:
11747:
11742:
11739:
11735:
11731:
11728:
11717:
11712:
11702:
11693:
11687:
11684:
11681:
11678:
11665:
11663:
11658:
11654:
11649:
11633:
11630:
11627:
11624:
11621:
11618:
11614:
11610:
11607:
11604:
11593:
11588:
11577:
11576:
11574:
11570:
11566:
11562:
11561:
11559:
11556:
11553:
11549:
11543:
11539:
11533:
11529:
11524:
11520:
11501:
11497:
11493:
11490:
11487:
11476:
11471:
11464:
11460:
11456:
11453:
11450:
11439:
11434:
11423:
11422:
11421:
11419:
11415:
11410:
11406:
11401:
11396:
11392:
11387:
11383:
11360:
11354:
11351:
11348:
11336:
11333:
11330:
11321:
11316:
11310:
11305:
11298:
11293:
11287:
11275:
11274:
11273:
11259:
11239:
11219:
11216:
11213:
11193:
11190:
11187:
11178:
11164:
11161:
11158:
11135:
11129:
11126:
11123:
11120:
11117:
11108:
11103:
11097:
11092:
11085:
11080:
11074:
11062:
11061:
11060:
11046:
11043:
11040:
11037:
11034:
11026:
11011:
10991:
10983:
10978:
10976:
10953:
10947:
10942:
10939:
10935:
10931:
10928:
10925:
10913:
10904:
10902:
10892:
10884:
10880:
10874:
10865:
10862:
10858:
10854:
10849:
10844:
10841:
10837:
10833:
10830:
10827:
10821:
10810:
10801:
10799:
10787:
10784:
10776:
10766:
10757:
10754:
10750:
10739:
10733:
10727:
10719:
10716:
10712:
10706:
10701:
10698:
10694:
10690:
10683:
10676:
10672:
10665:
10651:
10645:
10640:
10633:
10628:
10622:
10611:
10602:
10600:
10593:
10587:
10582:
10575:
10570:
10564:
10548:
10547:
10546:
10532:
10523:
10509:
10506:
10503:
10483:
10460:
10455:
10452:
10448:
10444:
10441:
10438:
10429:
10396:
10390:
10385:
10382:
10378:
10374:
10371:
10368:
10356:
10347:
10345:
10335:
10329:
10324:
10321:
10317:
10313:
10310:
10307:
10300:
10297:
10293:
10289:
10282:
10275:
10271:
10264:
10253:
10244:
10242:
10230:
10227:
10219:
10209:
10200:
10197:
10193:
10182:
10176:
10170:
10162:
10158:
10152:
10145:
10140:
10137:
10133:
10129:
10122:
10119:
10115:
10108:
10094:
10088:
10083:
10076:
10071:
10065:
10054:
10045:
10043:
10036:
10030:
10025:
10018:
10013:
10007:
9991:
9990:
9989:
9987:
9971:
9948:
9943:
9937:
9932:
9925:
9920:
9914:
9906:
9900:
9888:
9879:
9874:
9868:
9863:
9856:
9851:
9845:
9833:
9832:
9831:
9829:
9813:
9810:
9807:
9787:
9784:
9781:
9761:
9758:
9755:
9735:
9732:
9729:
9722:of dimension
9709:
9706:
9703:
9700:
9697:
9694:
9691:
9683:
9667:
9664:
9661:
9633:
9630:
9627:
9624:
9618:
9611:
9606:
9601:
9598:
9594:
9586:
9585:
9584:
9582:
9563:
9560:
9553:
9550:
9547:
9541:
9538:
9535:
9532:
9529:
9526:
9523:
9517:
9504:
9503:
9502:
9485:
9480:
9477:
9473:
9467:
9464:
9461:
9453:
9450:
9444:
9439:
9436:
9433:
9422:
9416:
9413:
9403:
9402:
9401:
9384:
9378:
9375:
9368:
9358:
9357:) submatrix.
9356:
9352:
9348:
9345:
9341:
9337:
9313:
9310:
9293:
9289:
9285:
9281:
9265:
9261:
9255:
9251:
9247:
9242:
9238:
9233:
9227:
9224:
9221:
9218:
9215:
9212:
9209:
9205:
9201:
9196:
9188:
9185:
9182:
9177:
9173:
9167:
9160:
9157:
9154:
9149:
9145:
9137:
9134:
9131:
9126:
9122:
9114:
9111:
9108:
9103:
9099:
9091:
9086:
9081:
9076:
9071:
9062:
9057:
9053:
9047:
9040:
9035:
9031:
9023:
9018:
9014:
9006:
9001:
8997:
8987:
8983:
8977:
8970:
8966:
8958:
8954:
8946:
8942:
8934:
8929:
8924:
8919:
8914:
8908:
8899:
8879:
8874:
8871:
8868:
8864:
8858:
8855:
8852:
8848:
8842:
8839:
8836:
8828:
8825:
8817:
8812:
8809:
8806:
8802:
8798:
8792:
8779:
8778:
8777:
8775:
8761:
8739:
8736:
8733:
8708:
8702:
8697:
8690:
8685:
8679:
8674:
8671:
8666:
8660:
8655:
8648:
8643:
8637:
8632:
8629:
8624:
8618:
8613:
8606:
8601:
8595:
8590:
8587:
8582:
8576:
8571:
8566:
8559:
8554:
8549:
8542:
8537:
8532:
8526:
8517:
8516:
8515:
8501:
8498:
8495:
8487:
8464:
8459:
8456:
8453:
8449:
8443:
8440:
8437:
8433:
8427:
8424:
8421:
8413:
8410:
8402:
8397:
8394:
8391:
8387:
8383:
8377:
8364:
8363:
8362:
8348:
8340:
8322:
8319:
8316:
8312:
8306:
8303:
8300:
8292:
8289:
8266:
8246:
8226:
8203:
8200:
8197:
8191:
8185:
8182:
8179:
8154:
8151:
8148:
8144:
8135:
8131:
8116:
8108:
8099:
8097:
8093:
8089:
8084:
8068:
8064:
8040:
8034:
8029:
8025:
8020:
8013:
8007:
8002:
7998:
7990:
7971:
7965:
7960:
7956:
7947:
7928:
7922:
7917:
7913:
7904:
7885:
7879:
7874:
7870:
7849:
7827:
7823:
7799:
7793:
7788:
7784:
7775:
7770:
7768:
7764:
7760:
7756:
7752:
7733:
7727:
7722:
7718:
7714:
7708:
7702:
7697:
7693:
7685:
7681:
7662:
7656:
7651:
7647:
7639:
7623:
7616:over a field
7603:
7577:
7574:
7563:
7551:
7542:
7532:
7527:
7523:
7518:
7515:
7511:
7507:
7496:
7495:
7494:
7491:
7487:
7471:
7462:
7448:
7428:
7405:
7385:
7359:
7347:
7338:
7332:
7329:
7316:
7315:
7314:
7312:
7296:
7276:
7268:
7258:
7256:
7252:
7247:
7243:
7219:
7210:
7206:
7195:
7191:
7180:
7179:
7178:
7176:
7160:
7152:
7142:
7129:
7126:
7117:
7114:
7108:
7105:
7102:
7099:
7093:
7090:
7082:
7074:
7071:
7063:
7045:
7027:
7019:
7011:
6998:
6996:
6978:
6970:
6967:
6959:
6946:
6944:
6926:
6918:
6910:
6897:
6878:
6870:
6862:
6850:
6847:
6846:
6843:
6827:
6824:
6819:
6810:
6808:
6805:
6803:
6800:
6798:
6795:
6792:
6791:
6788:
6773:
6767:
6764:
6759:
6754:
6747:
6742:
6737:
6730:
6725:
6722:
6717:
6711:
6706:
6703:
6695:
6693:
6678:
6672:
6669:
6664:
6659:
6652:
6647:
6642:
6635:
6630:
6627:
6622:
6616:
6611:
6608:
6600:
6598:
6583:
6577:
6574:
6569:
6564:
6557:
6552:
6547:
6540:
6535:
6530:
6527:
6521:
6516:
6513:
6505:
6489:
6483:
6480:
6475:
6470:
6463:
6458:
6453:
6446:
6441:
6438:
6433:
6430:
6424:
6419:
6416:
6409:
6406:
6405:
6390:
6366:
6361:
6355:
6352:
6347:
6342:
6339:
6332:
6327:
6322:
6315:
6310:
6307:
6302:
6299:
6293:
6288:
6285:
6278:
6277:
6276:
6262:
6253:
6228:
6220:
6204:
6199:
6196:
6192:
6186:
6181:
6178:
6175:
6171:
6167:
6162:
6159:
6155:
6151:
6146:
6142:
6136:
6132:
6128:
6122:
6096:
6093:
6090:
6070:
6067:
6064:
6044:
6041:
6036:
6033:
6029:
6020:
6004:
5996:
5992:
5988:
5985:
5981:
5977:
5974:
5958:
5948:
5944:
5940:
5937:
5934:
5929:
5925:
5921:
5916:
5912:
5908:
5903:
5899:
5895:
5890:
5886:
5877:
5867:
5863:
5859:
5856:
5853:
5848:
5844:
5840:
5835:
5831:
5827:
5822:
5818:
5814:
5809:
5805:
5796:
5793:
5783:
5779:
5775:
5772:
5767:
5763:
5759:
5754:
5750:
5746:
5741:
5737:
5733:
5728:
5724:
5698:
5688:
5684:
5680:
5677:
5674:
5669:
5665:
5661:
5658:
5655:
5650:
5646:
5642:
5639:
5636:
5631:
5627:
5618:
5615:
5605:
5601:
5597:
5594:
5591:
5586:
5582:
5578:
5575:
5570:
5566:
5562:
5559:
5556:
5551:
5547:
5533:
5518:
5498:
5495:
5492:
5469:
5458:
5454:
5450:
5444:
5441:
5427:
5423:
5422:
5421:
5413:
5410:
5405:
5403:
5387:
5384:
5381:
5355:
5352:
5342:
5338:
5334:
5331:
5328:
5325:
5322:
5319:
5316:
5313:
5310:
5307:
5304:
5299:
5295:
5282:
5281:
5279:
5278:
5271:
5244:
5240:
5236:
5233:
5230:
5227:
5224:
