19744:
24537:
25377:
10233:
702:
16684:
16691:
9834:
689:
24338:
1826:
1210:
25389:
24801:
8883:
19079:
16714:
1511:
25413:
16698:
10228:{\displaystyle {\begin{aligned}\det(A-\lambda I)&=\left|{\begin{bmatrix}2&0&0\\0&3&4\\0&4&9\end{bmatrix}}-\lambda {\begin{bmatrix}1&0&0\\0&1&0\\0&0&1\end{bmatrix}}\right|={\begin{vmatrix}2-\lambda &0&0\\0&3-\lambda &4\\0&4&9-\lambda \end{vmatrix}},\\&=(2-\lambda ){\bigl }=-\lambda ^{3}+14\lambda ^{2}-35\lambda +22.\end{aligned}}}
25401:
16705:
8618:
9552:
24214:
21253:
12524:
22173:(One knows, moreover, that by following Lagrange's method, one obtains for the general value of the principal variable a function in which there appear, together with the principal variable, the roots of a certain equation that I will call the "characteristic equation", the degree of this equation being precisely the order of the differential equation that must be integrated.)
21922:
12071:
12875:
11374:
11191:
718:. Points in the top half are moved to the right, and points in the bottom half are moved to the left, proportional to how far they are from the horizontal axis that goes through the middle of the painting. The vectors pointing to each point in the original image are therefore tilted right or left, and made longer or shorter by the transformation. Points
10688:
9286:
1821:{\displaystyle {\begin{bmatrix}A_{11}&A_{12}&\cdots &A_{1n}\\A_{21}&A_{22}&\cdots &A_{2n}\\\vdots &\vdots &\ddots &\vdots \\A_{n1}&A_{n2}&\cdots &A_{nn}\\\end{bmatrix}}{\begin{bmatrix}v_{1}\\v_{2}\\\vdots \\v_{n}\end{bmatrix}}={\begin{bmatrix}w_{1}\\w_{2}\\\vdots \\w_{n}\end{bmatrix}}}
3372:
12323:
17845:
8514:= . The red vectors are not parallel to either eigenvector, so, their directions are changed by the transformation. The lengths of the purple vectors are unchanged after the transformation (due to their eigenvalue of 1), while blue vectors are three times the length of the original (due to their eigenvalue of 3). See also:
17649:
14228:
9077:
8878:{\displaystyle {\begin{aligned}\det(A-\lambda I)&=\left|{\begin{bmatrix}2&1\\1&2\end{bmatrix}}-\lambda {\begin{bmatrix}1&0\\0&1\end{bmatrix}}\right|={\begin{vmatrix}2-\lambda &1\\1&2-\lambda \end{vmatrix}}\\&=3-4\lambda +\lambda ^{2}\\&=(\lambda -3)(\lambda -1).\end{aligned}}}
11883:
14051:
18771:
18953:
18645:
12665:
11196:
11013:
4212:
Because of the definition of eigenvalues and eigenvectors, an eigenvalue's geometric multiplicity must be at least one, that is, each eigenvalue has at least one associated eigenvector. Furthermore, an eigenvalue's geometric multiplicity cannot exceed its algebraic multiplicity. Additionally, recall
722:
the horizontal axis do not move at all when this transformation is applied. Therefore, any vector that points directly to the right or left with no vertical component is an eigenvector of this transformation, because the mapping does not change its direction. Moreover, these eigenvectors all have an
7986:
21212:. In this case, the term eigenvector is used in a somewhat more general meaning, since the Fock operator is explicitly dependent on the orbitals and their eigenvalues. Thus, if one wants to underline this aspect, one speaks of nonlinear eigenvalue problems. Such equations are usually solved by an
3545:
21991:
represent the general direction of variability in human pronunciations of a particular utterance, such as a word in a language. Based on a linear combination of such eigenvoices, a new voice pronunciation of the word can be constructed. These concepts have been found useful in automatic speech
16337:
The converse approach, of first seeking the eigenvectors and then determining each eigenvalue from its eigenvector, turns out to be far more tractable for computers. The easiest algorithm here consists of picking an arbitrary starting vector and then repeatedly multiplying it with the matrix
3001:
are all real numbers, then the coefficients of the characteristic polynomial will also be real numbers, but the eigenvalues may still have nonzero imaginary parts. The entries of the corresponding eigenvectors therefore may also have nonzero imaginary parts. Similarly, the eigenvalues may be
1205:
Eigenvalues and eigenvectors are often introduced to students in the context of linear algebra courses focused on matrices. Furthermore, linear transformations over a finite-dimensional vector space can be represented using matrices, which is especially common in numerical and computational
1359:
235:
the vectors upon which it acts. Its eigenvectors are those vectors that are only stretched, with neither rotation nor shear. The corresponding eigenvalue is the factor by which an eigenvector is stretched or squished. If the eigenvalue is negative, the eigenvector's direction is reversed.