5221:
5218:
5213:
5209:
5200:
5190:
5186:
5182:
5179:
5176:
5173:
5170:
5167:
5162:
5158:
5149:
5146:
5143:
5141:
5126:
5122:
5118:
5115:
5112:
5107:
5104:
5101:
5097:
5093:
5090:
5087:
5084:
5081:
5078:
5075:
5070:
5067:
5064:
5060:
5056:
5053:
5050:
5045:
5041:
5030:
5028:
5018:
5002:
5001:
4999:
4995:
4992:and a number
4991:
4987:
4984:
4968:
4965:
4962:
4959:
4956:
4953:
4948:
4944:
4936:
4920:
4912:
4908:
4907:
4900:
4897:
4881:
4861:
4858:
4854:
4851:
4848:
4835:
4834:
4833:
4831:
4827:
4809:
4805:
4797:
4796:column vector
4778:
4766:
4762:
4758:
4755:
4752:
4747:
4743:
4732:
4729:
4722:
4721:
4720:
4706:
4698:
4682:
4679:
4676:
4657:
4644:
4638:
4635:
4632:
4629:
4626:
4596:
4589:
4585:
4581:
4578:
4574:
4570:
4562:
4558:
4554:
4551:
4547:
4539:
4535:
4531:
4526:
4522:
4517:
4509:
4505:
4501:
4498:
4495:
4490:
4486:
4482:
4477:
4473:
4468:
4464:
4458:
4445:
4444:
4443:
4441:
4418:
4415:
4412:
4409:
4406:
4395:
4371:
4367:
4363:
4360:
4357:
4352:
4348:
4343:
4335:
4314:
4305:
4299:
4296:
4293:
4289:
4283:
4278:
4275:
4272:
4268:
4261:
4255:
4252:
4248:
4240:
4236:
4232:
4229:
4225:
4221:
4215:
4202:
4201:
4200:
4198:
4179:
4171:
4165:
4162:
4159:
4155:
4151:
4143:
4137:
4134:
4131:
4127:
4120:
4114:
4111:
4104:
4100:
4096:
4093:
4089:
4085:
4080:
4072:
4069:
4066:
4062:
4058:
4053:
4050:
4047:
4043:
4035:
4031:
4022:
4019:
4016:
4012:
4008:
4003:
4000:
3997:
3993:
3986:
3981:
3975:
3962:
3961:
3960:
3959:for the sum,
3958:
3939:
3934:
3926:
3923:
3920:
3916:
3912:
3907:
3904:
3901:
3897:
3889:
3885:
3876:
3873:
3870:
3866:
3862:
3857:
3854:
3851:
3847:
3840:
3835:
3832:
3825:
3824:
3823:
3820:
3807:
3804:
3784:
3781:
3778:
3758:
3735:
3729:
3726:
3704:
3700:
3691:
3672:
3666:
3663:
3660:
3657:
3651:
3645:
3642:
3636:
3630:
3610:
3603:
3584:
3581:
3578:
3575:
3572:
3569:
3566:
3555:
3551:
3535:
3532:
3529:
3517:
3513:
3508:
3501:
3494:
3490:
3486:
3484:
3480:
3476:
3472:
3467:
3463:
3459:
3455:
3433:
3430:
3427:
3424:
3421:
3418:
3415:
3412:
3409:
3406:
3403:
3400:
3397:
3394:
3391:
3388:
3385:
3382:
3379:
3376:
3373:
3370:
3367:
3364:
3361:
3356:
3350:
3345:
3340:
3333:
3328:
3323:
3316:
3311:
3306:
3300:
3291:
3290:
3289:
3283:
3272:
3262:
3260:
3256:
3237:
3232:
3224:
3221:
3218:
3214:
3208:
3201:
3198:
3195:
3191:
3183:
3180:
3177:
3173:
3165:
3160:
3155:
3150:
3141:
3138:
3135:
3131:
3125:
3118:
3115:
3112:
3108:
3100:
3097:
3094:
3090:
3080:
3077:
3074:
3070:
3064:
3057:
3054:
3051:
3047:
3039:
3036:
3033:
3029:
3022:
3013:
3012:
3011:
3009:
3005:
3000:
2998:
2980:
2977:
2974:
2970:
2946:
2941:
2933:
2930:
2927:
2923:
2917:
2910:
2907:
2904:
2900:
2892:
2889:
2886:
2882:
2874:
2869:
2864:
2859:
2850:
2847:
2844:
2840:
2834:
2827:
2824:
2821:
2817:
2809:
2806:
2803:
2799:
2789:
2786:
2783:
2779:
2773:
2766:
2763:
2760:
2756:
2748:
2745:
2742:
2738:
2731:
2726:
2723:
2716:
2715:
2714:
2712:
2708:
2704:
2703:square matrix
2700:
2690:
2688:
2684:
2680:
2676:
2672:
2669:is less than
2668:
2664:
2660:
2656:
2652:
2648:
2632:
2626:
2615:
2612:
2606:
2592:
2589:
2584:
2571:
2567:
2563:
2554:
2551:
2546:
2542:
2538:
2535:
2532:
2527:
2515:
2511:
2507:
2504:
2501:
2496:
2484:
2480:
2475:
2471:
2468:
2448:
2443:
2433:
2430:
2427:
2422:
2412:
2407:
2392:
2391:parallelotope
2389:-dimensional
2388:
2384:
2382:
2365:
2342:
2337:
2327:
2322:
2316:
2309:
2302:
2295:
2289:
2284:
2280:
2277:
2273:
2268:
2258:
2253:
2247:
2240:
2233:
2226:
2220:
2215:
2211:
2206:
2196:
2191:
2185:
2178:
2171:
2164:
2158:
2153:
2146:
2145:
2144:
2129:
2120:
2108:
2101:
2087:
2073:
2069:
2066:
2058:
2054:
2050:
2046:
2040:
2037:
2033:
2028:
2022:
2018:
2014:
2010:
2002:
1998:
1990:
1986:
1976:
1972:
1971:
1965:
1963:
1959:
1951:
1946:
1928:
1925:
1922:
1919:
1916:
1913:
1910:
1905:
1899:
1892:
1886:
1881:
1876:
1870:
1863:
1860:
1854:
1849:
1845:
1842:
1837:
1832:
1824:
1818:
1813:
1803:
1799:
1796:
1792:
1757:
1745:
1744:
1743:
1740:
1736:
1731:
1726:
1722:
1718:
1710:
1706:
1702:
1697:
1693:
1689:
1684:
1680:
1676:
1668:
1664:
1660:
1653:
1649:
1645:
1639:
1635:
1631:To show that
1629:
1627:
1623:
1619:
1614:
1603:
1599:
1593:
1589:
1585:
1577:
1573:
1569:
1565:
1557:
1553:
1543:
1539:
1538:parallelogram
1527:
1523:
1510:
1487:
1482:
1476:
1471:
1464:
1459:
1453:
1448:
1445:
1442:
1436:
1433:
1430:
1427:
1424:
1418:
1415:
1412:
1409:
1406:
1403:
1400:
1397:
1394:
1391:
1386:
1380:
1375:
1372:
1369:
1362:
1357:
1354:
1351:
1345:
1336:
1335:
1334:
1320:
1297:
1292:
1285:
1282:
1276:
1268:
1265:
1259:
1253:
1248:
1243:
1237:
1232:
1225:
1220:
1214:
1209:
1206:
1199:
1196:
1192:
1189:
1183:
1176:
1173:
1169:
1166:
1160:
1157:
1152:
1145:
1142:
1138:
1135:
1130:
1122:
1119:
1115:
1112:
1107:
1101:
1092:
1091:
1090:
1073:
1070:
1067:
1064:
1061:
1058:
1055:
1052:
1047:
1041:
1036:
1029:
1024:
1018:
1009:
1008:
1007:
991:
985:
980:
973:
968:
962:
953:
937:
934:
931:
903:
900:
897:
891:
888:
885:
879:
870:
867:
861:
858:
852:
847:
840:
837:
831:
824:
819:
813:
808:
803:
797:
794:
789:
782:
777:
771:
759:
758:
757:
756:For example,
740:
737:
734:
731:
728:
725:
722:
717:
711:
706:
699:
694:
688:
683:
678:
672:
667:
660:
655:
649:
637:
636:
635:
615:
609:
604:
597:
592:
586:
566:
564:
560:
556:
552:
548:
540:
536:
532:
529:-dimensional
524:
521:-dimensional
517:, the signed
516:
512:
508:
504:
503:Cramer's rule
500:
496:
491:
489:
485:
481:
477:
473:
469:
464:
458:
455:
448:
441:
437:
436:
435:
432:
428:
422:
420:
416:
412:
408:
400:
384:
381:
373:
368:
364:
344:
341:
338:
335:
332:
329:
326:
323:
320:
317:
314:
311:
308:
305:
302:
299:
296:
293:
290:
287:
284:
281:
278:
275:
272:
267:
261:
256:
251:
244:
239:
234:
227:
222:
217:
211:
202:
201:
200:
179:
176:
173:
170:
167:
164:
161:
156:
150:
145:
138:
133:
127:
118:
117:
116:
109:
107:
102:
100:
96:
92:
88:
84:
77:
69:
61:
54:
49:
48:square matrix
45:
41:
37:
33:
26:
22:
28101:Determinants
27990:Vector space
27934:
27722:Vector space
27602:
27582:
27561:
27527:
27496:
27492:
27463:
27437:
27421:
27412:
27408:
27398:
27371:
27348:, Springer,
27341:
27316:
27306:, retrieved
27299:the original
27266:
27250:
27231:
27213:
27195:
27172:
27131:
27122:
27119:Muir, Thomas
27111:the original
27095:
27076:
27071:
27052:, Springer,
27049:
27031:
27011:
26973:
26958:, Elsevier,
26955:
26929:
26925:
26886:math/0203276
26876:
26872:
26844:
26838:
26812:
26783:
26777:
26758:, Springer,
26755:
26723:
26719:
26677:
26673:
26643:
26631:
26597:
26593:
26583:
26577:Bareiss 1968
26568:
26557:. Retrieved
26550:the original
26527:
26514:
26503:
26491:
26480:
26459:
26454:, §1.1, §4.3
26447:
26442:, Lecture 1.