2993:
20296:
10526:
15825:
10894:
6099:
5474:
13225:
12660:
3174:
2799:
11562:
11471:
9257:
21310:, eigenvectors and eigenvalues are used as a method by which a mass of information of a clast fabric's constituents' orientation and dip can be summarized in a 3-D space by six numbers. In the field, a geologist may collect such data for hundreds or thousands of
16628:
Most numeric methods that compute the eigenvalues of a matrix also determine a set of corresponding eigenvectors as a by-product of the computation, although sometimes implementors choose to discard the eigenvector information as soon as it is no longer needed.
14106:
9675:
7840:
15374:
11006:
17055:
8924:
9547:{\displaystyle {\begin{aligned}(A-3I)\mathbf {v} _{\lambda =3}&={\begin{bmatrix}-1&1\\1&-1\end{bmatrix}}{\begin{bmatrix}v_{1}\\v_{2}\end{bmatrix}}={\begin{bmatrix}0\\0\end{bmatrix}}\\-1v_{1}+1v_{2}&=0;\\1v_{1}-1v_{2}&=0\end{aligned}}}
2544:
17702:
13920:
16923:
17494:
14837:
15459:
In theory, the coefficients of the characteristic polynomial can be computed exactly, since they are sums of products of matrix elements; and there are algorithms that can find all the roots of a polynomial of arbitrary degree to any required
3037:
with odd order has at least one real eigenvalue, whereas a real matrix with even order may not have any real eigenvalues. The eigenvectors associated with these complex eigenvalues are also complex and also appear in complex conjugate pairs.
12519:{\displaystyle \mathbf {v} _{\lambda _{1}}={\begin{bmatrix}1\\-1\\{\frac {1}{2}}\end{bmatrix}},\quad \mathbf {v} _{\lambda _{2}}={\begin{bmatrix}0\\1\\-3\end{bmatrix}},\quad \mathbf {v} _{\lambda _{3}}={\begin{bmatrix}0\\0\\1\end{bmatrix}},}
6231:
3756:
3401:
16642:
Eigenvectors and eigenvalues can be useful for understanding linear transformations of geometric shapes. The following table presents some example transformations in the plane along with their 2×2 matrices, eigenvalues, and eigenvectors.
17488:
16534:
8394:
1987:
22167:"On sait d'ailleurs qu'en suivant la méthode de Lagrange, on obtient pour valeur générale de la variable prinicipale une fonction dans laquelle entrent avec la variable principale les racines d'une certaine équation que j'appellerai l'
19709:
represented by the row-normalized adjacency matrix; however, the adjacency matrix must first be modified to ensure a stationary distribution exists. The second smallest eigenvector can be used to partition the graph into clusters, via
12184:
11744:
10515:
9823:
7537:
4452:
1244:
18651:
18836:
2877:
18531:
15919:
16011:
251:). In such an application, the largest eigenvalue is of particular importance, because it governs the long-term behavior of the system after many applications of the linear transformation, and the associated eigenvector is the
18517:
18413:
18343:
13886:
12066:{\displaystyle \mathbf {v} _{\lambda _{1}}={\begin{bmatrix}1\\0\\0\end{bmatrix}},\quad \mathbf {v} _{\lambda _{2}}={\begin{bmatrix}0\\1\\0\end{bmatrix}},\quad \mathbf {v} _{\lambda _{3}}={\begin{bmatrix}0\\0\\1\end{bmatrix}},}
7871:
22246:
15716:
10765:
12968:
13028:
18181:
18077:
18007:
5354:
15037:
13062:
11642:
7168:
Many disciplines traditionally represent vectors as matrices with a single column rather than as matrices with a single row. For that reason, the word "eigenvector" in the context of matrices almost always refers to a
4208:
15187:
12539:
10355:
12870:{\displaystyle \det(A-\lambda I)={\begin{vmatrix}2-\lambda &0&0&0\\1&2-\lambda &0&0\\0&1&3-\lambda &0\\0&0&1&3-\lambda \end{vmatrix}}=(2-\lambda )^{2}(3-\lambda )^{2}.}
10413:
10295:
19730:
between states of a system. In particular the entries are non-negative, and every row of the matrix sums to one, being the sum of probabilities of transitions from one state to some other state of the system. The
11369:{\displaystyle A{\begin{bmatrix}1\\\lambda _{3}\\\lambda _{2}\end{bmatrix}}={\begin{bmatrix}\lambda _{3}\\\lambda _{2}\\1\end{bmatrix}}=\lambda _{3}\cdot {\begin{bmatrix}1\\\lambda _{3}\\\lambda _{2}\end{bmatrix}}.}
11186:{\displaystyle A{\begin{bmatrix}1\\\lambda _{2}\\\lambda _{3}\end{bmatrix}}={\begin{bmatrix}\lambda _{2}\\\lambda _{3}\\1\end{bmatrix}}=\lambda _{2}\cdot {\begin{bmatrix}1\\\lambda _{2}\\\lambda _{3}\end{bmatrix}},}
6518:
4285:
12279:
11839:
16842:
16301:
14688:
7032:
16095:
4502:
18830:
2684:
833:
21374:
2170:
8623:
8582:
2870:
2661:
705:
A 2×2 real and symmetric matrix representing a stretching and shearing of the plane. The eigenvectors of the matrix (red lines) are the two special directions such that every point on them will just slide on
696:
the red arrow changes direction, but the blue arrow does not. The blue arrow is an eigenvector of this shear mapping because it does not change direction, and since its length is unchanged, its eigenvalue is
22097:"Théorem. 44. De quelque figure que soit le corps, on y peut toujours assigner un tel axe, qui passe par son centre de gravité, autour duquel le corps peut tourner librement & d'un mouvement uniforme."
15677:
9149:
2049:
21859:, from one person becoming infected to the next person becoming infected. In a heterogeneous population, the next generation matrix defines how many people in the population will become infected after time
21101:
correspond to the intensity transmittance associated with each eigenchannel. One of the remarkable properties of the transmission operator of diffusive systems is their bimodal eigenvalue distribution with
14961:
5953:
22891:
16977:
16774:
9573:
22199:"Insbesondere in dieser ersten Mitteilung gelange ich zu Formeln, die die Entwickelung einer willkürlichen Funktion nach gewissen ausgezeichneten Funktionen, die ich 'Eigenfunktionen' nenne, liefern: ..."
5164:
4345:
20991:. Even though multiple scattering repeatedly randomizes the waves, ultimately coherent wave transport through the system is a deterministic process which can be described by a field transmission matrix
9839:
7151:
714:, provides a simple illustration. Each point on the painting can be represented as a vector pointing from the center of the painting to that point. The linear transformation in this example is called a
21047:
form a set of disorder-specific input wavefronts which enable waves to couple into the disordered system's eigenchannels: the independent pathways waves can travel through the system. The eigenvalues,
10901:
10683:{\displaystyle {\begin{aligned}\lambda _{1}&=1\\\lambda _{2}&=-{\frac {1}{2}}+i{\frac {\sqrt {3}}{2}}\\\lambda _{3}&=\lambda _{2}^{*}=-{\frac {1}{2}}-i{\frac {\sqrt {3}}{2}}\end{aligned}}}
6911:
6378:
20208:
22099:(Theorem. 44. Whatever be the shape of the body, one can always assign to it such an axis, which passes through its center of gravity, around which it can rotate freely and with a uniform motion.)