26434:
26414:
26407:
26132:
26128:
26122:
26111:
26095:
26083:
26073:
26065:
26053:
26041:
26032:
26026:
26014:
26002:
25990:
25979:
25967:
25955:
25943:
25931:
25919:
25911:
25905:
25898:Kleiner 2007
25893:
25878:
25873:
25861:
25856:, p. 80
25854:Kleiner 2007
25849:
25837:. Retrieved
25833:the original
25822:
25810:
25787:
25781:
25769:
25757:
25724:
25720:
25713:
25701:
25696:, Def. 1.2.3
25685:
25677:MathOverflow
25676:
25669:
25660:
25641:
25637:
25627:
25606:
25594:
25559:
25555:
25549:
25516:
25512:
25502:
25490:
25478:
25466:
25449:
25437:
25429:
25421:
25409:
25394:
25386:
25374:. Retrieved
25370:
25361:
25343:
25336:
25324:
25254:
25248:
25243:
25239:
25234:
25230:
25221:
25217:
25215:
25097:
24857:
24552:
24491:
24369:
24361:
24339:
24233:
24222:
24209:graded rings
24163:
24151:
24147:
24136:
24129:
24126:
24102:
24090:
23990:
23982:
23969:reduced norm
23908:
23734:
23646:
23564:
23395:
23205:
23119:
22797:of (finite)
22720:
22528:
22525:
22255:
22221:of units in
22167:between the
22166:
22102:
21972:
21885:
21875:
21871:
21867:
21859:
21857:vector space
21854:
21809:
21655:
21652:
21648:
21643:
21639:
21635:
21629:
21612:
21605:
21454:is given by
21450:
21446:
21442:
21437:
21433:
21429:
21425:
21421:
21414:
21404:
21290:
21286:
21283:
21222:
21212:
21015:
20708:is given by
20558:
20548:
20532:
20524:
20520:
20518:
20505:
20497:
20485:
20481:
20477:
20473:
20467:
20387:
19873:lies in the
19743:
19684:
19589:
19560:
19214:
19019:
19011:Applications
18969:Spottiswoode
18953:persymmetric
18942:
18925:
18919:
18910:
18896:
18892:
18887:
18886:columns and
18883:
18877:
18869:discriminant
18859:
18837:
18833:plane curves
18814:
18803:
18797:
18504:
18403:Lie algebras
18400:
18272:
18078:
17940:, these are
17919:
17757:denotes the
17724:
17546:
17522:
17364:
17358:between two
17350:
17346:
17343:
17237:
17234:
17222:
17218:
17211:
17207:
17195:power series
17192:
17023:
17020:
16841:
16831:
16685:
16678:
16671:
16667:
16657:
16650:
16643:
16639:
16630:
16627:
16493:
16489:
16484:
16480:
16476:
16470:
16397:
16394:
16105:
15912:
15908:
15888:
15331:
15324:
15317:
15314:
15247:is given by
15242:
15158:
15074:
14987:
14981:
14913:
14648:
14488:
14416:
14293:
14286:
14176:
14096:
14016:
13912:
13684:real. Since
13496:
13492:
13418:
13239:
13025:
12818:
12627:
12472:
12280:
12042:
12037:
12033:
12005:
11661:
11656:
11652:
11572:
11571:, each with
11568:
11564:
11557:
11551:
11547:
11541:
11537:
11531:
11527:
11522:
11518:
11516:
11417:
11413:
11408:
11404:
11399:
11398:matrix, and
11394:
11390:
11385:
11380:
11179:
11150:
10981:
10979:
10972:
10524:
10415:
9963:
9682:block matrix
9653:
9578:
9500:
9364:
9354:
9350:
9343:
9339:
9291:
9287:
9286:to write an
9279:
8894:
8753:
8725:
8481:
8479:
8341:. For every
8136:. The minor
8105:
8091:
8085:
7771:
7758:
7751:matrix group
7595:
7463:
7377:
7266:
7264:
7254:
7250:
7245:
7241:
7238:
7174:
7148:
7050:
6999:
6947:
6898:
6811:
6806:
6801:
6796:
6696:
6601:
6506:
6249:
5994:alternating.
5990:
5979:
5419:
5406:
5401:
5372:
5275:
4997:
4993:
4989:
4985:
4910:
4904:
4832:-th column.
4829:
4825:
4793:
4696:
4668:
4611:
4439:
4331:
4194:
3954:
3821:
3550:permutations
3521:
3515:
3511:
3497:
3482:
3478:
3474:
3470:
3465:
3461:
3457:
3453:
3451:
3281:
3279:
3254:
3252:
3007:
3003:
3001:
2962:The entries
2961:
2710:
2706:
2698:
2696:
2678:
2670:
2666:
2662:
2654:
2653:produced by
2646:
2590:
2585:
2386:
2380:
2357:
2056:
2048:
2044:
2041:
2035:
2031:
2026:
2020:
2016:
2012:
2008:
2000:
1996:
1988:
1984:
1974:
1968:
1966:
1957:
1955:
1738:
1734:
1724:
1720:
1716:
1708:
1704:
1700:
1687:
1682:
1678:
1674:
1666:
1662:
1658:
1651:
1647:
1643:
1637:
1633:
1630:
1615:
1601:
1597:
1594:
1587:
1583:
1575:
1571:
1567:
1563:
1555:
1551:
1515:
1312:
1088:
923:
755:
572:
535:endomorphism
495:coefficients
492:
483:
472:endomorphism
465:
462:
430:
426:
423:
366:
362:
359:
194:
110:
103:
75:
67:
59:
52:
35:
29:
27970:Multivector
27935:Determinant
27892:Dot product
27737:Linear span
25960:Cajori 1993
25785:Cajori, F.
25692:, §VIII.2,
25644:: 332–341.
25562:: 202–218.
25414:Harris 2014
24219:Calculation
22795:free module
21615:Cartography
21419:open subset
21018:tetrahedron
20885:dimensional
20863:, then the
20490:orientation
20476:vectors in
20448:polynomials
18993:Christoffel
18938:Cayley 1841
18932:(1839) and
18839:Vandermonde
18531:-matrix as
15199:satisfying
14555:eigenvalues
14100:eigenvalues
14019:eigenvalues
9284:generalized
8130:recursively
8092:rectangular
7765:), and the
7682:called the
6848:Determinant
6793:Obtained by
6057:, whenever
5277:alternating
4906:multilinear
4197:pi notation
2659:orientation
2461:the region
2027:signed area
1754:Signed area
1618:signed area
1542:unit square
1522:linear maps
539:orientation
511:eigenvalues
484:determinant
99:isomorphism
36:determinant
32:mathematics
21:Risk factor
28095:Categories
28004:Direct sum
27839:Invertible
27742:Linear map
27308:2020-06-04
26932:: 98–109.
26726:(2): 3–7,
26622:See also:
26618:References
26559:2011-01-22
26471:1812.02056
25887:37.0181.02
25839:24 January
25790:p. 80
25569:1805.06027
25426:Serge Lang
24858:The order
24172:Berezinian
23792:quaternion
22913:. The map
22715:See also:
22217:) and the
22098:unimodular
21646:such that
21022:skew lines
20887:volume of
20674:measurable
18985:Wronskians
18961:circulants
18926:alternants
18505:Writing a
18407:Lie groups
17549:polynomial
17543:Derivative
16479:arguments
14601:such that
12457:since the
9986:invertible
8774:-th column
7757:(which if
7490:invertible
4824:(for each
4794:where the
3554:signatures
3552:and their
2693:Definition
2687:one-to-one
1962:equi-areal
1713:, so that
199:matrix is
115:matrix is
95:invertible
83:linear map
28034:Numerical
27797:Transpose
27584:MathWorld
27568:EMS Press
27513:123637858
27415:: 267–271
27374:, Dover,
27171:(2018) .