20760:
19244:
for the space of the observed data: In this basis, the largest eigenvalues correspond to the principal components that are associated with most of the covariability among a number of observed data.
17326:
5086:
15092:
8163:
can therefore be decomposed into a matrix composed of its eigenvectors, a diagonal matrix with its eigenvalues along the diagonal, and the inverse of the matrix of eigenvectors. This is called the
726:
Linear transformations can take many different forms, mapping vectors in a variety of vector spaces, so the eigenvectors can also take many forms. For example, the linear transformation could be a
17225:
13738:
6801:
659:
21099:
21045:
18841:
18656:
18536:
18458:
18354:
18284:
18122:
18018:
17948:
17707:
17499:
17415:
15939:
15844:
14111:
13925:
10531:
9291:
7270:
5359:
3406:
896:
11476:
11385:
7339:
1413:
489:
179:
18239:
13488:
5832:
5259:
4725:
17167:
11880:
Each diagonal element corresponds to an eigenvector whose only nonzero component is in the same row as that diagonal element. In the example, the eigenvalues correspond to the eigenvectors,
7758:
24236:
20140:
19061:
17903:
17696:
17404:
15202:
4973:
3367:{\displaystyle \det(A-\lambda I)=(\lambda _{1}-\lambda )^{\mu _{A}(\lambda _{1})}(\lambda _{2}-\lambda )^{\mu _{A}(\lambda _{2})}\cdots (\lambda _{d}-\lambda )^{\mu _{A}(\lambda _{d})}.}
1506:
21427:
16983:
5329:
1451:
17840:{\displaystyle {\begin{aligned}\lambda _{1}&=e^{\varphi }\\&=\cosh \varphi +\sinh \varphi \\\lambda _{2}&=e^{-\varphi }\\&=\cosh \varphi -\sinh \varphi \end{aligned}}}
21762:
20470:
15398:
The classical method is to first find the eigenvalues, and then calculate the eigenvectors for each eigenvalue. It is in several ways poorly suited for non-exact arithmetics such as
13569:
6939:
4600:
3682:
19016:
5918:
5875:
2436:
20368:
4813:
4558:
4381:
4092:
21709:
19620:
16417:
15600:.) Therefore, for matrices of order 5 or more, the eigenvalues and eigenvectors cannot be obtained by an explicit algebraic formula, and must therefore be computed by approximate
1831:
20907:
19430:
12100:
11660:
10431:
9739:
8432:
7476:
2305:
20962:
20798:
20695:
17644:{\displaystyle {\begin{aligned}\lambda _{1}&=e^{i\theta }\\&=\cos \theta +i\sin \theta \\\lambda _{2}&=e^{-i\theta }\\&=\cos \theta -i\sin \theta \end{aligned}}}
17268:
17105:
16848:
13371:
8155:
8100:
21656:
19814:
14223:{\displaystyle {\begin{aligned}T(\mathbf {u} +\mathbf {v} )&=\lambda (\mathbf {u} +\mathbf {v} ),\\T(\alpha \mathbf {v} )&=\lambda (\alpha \mathbf {v} ).\end{aligned}}}
21514:
21485:
21456:
19996:
19858:
16566:
13423:
8193:
and Λ represent the same linear transformation expressed in two different bases. The eigenvectors are used as the basis when representing the linear transformation as Λ.
6700:
5711:
5636:
5604:
3871:
762:
21166:
21133:
9128:
7623:
7594:
239:
The eigenvectors and eigenvalues of a linear transformation serve to characterize it, and so they play important roles in all the areas where linear algebra is applied, from
16386:
15835:
22209:"Dieser Erfolg ist wesentlich durch den Umstand bedingt, daß ich nicht, wie es bisher geschah, in erster Linie auf den Beweis für die Existenz der Eigenwerte ausgehe, ... "
19148:
18447:
18273:
8041:
5551:
5031:
389:
21011:
17357:
16592:
16443:
14707:
8416:
5286:
445:
411:
364:
315:
99:
23323:
20023:
9072:{\displaystyle (A-I)\mathbf {v} _{\lambda =1}={\begin{bmatrix}1&1\\1&1\end{bmatrix}}{\begin{bmatrix}v_{1}\\v_{2}\end{bmatrix}}={\begin{bmatrix}0\\0\end{bmatrix}}}
289:
21168:. Furthermore, one of the striking properties of open eigenchannels, beyond the perfect transmittance, is the statistically robust spatial profile of the eigenchannels.