27154:54:75–90
27121:(1960) ,
26854:1206.7067
26831:248917264
26803:1813/6003
26508:Rote 2001
26019:Lang 1985
26007:Lang 2002
25995:Lang 1985
25948:Eves 1990
25815:Eves 1990
25749:120467300
25690:Lang 1985
25618:1410.1958
25586:119272194
25513:Math. Gaz
25497:, §0.8.2.
25442:Lang 1987
25329:Lang 1985
25161:
25110:
25063:
24988:for some
24966:≥
24869:
24823:⋅
24808:ε
24747:−
24704:ε
24465:
24426:
24291:×
24278:factorial
24052:
24043:
24034:
23944:→
23769:
23761:×
23714:→
23607:character
23586:σ
23580:
23567:permanent
23510:⋀
23452:⋀
23368:∧
23365:⋯
23362:∧
23349:⋅
23316:∧
23313:⋯
23310:∧
23278:⋀
23246:∈
23159:⋀
23129:⋀
23085:∧
23082:⋯
23079:∧
23063:∧
23047:↦
23030:∧
23027:⋯
23024:∧
23011:∧
22985:⋀
22981:→
22965:⋀
22949:⋀
22872:⋀
22738:→
22680:→
22631:×
22623:−
22359:
22346:→
22334:
22309:
22276:→
22182:×
22147:×
22139:→
22127:
22003:
21995:×
21984:∈
21973:A matrix
21837:→
21760:−
21699:−
21557:
21520:ϕ
21511:∫
21484:ϕ
21466:∫
21382:≤
21370:≤
21346:∂
21331:∂
21254:→
21184:→
21169::
21124:−
21112:−
21100:−
21083:⋅
20989:
20930:
20828:×
20793:→
20648:⊂
20593:→
20444:resultant
20419:−
20345:−
20324:⋯
20302:−
20264:−
20241:⋮
20236:⋱
20231:⋮
20226:⋮
20195:⋯
20119:⋯
20038:…
20006:Wronskian
19985:−
19940:…
19891:×
19822:∈
19748:vectors:
19696:
19628:
19610:
19501:…
19461:−
19448:…
19395:∑
19368:…
19358:…
19194:…
18997:Frobenius
18977:Pfaffians
18930:Sylvester
18823:in 1693.
18766:×
18731:∇
18718:×
18683:∇
18670:×
18635:∇
18516:×
18373:ϵ
18359:ϵ
18347:
18326:ϵ
18247:ϵ
18233:ϵ
18217:−
18204:
18173:ϵ
18159:ϵ
18141:
18132:
18111:−
18102:ϵ
18043:−
17996:
17971:∂
17954:∂
17894:α
17872:−
17859:
17835:α
17736:
17699:α
17673:
17662:
17650:α
17569:×
17485:
17467:≤
17455:
17439:≤
17412:≤
17397:−
17385:
17323:−
17314:
17308:≤
17293:
17287:≤
17274:−
17266:−
17255:
17145:
17110:−
17099:∞
17084:∑
17080:−
17054:∞
17039:∑
16973:
16948:−
16937:∞
16922:∑
16918:−
16892:∞
16877:∑
16793:∑
16777:
16732:∑
16594:…
16528:−
16492:– 1)! tr(
16417:∑
16345:
16278:−
16252:∏
16221:⋯
16182:≥
16166:…
16133:∑
16079:≤
16073:≤
16040:
16023:−
15995:∑
15981:−
15970:−
15897:, or the
15844:
15835:−
15823:
15796:
15755:
15717:
15687:
15678:−
15655:
15579:
15549:
15531:
15522:−
15499:
15423:
15417:−
15394:
15285:
15276:
15213:
15189:logarithm
15183:) of exp(
15159:Here exp(
15126:
15111:
15093:
15048:
15039:
15018:
14873:⋯
14830:⋮
14825:⋱
14820:⋮
14815:⋮
14790:⋯
14729:⋯
14625:λ
14613:χ
14589:λ
14465:−
14459:⋅
14429:χ
14263:λ
14259:⋯
14250:λ
14240:λ
14227:λ
14206:∏
14142:λ
14135:…
14123:λ
14110:λ
14056:×
13723:, taking
13609:−
13508:×
13458:−
13309:−
13202:−
13179:−
12988:−
12965:−
12942:−
12919:−
12743:−
12589:−
12484:×
12393:≥
12341:×
12307:×
12294:Hermitian
12235:≥
12153:≥
11879:
11838:−
11740:−
11334:−
11162:×
11124:−
10940:−
10929:−
10863:−
10842:−
10831:−
10785:−
10755:−
10734:⏟
10717:−
10699:−
10691:−
10522:-matrix.
10507:×
10453:−
10442:−
10383:−
10372:−
10322:−
10311:−
10298:−
10228:−
10198:−
10177:⏟
10138:−
10130:−
10120:−
9811:×
9785:×
9759:×
9733:×
9665:×
9628:
9599:−
9551:
9536:
9451:−
9417:
9379:
9347:submatrix
9248:−
9225:≤
9213:≤
9206:∏
9186:−
9168:⋯
9158:−
9135:−
9112:−
9092:⋮
9087:⋱
9082:⋮
9077:⋮
9072:⋮
9048:⋯
8978:⋯
8930:⋯
8826:−
8803:∑
8737:×
8630:−
8411:−
8388:∑
8290:−
8201:−
8192:×
8183:−
8069:×
8035:
8008:
7966:
7923:
7880:
7828:×
7794:
7728:
7715:⊂
7703:
7657:
7575:−
7516:−
7464:A matrix
7151:transpose
7145:Transpose
7115:−
7109:⋅
7103:⋅
7094:−
7075:−
6971:−
6820:−
6765:−
6723:−
6670:−
6628:−
6575:−
6528:−
6481:−
6439:−
6431:−
6353:−
6340:−
6308:−
6300:−
6229:σ
6172:∏
6152:⋯
5938:…
5857:…
5797:−
5773:…
5678:…
5659:…
5640:…
5619:−
5595:…
5579:…
5560:…
5496:×
5385:×
5332:…
5320:…
5308:…
5234:…
5222:…
5183:…
5171:…
5150:⋅
5116:…
5082:⋅
5068:−
5054:…
4960:⋅
4909:: if the
4756:…
4680:×
4633:…
4571:⋯
4532:⋯
4518:ε
4499:…
4469:∑
4413:…
4361:…
4344:ε
4300:σ
4269:∏
4262:σ
4256:
4233:∈
4230:σ
4226:∑
4166:σ
4152:⋯
4138:σ
4121:σ
4115:
4097:∈
4094:σ
4090:∑
4059:…
4036:⋮
4032:⋮
4009:…
3913:…
3890:⋮
3886:⋮
3863:…
3805:−
3759:σ
3736:σ
3730:
3667:σ
3661:…
3646:σ
3631:σ
3611:σ
3579:…
3533:×
3422:−
3410:−
3398:−
3209:⋯
3166:⋮
3161:⋱
3156:⋮
3151:⋮
3126:⋯
3065:⋯
2918:⋯
2875:⋮
2870:⋱
2865:⋮
2860:⋮
2835:⋯
2774:⋯
2709:rows and
2616:±
2561:∀
2552:≤
2539:≤
2533:∣
2505:⋯
2431:…
2310:⋮
2278:…
2241:⋮
2179:⋮
2109:⋯
2025:) is the
1920:−
1882:⋅
1861:−
1843:θ
1814:⊥
1797:θ
1449:⋅
1431:−
1404:−
1373:⋅
1355:⋅
1184:−
1062:−
935:×
901:−
889:⋅
880:−
868:−
862:⋅
838:−
795:−
732:−
399:factorial
333:−
321:−
309:−
171:−
28080:Category
28019:Subspace
28014:Quotient
27965:Bivector
27879:Bilinear
27821:Matrices
27696:Glossary
27623:Archived
27536:citation
27522:(1772),
27487:(1841),
27069:(1998),
27010:(2009),
26944:Archived
26847:: 1–16,
26744:archived
26740:62780452
26711:(1990),
26698:archived
26648:Springer
26642:(2015).
26575:, §1.1,
26106:, §III.5
26102:, §5.2,
26090:, §III.8
25997:, §VII.3
25541:41879675
25473:, §0.8.7
25376:16 March
25353:Archived
25331:, §VII.1
25267:See also
25238:, where
24946:, where
24308:of order
24158:and the
23965:Pfaffian
23670:that is
23647:For any
23603:immanant
22256:Given a
21445: :
21026:vertices
20843:-matrix
20545:rotation
20470:sequence
20209:′
20180:′
20156:′
19001:Hessians
18973:Glaisher
18945:Lebesgue
18916:Jacobian
18851:Lagrange
17759:adjugate
14934:between
14021:and the
11535:are the
11059:), then
8339:cofactor
7680:subgroup
5485:(for an
5428:, i.e.,
5400:-matrix
4874:, where
4695:-matrix
3518:matrices
1970:bivector
1846:′
1725:θ′
1286:′
1269:′
1200:′
1177:′
1146:′
1123:′
553:and the
547:calculus
541:and the
515:geometry
44:function
42:-valued
27691:Outline
27456:1104435
27402:, Paris
27390:3363427
27295:1911585
27232:Algebra
27159:0019078
27000:2347309
26911:2104048
26903:4145188
26694:2004533
26048:, §11.4
26009:, §IV.8
25729:Bibcode
25533:3620776
24407:or the
24142:with a
23609:of the
21417:and an
20672:is any
20498:product
20004:), the
19878:spanned
18965:Catalan
18873:quantic
18821:Leibniz
18810:Cardano
18794:History
16684:, ...,
16656:, ...,
14509:is the
14177:product
14167:occurs
14073:complex
12036:and of
11660:matrix
11027:(i.e.,
11025:commute
6246:Example
6021:, i.e.
5991:another
5511:matrix
4981:of two
2673:. This
2385:to the
2143:, then
2055:matrix
577:matrix
25:Epitope
27975:Tensor
27787:Kernel
27717:Vector
27712:Scalar
27511:
27474:
27454:
27444:
27388:
27378:
27360:
27323:
27293:
27283:
27238:
27220:
27202:
27183:
27138:
27103:
27083:
27056:
27038:
27020:
26998:
26988:
26962:
26909:
26901:
26829:
26819:
26762:
26738:
26692:
26654:
26542:
26498:, §1.1
26422:
25900:, §5.2
25885:
25776:, §6.6
25747:
25584:
25539:
25531:
25416:, §4.7
24403:, the
24105:factor
23206:define
22753:of an
22068:. For
21225:. For
20986:volume
20927:volume
20677:subset
20637:, and
20000:times
19215:where
18957:Hankel
18934:Cayley
18902:Cauchy
18847:minors
18583:where
18273:using
17725:where
17365:Also,
17021:where
16088:
16064:
16061:
15959:
15956:
15953:
15911:= (−1)
15817:
15543:
15329:, and
14489:Here,
14045:be an
11517:where
8486:th row
8134:minors
6407:Matrix
4894:is an
4394:tuples
4195:Using
3481:gives
3437:
2588:signed
2558:
2042:If an
1979:(0, 0)
1723:| cos
1696:cosine
1681:| sin
1580:, and
1546:(0, 0)
523:volume
79:|
73:|
40:scalar
34:, the
27844:Minor
27829:Block
27767:Basis
27509:S2CID
27302:(PDF)
27263:(PDF)
26947:(PDF)
26922:(PDF)
26899:JSTOR
26881:arXiv
26849:arXiv
26747:(PDF)
26736:S2CID
26716:(PDF)
26701:(PDF)
26690:JSTOR
26670:(PDF)
26553:(PDF)
26524:(PDF)
26466:arXiv
25745:S2CID
25613:arXiv
25582:S2CID
25564:arXiv
25537:S2CID
25529:JSTOR
25401:10–17
25317:Notes
25074:2.376
24553:of a
24411:(for
24115:on a
23971:of a
23959:of a
23531:with
21933:field
21864:basis
20541:3 × 3
20537:2 × 2
20482:basis
19875:plane
19732:, or
19005:Trudi
18991:) by
18949:Hesse
18880:Binet
18871:of a
18861:Gauss
17210:>
16496:) as
14984:trace
14978:Trace
14579:roots
13644:with
11402:, an
11388:, an
10496:is a
10476:when
9830:, is
9353:) x (
9336:terms
7944:is a
7776:from
7486:field
7441:when
6017:is a
5980:other
3600:is a
3505:3 × 3
3286:3 × 3
2705:with
2701:be a
2677:that
2675:means
2383:-cube
2015:) ∧ (
575:2 × 2
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525:of a
513:. In
497:in a
480:basis
474:of a
397:(the
197:3 × 3
113:2 × 2
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71:, or
38:is a
27999:Dual
27854:Rank
27542:link
27497:1841
27472:ISBN
27442:ISBN
27376:ISBN
27358:ISBN
27321:ISBN
27281:ISBN
27236:ISBN
27218:ISBN
27200:ISBN
27181:ISBN
27136:ISBN
27101:ISBN
27081:ISBN
27054:ISBN
27036:ISBN
27018:ISBN
26986:ISBN
26960:ISBN
26827:OCLC
26817:ISBN
26760:ISBN
26652:ISBN
26540:ISBN
26420:ISBN
25841:2012
25378:2018
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24999:>
24656:and
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23911:norm
23909:the
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22860:-th
22799:rank
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21638:and
20531:(if
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16665:and
16488:= −(
15889:cf.
14982:The
14954:and
13952:and
13752:and
13664:and
12326:and
12086:and
11973:and
11930:and
11545:and
11526:and
11416:and
11252:and
11206:and
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11004:and
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3456:has
3280:The
2697:Let
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1993:and
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66:det
58:det(
27501:doi
27426:doi
27350:doi
27271:doi
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26877:111
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26798:hdl
26788:doi
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19756:det
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18742:det
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18090:det
18021:det
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17818:det
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17733:adj
17670:adj
17633:det
17586:to
17415:det
17296:det
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16855:det
16670:= (
16642:= (
16507:det
16475:of
16403:≥ 0
16117:det
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15612:det
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15334:= 4
15327:= 3
15320:= 2
15273:exp
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19524:=
19519:]
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19488:i
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19340:[
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19316:(
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19224:A
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19163:)
19160:A
19157:(
19149:)
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19140:A
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19090:)
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18897:n
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18770:b
18762:a
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18751:)
18748:A
18745:(
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18714:c
18710:=
18703:)
18700:A
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18674:c
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18662:=
18655:)
18652:A
18649:(
18640:a
18603:c
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18591:a
18569:]
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18547:[
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18539:A
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18450:=
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18422:n
18386:.
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18365:O
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18356:)
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18341:+
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18335:=
18332:)
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18323:+
18320:I
18317:(
18291:I
18288:=
18285:A
18269:,
18256:)
18251:2
18243:(
18239:O
18236:+
18229:)
18225:X
18220:1
18213:A
18208:(
18198:)
18195:A
18192:(
18186:=
18182:)
18177:2
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18165:O
18162:+
18156:)
18153:X
18150:)
18147:A
18144:(
18135:(
18126:=
18123:)
18120:A
18117:(
18108:)
18105:X
18099:+
18096:A
18093:(
18064:.
18059:i
18056:j
18051:)
18046:1
18039:A
18035:(
18030:)
18027:A
18024:(
18018:=
18013:i
18010:j
18006:)
18002:A
17999:(
17990:=
17982:j
17979:i
17975:A
17966:)
17963:A
17960:(
17928:A
17905:.
17901:)
17891:d
17886:A
17883:d
17875:1
17868:A
17863:(
17853:)
17850:A
17847:(
17841:=
17832:d
17827:)
17824:A
17821:(
17815:d
17789:A
17769:A
17745:)
17742:A
17739:(
17710:.
17706:)
17696:d
17691:A
17688:d
17682:)
17679:A
17676:(
17666:(
17656:=
17647:d
17642:)
17639:A
17636:(
17630:d
17595:R
17572:n
17566:n
17561:R
17508:.
17502:)
17497:2
17493:A
17489:(
17477:n
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17464:)
17461:A
17458:(
17447:n
17444:1
17433:n
17430:1
17425:)
17421:A
17418:(
17405:)
17400:1
17393:A
17389:(
17378:n
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17347:A
17329:)
17326:I
17320:A
17317:(
17305:)
17302:A
17299:(
17283:)
17277:1
17270:A
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17238:A
17225:)
17219:I
17212:n
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17127:s
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17117:)
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17107:(
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17091:=
17088:j
17076:(
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17046:=
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17024:I
17006:,
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16995:)
16990:)
16985:j
16981:A
16977:(
16965:j
16959:j
16955:)
16951:1
16945:(
16932:1
16929:=
16926:j
16914:(
16906:!
16903:k
16899:1
16887:0
16884:=
16881:k
16873:=
16870:)
16867:A
16864:+
16861:I
16858:(
16842:A
16834:n
16817:.
16812:I
16807:I
16803:A
16797:I
16789:=
16786:)
16783:A
16780:(
16771:,
16766:K
16761:J
16757:B
16751:I
16746:K
16742:A
16736:K
16728:=
16723:I
16718:J
16714:)
16710:B
16707:A
16704:(
16691:)
16688:r
16686:j
16682:2
16679:j
16675:1
16672:j
16668:J
16663:)
16660:r
16658:i
16654:2
16651:i
16647:1
16644:i
16640:I
16633:J
16631:A
16613:.
16610:)
16605:n
16601:s
16597:,
16591:,
16586:2
16582:s
16578:,
16573:1
16569:s
16565:(
16560:n
16556:B
16549:!
16546:n
16539:n
16535:)
16531:1
16525:(
16519:=
16516:)
16513:A
16510:(
16494:A
16490:l
16485:l
16481:s
16477:n
16456:.