18111:
17937:
16210:
16161:
13822:
6968:
4679:
24330:
20633:
20590:
20528:
15500:
15454:
13662:
7407:
7205:
6749:
5786:
22319:
19549:
10749:
6134:
5106:
211:
143:
22202:(In particular, in this first report I arrive at formulas that provide the development of an arbitrary function in terms of some distinctive functions, which I call
20043:
15616:
Once the (exact) value of an eigenvalue is known, the corresponding eigenvectors can be found by finding nonzero solutions of the eigenvalue equation, that becomes a
15556:
there is no general, explicit and exact algebraic formula for the roots of a polynomial with degree 5 or more. (Generality matters because any polynomial with degree
7471:
7361:
5213:
4911:
4882:
19484:
16245:
14046:{\displaystyle {\begin{aligned}T(\mathbf {x} +\mathbf {y} )&=T(\mathbf {x} )+T(\mathbf {y} ),\\T(\alpha \mathbf {x} )&=\alpha T(\mathbf {x} ),\end{aligned}}}
6458:
4751:
21911:
21884:
21857:
21830:
21803:
21603:
21576:
21549:
21065:
19511:
19116:
17410:
16471:
15711:
8492:
8336:
6575:
6302:
21805:) is a fundamental number in the study of how infectious diseases spread. If one infectious person is put into a population of completely susceptible people, then
16042:
7081:
20655:
can be represented as a one-dimensional array (i.e., a vector) and a matrix respectively. This allows one to represent the Schrödinger equation in a matrix form.
19358:
18766:{\displaystyle {\begin{aligned}\mathbf {u} _{1}&={\begin{bmatrix}1\\-i\end{bmatrix}}\\\mathbf {u} _{2}&={\begin{bmatrix}1\\+i\end{bmatrix}}\end{aligned}}}
11581:
6827:
18948:{\displaystyle {\begin{aligned}\mathbf {u} _{1}&={\begin{bmatrix}1\\1\end{bmatrix}}\\\mathbf {u} _{2}&={\begin{bmatrix}1\\-1\end{bmatrix}}\end{aligned}}}
15523:
13252:
10519:
This matrix shifts the coordinates of the vector up by one position and moves the first coordinate to the bottom. Its characteristic polynomial is 1 −
4389:
4141:
338:
22283:
22212:(This success is mainly attributable to the fact that I do not, as it has happened until now, first of all aim at a proof of the existence of eigenvalues, ... )
20927:
20862:
20842:
20822:
20653:
20552:
20493:
20412:
20071:
19947:
19923:
19903:
19880:
19680:
19660:
19640:
19569:
19454:
19324:
18640:{\displaystyle {\begin{aligned}\mathbf {u} _{1}&={\begin{bmatrix}1\\0\end{bmatrix}}\\\mathbf {u} _{2}&={\begin{bmatrix}0\\1\end{bmatrix}}\end{aligned}}}
16463:
16321:
16184:
16135:
16115:
15598:
15574:
15550:
15428:
10708:
8328:
7663:
7643:
7565:
7451:
7427:
7381:
7300:
7225:
7052:
6847:
6723:
6653:
6626:
6595:
6545:
6428:
6401:
6322:
6275:
6255:
6127:
5946:
5760:
5731:
5676:
5656:
5575:
5518:
5498:
5349:
5187:
4993:
4853:
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4620:
4522:
119:
23997:
22122:(However, it is not inconsistent be three such positions of the plane HM, because in cubic equations, can be three roots, and three values of the tangent t.)
15390:
The calculation of eigenvalues and eigenvectors is a topic where theory, as presented in elementary linear algebra textbooks, is often very far from practice.
6632:, positive-semidefinite, negative-definite, or negative-semidefinite, then every eigenvalue is positive, non-negative, negative, or non-positive, respectively.
4220:
19063:; and all eigenvectors have non-real entries. Indeed, except for those special cases, a rotation changes the direction of every nonzero vector in the plane.