16453:n
16450:=
16445:l
16441:k
16437:l
16432:n
16427:1
16424:=
16421:l
16400:l
16398:k
16380:,
16373:l
16369:k
16363:)
16358:l
16354:A
16350:(
16336:!
16331:l
16327:k
16319:l
16315:k
16310:l
16302:1
16299:+
16294:l
16290:k
16285:)
16281:1
16275:(
16267:n
16262:1
16259:=
16256:l
16243:n
16240:=
16235:n
16231:k
16227:n
16224:+
16218:+
16213:2
16209:k
16205:2
16202:+
16197:1
16193:k
16185:0
16177:n
16173:k
16169:,
16163:,
16158:2
16154:k
16150:,
16145:1
16141:k
16129:=
16126:)
16123:A
16120:(
16091:.
16085:)
16082:n
16076:m
16070:1
16067:(
16057:)
16052:k
16048:A
16044:(
16032:k
16029:+
16026:m
16020:n
16016:c
16010:m
16005:1
16002:=
15999:k
15989:m
15986:1
15978:=
15973:m
15967:n
15963:c
15950:;
15947:1
15944:=
15939:n
15935:c
15916:0
15913:c
15909:A
15903:n
15870:.
15866:)
15861:)
15856:4
15852:A
15848:(
15838:6
15832:)
15829:A
15826:(
15813:)
15808:3
15804:A
15800:(
15790:8
15787:+
15782:2
15777:)
15772:)
15767:2
15763:A
15759:(
15748:(
15743:3
15740:+
15735:2
15730:)
15726:)
15723:A
15720:(
15710:(
15704:)
15699:2
15695:A
15691:(
15681:6
15673:4
15668:)
15664:)
15661:A
15658:(
15648:(
15642:(
15633:1
15628:=
15621:)
15618:A
15615:(
15605:,
15601:)
15596:)
15591:3
15587:A
15583:(
15573:2
15570:+
15566:)
15561:2
15557:A
15553:(
15540:)
15537:A
15534:(
15525:3
15517:3
15512:)
15508:)
15505:A
15502:(
15492:(
15486:(
15480:6
15477:1
15472:=
15465:)
15462:A
15459:(
15449:,
15445:)
15440:)
15435:2
15431:A
15427:(
15412:2
15407:)
15403:)
15400:A
15397:(
15387:(
15381:(
15375:2
15372:1
15367:=
15360:)
15357:A
15354:(
15332:n
15325:n
15318:n
15300:.
15297:)
15294:)
15291:L
15288:(
15279:(
15270:=
15267:)
15264:A
15261:(
15245:A
15228:A
15225:=
15222:)
15219:L
15216:(
15197:L
15193:A
15185:A
15181:λ
15177:A
15173:λ
15169:A
15161:A
15144:.
15141:)
15138:)
15135:)
15132:A
15129:(
15120:(
15114:(
15105:=
15102:)
15099:A
15096:(
15077:A
15060:)
15057:)
15054:A
15051:(
15042:(
15033:=
15030:)
15027:)
15024:A
15021:(
15012:(
14996:A
14992:A
14988:A
14962:n
14942:1
14922:k
14897:]
14889:k
14886:,
14883:k
14879:a
14866:2
14863:,
14860:k
14856:a
14848:1
14845:,
14842:k
14838:a
14806:k
14803:,
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14780:,
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14765:1
14762:,
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14745:k
14742:,
14739:1
14735:a
14722:2
14719:,
14716:1
14712:a
14704:1
14701:,
14698:1
14694:a
14687:[
14677:k
14673:A
14631:=
14628:)
14622:(
14617:A
14565:A
14541:A
14521:I
14497:t
14474:.
14471:)
14468:A
14462:I
14456:t
14453:(
14447:=
14444:)
14441:t
14438:(
14433:A
14402:A
14382:0
14362:A
14342:A
14322:0
14302:A
14272:.
14267:n
14254:2
14244:1
14236:=
14231:i
14221:n
14216:1
14213:=
14210:i
14202:=
14199:)
14196:A
14193:(
14173:A
14169:μ
14165:μ
14146:n
14138:,
14132:,
14127:2
14119:,
14114:1
14097:n
14083:A
14059:n
14053:n
14033:A
13992:1
13989:=
13986:)
13982:i
13978:(
13972:=
13969:)
13966:I
13963:(
13940:0
13937:=
13934:)
13930:i
13926:(
13898:.
13893:2
13889:b
13885:+
13880:2
13876:a
13872:=
13869:)
13865:i
13861:(
13853:2
13849:b
13845:+
13842:)
13839:I
13836:(
13828:2
13824:a
13820:=
13817:)
13813:i
13809:b
13806:+
13803:I
13800:a
13797:(
13770:i
13766:b
13763:=
13760:B
13740:I
13737:a
13734:=
13731:A
13711:0
13708:=
13705:)
13701:i
13697:(
13672:b
13652:a
13630:)
13624:0
13619:1
13612:1
13604:0
13598:(
13593:b
13590:+
13585:)
13579:1
13574:0
13567:0
13562:1
13556:(
13551:a
13544:i
13540:b
13537:+
13534:I
13531:a
13511:2
13505:2
13478:.
13475:)
13472:B
13469:A
13466:(
13455:)
13452:B
13449:(
13441:)
13438:A
13435:(
13404:)
13395:B
13385:A
13381:+
13372:B
13362:A
13358:+
13349:B
13339:A
13335:+
13326:B
13316:A
13312:(
13306:)
13297:B
13293:+
13284:B
13280:(
13277:)
13268:A
13264:+
13255:A
13251:(
13225:.
13216:A
13206:B
13193:B
13183:A
13170:A
13160:B
13156:+
13147:B
13137:A
13133:+
13130:)
13127:B
13124:(
13118:+
13115:)
13112:A
13109:(
13083:B
13063:)
13060:A
13057:(
13034:A
13011:.
13002:B
12992:B
12979:B
12969:A
12956:A
12946:B
12933:A
12923:A
12910:B
12900:B
12896:+
12887:B
12877:A
12873:+
12864:A
12854:B
12850:+
12841:A
12831:A
12804:.
12801:)
12792:B
12788:+
12779:A
12775:(
12772:)
12763:B
12759:+
12750:A
12746:(
12740:)
12731:B
12727:+
12718:A
12714:(
12711:)
12702:B
12698:+
12689:A
12685:(
12660:j
12657:i
12653:B
12649:,
12644:j
12641:i
12637:A
12609:.
12606:)
12603:B
12600:A
12597:(
12586:)
12583:B
12580:(
12572:)
12569:A
12566:(
12558:+
12555:)
12552:B
12549:(
12543:+
12540:)
12537:A
12534:(
12528:=
12525:)
12522:B
12519:+
12516:A
12513:(
12487:2
12481:2
12459:n
12445:,
12439:n
12434:)
12431:B
12428:(
12419:+
12413:n
12408:)
12405:A
12402:(
12387:n
12382:)
12379:B
12376:+
12373:A
12370:(
12344:n
12338:n
12328:B
12324:A
12310:n
12304:n
12286:n
12266:.
12262:)
12259:B
12256:(
12250:+
12247:)
12244:A
12241:(
12232:)
12229:B
12226:+
12223:A
12220:(
12196:,
12192:)
12189:C
12186:+
12183:B
12180:(
12174:+
12171:)
12168:C
12165:+
12162:A
12159:(
12150:)
12147:C
12144:(
12138:+
12135:)
12132:C
12129:+
12126:B
12123:+
12120:A
12117:(
12094:C
12074:B
12054:A
12038:B
12034:A
12020:B
12017:+
12014:A
11984:A
11981:B
11961:B
11958:A
11938:B
11918:A
11895:.
11892:c
11888:)
11885:X
11882:(
11872:r
11869:+
11866:)
11863:X
11860:(
11854:=
11850:)
11846:c
11841:1
11834:X
11830:r
11827:+
11824:1
11820:(
11813:)
11810:X
11807:(
11801:=
11798:)
11795:r
11792:c
11789:+
11786:X
11783:(
11756:,
11752:)
11748:A
11743:1
11736:X
11732:B
11729:+
11723:n
11718:I
11713:(
11706:)
11703:X
11700:(
11694:=
11691:)
11688:B
11685:A
11682:+
11679:X
11676:(
11662:X
11657:m
11653:m
11634:.
11631:c
11628:r
11625:+
11622:1
11619:=
11615:)
11611:r
11608:c
11605:+
11599:m
11594:I
11589:(
11573:m
11569:r
11565:c
11552:n
11548:n
11542:m
11538:m
11532:n
11528:I
11523:m
11519:I
11502:,
11498:)
11494:A
11491:B
11488:+
11482:n
11477:I
11472:(
11465:=
11461:)
11457:B
11454:A
11451:+
11445:m
11440:I
11435:(
11418:B
11414:A
11409:m
11405:n
11400:B
11395:n
11391:m
11386:A
11361:.
11358:)
11355:B
11352:+
11349:A
11346:(
11340:)
11337:B
11331:A
11328:(
11322:=
11317:)
11311:A
11306:B
11299:B
11294:A
11288:(
11260:B
11240:A
11220:C
11217:=
11214:B
11194:D
11191:=
11188:A
11165:2
11159:2
11136:.