12195:
11755:
15930:
7981:{\displaystyle AQ={\begin{bmatrix}\lambda _{1}\mathbf {v} _{1}&\lambda _{2}\mathbf {v} _{2}&\cdots &\lambda _{n}\mathbf {v} _{n}\end{bmatrix}}.}
18453:
18349:
18279:
247:. In particular, it is often the case that a system is represented by a linear transformation whose outputs are fed as inputs to the same transformation (
23613:
Knox-Robinson, C.; Gardoll, Stephen J. (1998), "GIS-stereoplot: an interactive stereonet plotting module for ArcView 3.0 geographic information system",
20988:
3540:{\displaystyle {\begin{aligned}1&\leq \mu _{A}(\lambda _{i})\leq n,\\\mu _{A}&=\sum _{i=1}^{d}\mu _{A}\left(\lambda _{i}\right)=n.\end{aligned}}}
3052:
of a matrix is the list of eigenvalues, repeated according to multiplicity; in an alternative notation the set of eigenvalues with their multiplicities.
771:
22194:
22186:
20662:
is often used in this context. A vector, which represents a state of the system, in the
Hilbert space of square integrable functions is represented by
14455:) does not exist. The converse is true for finite-dimensional vector spaces, but not for infinite-dimensional vector spaces. In general, the operator (
8525:
2604:
1065:
developed
Fourier's ideas further, and brought them to the attention of Cauchy, who combined them with his own ideas and arrived at the fact that real
516:
of the vector space. Hence, in a finite-dimensional vector space, it is equivalent to define eigenvalues and eigenvectors using either the language of
19735:
gives sufficient conditions for a Markov chain to have a unique dominant eigenvalue, which governs the convergence of the system to a steady state.
15623:
13571:
is the eigenfunction of the derivative operator. In this case the eigenfunction is itself a function of its associated eigenvalue. In particular, for
12910:
20374:(increasing across: s, p, d, ...). The illustration shows the square of the absolute value of the wavefunctions. Brighter areas correspond to higher
12973:
3029:, namely with the two members of each pair having imaginary parts that differ only in sign and the same real part. If the degree is odd, then by the
18117:
18013:
17943:
14318:, i.e., the maximum number of linearly independent eigenvectors associated with that eigenvalue. By the definition of eigenvalues and eigenvectors,
12970:
and is therefore 1-dimensional. Similarly, the geometric multiplicity of the eigenvalue 3 is 1 because its eigenspace is spanned by just one vector
21271:
14970:
1354:{\displaystyle \mathbf {x} ={\begin{bmatrix}1\\-3\\4\end{bmatrix}}\quad {\mbox{and}}\quad \mathbf {y} ={\begin{bmatrix}-20\\60\\-80\end{bmatrix}}.}
20223:
so that the system can be represented as linear summation of the eigenvectors. The eigenvalue problem of complex structures is often solved using
2988:{\displaystyle \mathbf {v} _{\lambda =1}={\begin{bmatrix}1\\-1\end{bmatrix}},\quad \mathbf {v} _{\lambda =3}={\begin{bmatrix}1\\1\end{bmatrix}}.}
15100:
10302:
24837:
22664:
20283:
tensor with the eigenvalues on the diagonal and eigenvectors as a basis. Because it is diagonal, in this orientation, the stress tensor has no
19682:
th smallest eigenvalue of the
Laplacian. The first principal eigenvector of the graph is also referred to merely as the principal eigenvector.
10363:
10245:
22120:"Non autem repugnat tres esse eiusmodi positiones plani HM, quia in aequatione cubica radices tres esse possunt, et tres tangentis t valores."
20149:
15472:). Even for matrices whose elements are integers the calculation becomes nontrivial, because the sums are very long; the constant term is the
15605:
13039:
is 2, which is the smallest it could be for a matrix with two distinct eigenvalues. Geometric multiplicities are defined in a later section.
6463:
16780:
21314:
in a soil sample, which can only be compared graphically such as in a Tri-Plot (Sneed and Folk) diagram, or as a
Stereonet on a Wulff Net.
16250:
14570:
12883:
of each eigenvalue is 2; in other words they are both double roots. The sum of the algebraic multiplicities of all distinct eigenvalues is
6973:
619:
22154:, pp. 807–808 Augustin Cauchy (1839) "Mémoire sur l'intégration des équations linéaires" (Memoir on the integration of linear equations),
21524:. Dip is measured as the eigenvalue, the modulus of the tensor: this is valued from 0° (no dip) to 90° (vertical). The relative values of
16047:
15820:{\displaystyle {\begin{bmatrix}4&1\\6&3\end{bmatrix}}{\begin{bmatrix}x\\y\end{bmatrix}}=6\cdot {\begin{bmatrix}x\\y\end{bmatrix}}}
10889:{\displaystyle A{\begin{bmatrix}5\\5\\5\end{bmatrix}}={\begin{bmatrix}5\\5\\5\end{bmatrix}}=1\cdot {\begin{bmatrix}5\\5\\5\end{bmatrix}}.}
4457:
23336:"On the definition and the computation of the basic reproduction ratio R0 in models for infectious diseases in heterogeneous populations"
22083:
In 1751, Leonhard Euler proved that any body has a principal axis of rotation: Leonhard Euler (presented: October 1751; published: 1760)
18777:
14511:
One can generalize the algebraic object that is acting on the vector space, replacing a single operator acting on a vector space with an
7236:
862:
21320:
7305:
6094:{\displaystyle \operatorname {tr} (A)=\sum _{i=1}^{n}a_{ii}=\sum _{i=1}^{n}\lambda _{i}=\lambda _{1}+\lambda _{2}+\cdots +\lambda _{n}.}
2120:
1382:
19183:
19150:
direction and of 1 in the orthogonal direction. The vectors shown are unit eigenvectors of the (symmetric, positive-semidefinite)
13428:
5469:{\displaystyle {\begin{aligned}\gamma _{A}&=\sum _{i=1}^{d}\gamma _{A}(\lambda _{i}),\\d&\leq \gamma _{A}\leq n,\end{aligned}}}
2823:
13220:{\displaystyle |v_{i,j}|^{2}={\frac {\prod _{k}{(\lambda _{i}-\lambda _{k}(M_{j}))}}{\prod _{k\neq i}{(\lambda _{i}-\lambda _{k})}}},}
13055:
th component of a normalized eigenvector can be calculated using only the matrix eigenvalues and the eigenvalues of the corresponding
2007:
24332:
size (for a square matrix), then fill out the entries numerically and click on the Go button. It can accept complex numbers as well.)
23495:
Graham, D.; Midgley, N. (2000), "Graphical representation of particle shape using triangular diagrams: an Excel spreadsheet method",
19154:
scaled by the square root of the corresponding eigenvalue. Just as in the one-dimensional case, the square root is taken because the
14849:
16929:
16726:
16338:(optionally normalizing the vector to keep its elements of reasonable size); this makes the vector converge towards an eigenvector.
1164:. For some time, the standard term in English was "proper value", but the more distinctive term "eigenvalue" is the standard today.
24395:
24294:
24288:
21317:
The output for the orientation tensor is in the three orthogonal (perpendicular) axes of space. The three eigenvectors are ordered
20080:
12655:{\displaystyle A={\begin{bmatrix}2&0&0&0\\1&2&0&0\\0&1&3&0\\0&0&1&3\end{bmatrix}},}
5111:
4292:
21832:
is the average number of people that one typical infectious person will infect. The generation time of an infection is the time,
19769:) of vibration, and the eigenvectors are the shapes of these vibrational modes. In particular, undamped vibration is governed by
14346:
eigenvalues are always linearly independent. Therefore, the sum of the dimensions of the eigenspaces cannot exceed the dimension
7086:
1475:
24728:
24342:
22131:
Report of the Thirty-second meeting of the
British Association for the Advancement of Science; held at Cambridge in October 1862
24786:
23518:
23282:
20213:
6852:
6327:
25459:
23904:
Van
Mieghem, Piet (18 January 2014). "Graph eigenvectors, fundamental weights and centrality metrics for nodes in networks".