11133:)
11130:C
11127:B
11121:D
11118:A
11115:(
11109:=
11104:)
11098:D
11093:C
11086:B
11081:A
11075:(
11047:C
11044:D
11041:=
11038:D
11035:C
11012:D
10992:C
10954:.
10951:)
10948:C
10943:1
10936:D
10932:B
10926:A
10923:(
10917:)
10914:D
10911:(
10905:=
10893:)
10885:n
10881:I
10875:0
10866:1
10859:D
10855:B
10850:C
10845:1
10838:D
10834:B
10828:A
10822:(
10814:)
10811:D
10808:(
10802:=
10788:1
10781:)
10777:D
10771:(
10767:=
10763:)
10758:1
10751:D
10747:(
10740:=
10728:)
10720:1
10713:D
10707:C
10702:1
10695:D
10684:0
10677:m
10673:I
10666:(
10652:)
10646:D
10641:C
10634:B
10629:A
10623:(
10615:)
10612:D
10609:(
10603:=
10594:)
10588:D
10583:C
10576:B
10571:A
10565:(
10533:D
10510:1
10504:1
10484:D
10464:)
10461:B
10456:1
10449:A
10445:C
10439:D
10436:(
10433:)
10430:A
10427:(
10397:,
10394:)
10391:B
10386:1
10379:A
10375:C
10369:D
10366:(
10360:)
10357:A
10354:(
10348:=
10336:)
10330:B
10325:1
10318:A
10314:C
10308:D
10301:1
10294:A
10290:C
10283:0
10276:m
10272:I
10265:(
10257:)
10254:A
10251:(
10245:=
10231:1
10224:)
10220:A
10214:(
10210:=
10206:)
10201:1
10194:A
10190:(
10183:=
10171:)
10163:n
10159:I
10153:0
10146:B
10141:1
10134:A
10123:1
10116:A
10109:(
10095:)
10089:D
10084:C
10077:B
10072:A
10066:(
10058:)
10055:A
10052:(
10046:=
10037:)
10031:D
10026:C
10019:B
10014:A
10008:(
9972:A
9949:.
9944:)
9938:D
9933:0
9926:B
9921:A
9915:(
9907:=
9904:)
9901:D
9898:(
9892:)
9889:A
9886:(
9880:=
9875:)
9869:D
9864:C
9857:0
9852:A
9846:(
9814:n
9808:n
9788:m
9782:n
9762:n
9756:m
9736:m
9730:m
9710:D
9707:,
9704:C
9701:,
9698:B
9695:,
9692:A
9668:2
9662:2
9634:.
9631:A
9619:A
9612:1
9607:=
9602:1
9595:A
9564:.
9561:A
9557:)
9554:A
9545:(
9542:=
9539:A
9530:A
9527:=
9524:I
9521:)
9518:A
9512:(
9486:.
9481:i
9478:j
9474:M
9468:j
9465:+
9462:i
9458:)
9454:1
9448:(
9445:=
9440:j
9437:,
9434:i
9430:)
9426:)
9423:A
9420:(
9411:(
9388:)
9385:A
9382:(
9344:k
9340:k
9319:)
9314:k
9311:n
9306:(
9292:n
9288:n
9280:n
9266:.
9262:)
9256:i
9252:x
9243:j
9239:x
9234:(
9228:n
9222:j
9216:i
9210:1
9202:=
9197:|
9189:1
9183:n
9178:n
9174:x
9161:1
9155:n
9150:3
9146:x
9138:1
9132:n
9127:2
9123:x
9115:1
9109:n
9104:1
9100:x
9063:2
9058:n
9054:x
9041:2
9036:3
9032:x
9024:2
9019:2
9015:x
9007:2
9002:1
8998:x
8988:n
8984:x
8971:3
8967:x
8959:2
8955:x
8947:1
8943:x
8935:1
8925:1
8920:1
8915:1
8909:|
8880:.
8875:j
8872:,
8869:i
8865:M
8859:j
8856:,
8853:i
8849:a
8843:j
8840:+
8837:i
8833:)
8829:1
8823:(
8818:n
8813:1
8810:=
8807:i
8799:=
8796:)
8793:A
8790:(
8762:j
8740:2
8734:2
8709:|
8703:h
8698:g
8691:e
8686:d
8680:|
8675:c
8672:+
8667:|
8661:i
8656:g
8649:f
8644:d
8638:|
8633:b
8625:|
8619:i
8614:h
8607:f
8602:e
8596:|
8591:a
8588:=
8583:|
8577:i
8572:h
8567:g
8560:f
8555:e
8550:d
8543:c
8538:b
8533:a
8527:|
8502:1
8499:=
8496:i
8484:i
8465:,
8460:j
8457:,
8454:i
8450:M
8444:j
8441:,
8438:i
8434:a
8428:j
8425:+
8422:i
8418:)
8414:1
8408:(
8403:n
8398:1
8395:=
8392:j
8384:=
8381:)
8378:A
8375:(
8349:i
8323:j
8320:,
8317:i
8313:M
8307:j
8304:+
8301:i
8297:)
8293:1
8287:(
8267:j
8247:i
8227:A
8207:)
8204:1
8198:n
8195:(
8189:)
8186:1
8180:n
8177:(
8155:j
8152:,
8149:i
8145:M
8117:A
8065:K
8044:)
8041:K
8038:(
8030:n
8021:/
8017:)
8014:K
8011:(
8003:n
7975:)
7972:K
7969:(
7961:n
7932:)
7929:K
7926:(
7918:n
7889:)
7886:K
7883:(
7875:n
7850:K
7824:K
7803:)
7800:K
7797:(
7789:n
7759:n
7737:)
7734:K
7731:(
7723:n
7712:)
7709:K
7706:(
7698:n
7666:)
7663:K
7660:(
7652:n
7624:K
7604:n
7592:.
7578:1
7571:]
7567:)
7564:A
7561:(
7555:[
7552:=
7546:)
7543:A
7540:(
7533:1
7528:=
7524:)
7519:1
7512:A
7508:(
7472:A
7449:A
7429:B
7406:A
7386:B
7363:)
7360:B
7357:(
7351:)
7348:A
7345:(
7339:=
7336:)
7333:B
7330:A
7327:(
7297:B
7277:A
7255:n
7251:n
7246:n
7242:n
7235:.
7223:)
7220:A
7217:(
7211:=
7207:)
7201:T
7196:A
7192:(
7175:A
7161:A
7127:=
7124:)
7121:)
7118:1
7112:(
7106:3
7097:(
7091:=
7087:|
7083:E
7079:|
7072:=
7068:|
7064:A
7060:|
7032:|
7028:D
7024:|
7020:=
7016:|
7012:E
7008:|
6983:|
6979:C
6975:|
6968:=
6964:|
6960:D
6956:|
6931:|
6927:C
6923:|
6919:=
6915:|
6911:B
6907:|
6883:|
6879:B
6875:|
6871:=
6867:|
6863:A
6859:|
6828:3
6774:]
6768:1
6760:0
6755:0
6748:4
6743:3
6738:0
6731:2
6726:3
6712:[
6707:=
6704:E
6679:]
6673:1
6665:0
6660:0
6653:4
6648:3
6636:2
6631:3
6623:5
6617:[
6612:=
6609:D
6584:]
6578:1
6570:0
6565:0
6558:4
6548:3
6541:2
6536:5
6531:3
6522:[
6517:=
6514:C
6490:]
6484:1
6476:3
6471:0
6464:4
6459:1
6454:3
6447:2
6442:1
6434:3
6425:[
6420:=
6417:B
6391:A
6367:.
6362:]
6356:1
6348:3
6343:3
6333:4
6328:1
6323:2
6316:2
6311:1
6303:2
6294:[
6289:=
6286:A
6263:A
6205:.
6200:i
6197:i
6193:a
6187:n
6182:1
6179:=
6176:i
6168:=
6163:n
6160:n
6156:a
6143:a
6133:a
6129:=
6126:)
6123:A
6120:(
6097:j
6091:i
6071:j
6065:i
6045:0
6042:=
6037:j
6034:i
6030:a
6005:A
5959:.
5955:|
5949:n
5945:a
5941:,
5935:,
5930:4
5926:a
5922:,
5917:3
5913:a
5909:,
5904:2
5900:a
5896:,
5891:1
5887:a
5882:|
5878:=
5874:|
5868:n
5864:a
5860:,
5854:,
5849:4
5845:a
5841:,
5836:2
5832:a
5828:,
5823:3
5819:a
5815:,
5810:1
5806:a
5801:|
5794:=
5790:|
5784:n
5780:a
5776:,
5768:4
5764:a
5760:,
5755:2
5751:a
5747:,
5742:1
5738:a
5734:,
5729:3
5725:a
5720:|
5699:.
5695:|
5689:n
5685:a
5681:,
5675:,
5670:j
5666:a
5662:,
5656:,
5651:i
5647:a
5643:,
5637:,
5632:1
5628:a
5623:|
5616:=
5612:|
5606:n
5602:a
5598:,
5592:,
5587:i
5583:a
5576:,
5571:j
5567:a
5563:,
5557:,
5552:1
5548:a
5543:|
5519:A
5499:n
5493:n
5473:)
5470:A
5467:(
5459:n
5455:c
5451:=
5448:)
5445:A
5442:c
5439:(
5402:A
5388:n
5382:n
5353:=
5349:|
5343:n
5339:a
5335:,
5329:,
5326:v
5323:,
5317:,
5314:v
5311:,
5305:,
5300:1
5296:a
5291:|
5251:|
5245:n
5241:a
5237:,
5231:,
5228:w
5225:,
5219:,
5214:1
5210:a
5205:|
5201:+
5197:|
5191:n
5187:a
5180:,
5177:v
5174:,
5168:,
5163:1
5159:a
5154:|
5147:r
5144:=
5133:|
5127:n
5123:a
5119:,
5113:,
5108:1
5105:+
5102:j
5098:a
5094:,
5091:w
5088:+
5085:v
5079:r
5076:,
5071:1
5065:j
5061:a
5057:,
5051:,
5046:1
5042:a
5036:|
5031:=
5023:|
5019:A
5015:|
4998:A
4994:r
4990:w
4986:v
4969:w
4966:+
4963:v
4957:r
4954:=
4949:j
4945:a
4921:A
4911:j
4898:.
4882:I
4862:1
4859:=
4855:)
4852:I
4849:(
4830:i
4826:i
4810:i
4806:a
4779:,
4774:)
4767:n
4763:a
4759:,
4753:,
4748:1
4744:a
4738:(
4733:=
4730:A
4707:n
4697:A
4683:n
4677:n
4645:.
4642:}
4639:n
4636:,
4630:,
4627:1
4624:{
4614:n
4597:,
4590:n
4586:i
4582:,
4579:n
4575:a
4563:1
4559:i
4555:,
4552:1
4548:a
4540:n
4536:i
4527:1
4523:i
4510:n
4506:i
4502:,
4496:,
4491:2
4487:i
4483:,
4478:1
4474:i
4465:=
4462:)
4459:A
4456:(
4436:0
4422:}
4419:n
4416:,
4410:,
4407:1
4404:{
4392:-
4390:n
4372:n
4368:i
4364:,
4358:,
4353:1
4349:i
4328:.
4315:)
4309:)
4306:i
4303:(
4297:,
4294:i
4290:a
4284:n
4279:1
4276:=
4273:i
4265:)
4259:(
4249:(
4241:n
4237:S
4222:=
4219:)
4216:A
4213:(
4180:.
4175:)
4172:n
4169:(
4163:,
4160:n
4156:a
4147:)
4144:1
4141:(
4135:,
4132:1
4128:a
4124:)
4118:(
4105:n
4101:S
4086:=
4081:|
4073:n
4070:,
4067:n
4063:a
4054:1
4051:,
4048:n
4044:a
4023:n
4020:,
4017:1
4013:a
4004:1
4001:,
3998:1
3994:a
3987:|
3982:=
3979:)
3976:A
3973:(
3940:,
3935:]
3927:n
3924:,
3921:n
3917:a
3908:1
3905:,
3902:n
3898:a
3877:n
3874:,
3871:1
3867:a
3858:1
3855:,
3852:1
3848:a
3841:[
3836:=
3833:A
3785:,
3782:1
3779:+
3739:)
3733:(
3705:n
3701:S
3676:)
3673:n
3670:(
3664:,
3658:,
3655:)
3652:2
3649:(
3643:,
3640:)
3637:1
3634:(
3588:}
3585:n
3582:,
3576:,
3573:2
3570:,
3567:1
3564:{
3536:n
3530:n
3516:n
3512:n
3466:i
3462:d
3458:b
3434:.
3431:h
3428:f
3425:a
3419:i
3416:d
3413:b
3407:g
3404:e
3401:c
3395:h
3392:d
3389:c
3386:+
3383:g
3380:f
3377:b
3374:+
3371:i
3368:e
3365:a
3362:=
3357:|
3351:i
3346:h
3341:g
3334:f
3329:e
3324:d
3317:c
3312:b
3307:a
3301:|
3255:A
3238:.
3233:|
3225:n
3222:,
3219:n
3215:a
3202:2
3199:,
3196:n
3192:a
3184:1
3181:,
3178:n
3174:a
3142:n
3139:,
3136:2
3132:a
3119:2
3116:,
3113:2
3109:a
3101:1
3098:,
3095:2
3091:a
3081:n
3078:,
3075:1
3071:a
3058:2
3055:,
3052:1
3048:a
3040:1
3037:,
3034:1
3030:a
3023:|
3008:A
3004:A
2981:1
2978:,
2975:1
2971:a
2947:.
2942:]
2934:n
2931:,
2928:n
2924:a
2911:2
2908:,
2905:n
2901:a
2893:1
2890:,
2887:n
2883:a
2851:n
2848:,
2845:2
2841:a
2828:2
2825:,
2822:2
2818:a
2810:1
2807:,
2804:2
2800:a
2790:n
2787:,
2784:1
2780:a
2767:2
2764:,
2761:1
2757:a
2749:1
2746:,
2743:1
2739:a
2732:[
2727:=
2724:A
2711:n
2707:n
2699:A
2679:A
2671:n
2667:A
2663:n
2655:A
2647:n
2633:,
2630:)
2627:P
2624:(
2613:=
2610:)
2607:A
2604:(
2591:n
2572:.
2568:}
2564:i
2555:1
2547:i
2543:c
2536:0
2528:n
2523:a
2516:n
2512:c
2508:+
2502:+
2497:1
2492:a
2485:1
2481:c
2476:{
2472:=
2469:P
2449:,
2444:n
2439:a
2434:,
2428:,
2423:2
2418:a
2413:,
2408:1
2403:a
2387:n
2381:n
2366:A
2343:.
2338:n
2333:a
2328:=
2323:)
2317:1
2303:0
2296:0
2290:(
2285:A
2281:,
2274:,
2269:2
2264:a
2259:=
2254:)
2248:0
2234:1
2227:0
2221:(
2216:A
2212:,
2207:1
2202:a
2197:=
2192:)
2186:0
2172:0
2165:1
2159:(
2154:A
2130:]
2121:n
2116:a
2102:2
2097:a
2088:1
2083:a
2074:[
2070:=
2067:A
2057:A
2049:n
2045:n
2023:)
2021:d
2017:c
2013:b
2009:a
2007:(
2003:)
2001:d
1997:c
1995:(
1991:)
1989:b
1985:a
1983:(
1958:A
1929:.
1926:c
1923:b
1917:d
1914:a
1911:=
1906:)
1900:d
1893:c
1887:(
1877:)
1871:a
1864:b
1855:(
1850:=
1833:|
1829:v
1825:|
1819:|
1809:u
1804:|
1800:=
1788:|
1783:v
1778:|
1772:|
1767:u
1762:|
1758:=
1721:v
1717:u
1715:|
1711:)
1709:a
1705:b
1701:u
1688:θ
1683:θ
1679:v
1675:u
1673:|
1669:)
1667:d
1663:c
1659:v
1654:)
1652:b
1648:a
1644:u
1611:A
1607:A
1590:)
1588:d
1584:c
1582:(
1578:)
1576:d
1572:b
1568:c
1564:a
1562:(
1558:)
1556:b
1552:a
1550:(
1534:A
1530:A
1518:A
1488:.
1483:|
1477:d
1472:c
1465:b
1460:a
1454:|
1446:r
1443:=
1440:)
1437:c
1434:b
1428:d
1425:a
1422:(
1419:r
1416:=
1413:c
1410:r
1407:b
1401:d
1398:a
1395:r
1392:=
1387:|
1381:d
1376:c
1370:r
1363:b
1358:a
1352:r
1346:|
1321:r
1298:.
1293:|
1283:d
1277:c
1266:b
1260:a
1254:|
1249:+
1244:|
1238:d
1233:c
1226:b
1221:a
1215:|
1210:=
1207:c
1204:)
1197:b
1193:+
1190:b
1187:(
1181:)
1174:d
1170:+
1167:d
1164:(
1161:a
1158:=
1153:|
1143:d
1139:+
1136:d
1131:c
1120:b
1116:+
1113:b
1108:a
1102:|
1071:=
1068:a
1065:b
1059:b
1056:a
1053:=
1048:|
1042:b
1037:a
1030:b
1025:a
1019:|
992:)
986:1
981:0
974:0
969:1
963:(
938:2
932:2
898:=
895:)
892:1
886:7
883:(
877:)
874:)
871:4
865:(
859:3
856:(
853:=
848:|
841:4
832:1
825:7
820:3
814:|
809:=
804:)
798:4
790:1
783:7
778:3
772:(
741:.
738:c
735:b
729:d
726:a
723:=
718:|
712:d
707:c
700:b
695:a
689:|
684:=
679:)
673:d
668:c
661:b
656:a
650:(
616:)
610:d
605:c
598:b
593:a
587:(
543:n
527:n
519:n
453:.
446:.
444:1
431:n
427:n
403:n
385:!
382:n
367:n
363:n
345:.
342:h
339:f
336:a
330:i
327:d
324:b
318:g
315:e
312:c
306:h
303:d
300:c
297:+
294:g
291:f
288:b
285:+
282:i
279:e
276:a
273:=
268:|
262:i
257:h
252:g
245:f
240:e
235:d
228:c
223:b
218:a
212:|
180:,
177:c
174:b
168:d
165:a
162:=
157:|
151:d
146:c
139:b
134:a
128:|
76:A
68:A
62:)
60:A
53:A
27:.
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