23781:
23721:
23497:
23486:
20703:
17274:
13514:
5036:
24017:
23645:(2000), "Mathematical Handbook for Scientists and Engineers: Definitions, Theorems, and Formulas for Reference and Review",
15046:
6520:
and each eigenvalue's geometric multiplicity coincides. Moreover, since the characteristic polynomial of the inverse is the
4727:. This can be seen by evaluating what the left-hand side does to the first column basis vectors. By reorganizing and adding
2794:{\displaystyle \det(A-\lambda I)={\begin{vmatrix}2-\lambda &1\\1&2-\lambda \end{vmatrix}}=3-4\lambda +\lambda ^{2}.}
25200:
17173:
13698:
11557:{\displaystyle \mathbf {v} _{\lambda _{3}}={\begin{bmatrix}1&\lambda _{3}&\lambda _{2}\end{bmatrix}}^{\textsf {T}}}
11466:{\displaystyle \mathbf {v} _{\lambda _{2}}={\begin{bmatrix}1&\lambda _{2}&\lambda _{3}\end{bmatrix}}^{\textsf {T}}}
6754:
3055:
An important quantity associated with the spectrum is the maximum absolute value of any eigenvalue. This is known as the
1160:, which means "own", to denote eigenvalues and eigenvectors in 1904, though he may have been following a related usage by
23702:
22036:
21070:
21016:
9252:{\displaystyle \mathbf {v} _{\lambda =1}={\begin{bmatrix}v_{1}\\-v_{1}\end{bmatrix}}={\begin{bmatrix}1\\-1\end{bmatrix}}}
19022:
is not an integer multiple of 180°. Therefore, except for these special cases, the two eigenvalues are complex numbers,
13326:
3167:
linear terms with some terms potentially repeating, the characteristic polynomial can also be written as the product of
1057:
458:
148:
25345:
24830:
23763:
23586:
19772:
19083:
18195:
15039:
in terms of its once-lagged value, and taking the characteristic equation of this system's matrix. This equation gives
14842:
which can be found by stacking into matrix form a set of equations consisting of the above difference equation and the
9670:{\displaystyle \mathbf {v} _{\lambda =3}={\begin{bmatrix}v_{1}\\v_{1}\end{bmatrix}}={\begin{bmatrix}1\\1\end{bmatrix}}}
7835:{\displaystyle Q={\begin{bmatrix}\mathbf {v} _{1}&\mathbf {v} _{2}&\cdots &\mathbf {v} _{n}\end{bmatrix}}.}
5791:
5218:
4684:
20:
24282:
19959:
19819:
17111:
25417:
24262:
24193:
24175:
23850:
23662:
23604:
23461:
23397:
23272:
23143:
23117:
21289:
19201:
of eigenvectors, each of which has a nonnegative eigenvalue. The orthogonal decomposition of a PSD matrix is used in
9081:
925:
The set of all eigenvectors of a linear transformation, each paired with its corresponding eigenvalue, is called the
524:
59:
15369:{\displaystyle {\frac {d^{k}x}{dt^{k}}}+a_{k-1}{\frac {d^{k-1}x}{dt^{k-1}}}+\cdots +a_{1}{\frac {dx}{dt}}+a_{0}x=0.}
8586:
The figure on the right shows the effect of this transformation on point coordinates in the plane. The eigenvectors
24881:
24776:
23859:
Sneed, E. D.; Folk, R. L. (1958), "Pebbles in the lower
Colorado River, Texas, a study of particle morphogenesis",
21981:
20379:
19025:
16602:
Efficient, accurate methods to compute eigenvalues and eigenvectors of arbitrary matrices were not known until the
11001:{\displaystyle \lambda _{2}\lambda _{3}=1,\quad \lambda _{2}^{2}=\lambda _{3},\quad \lambda _{3}^{2}=\lambda _{2}.}
2324:
72:
21516:
is the tertiary, in terms of strength. The clast orientation is defined as the direction of the eigenvector, on a
17050:{\displaystyle {\begin{bmatrix}\cosh \varphi &\sinh \varphi \\\sinh \varphi &\cosh \varphi \end{bmatrix}}}
25295:
24738:
24674:
20496:
17859:
17655:
17363:
15464:. However, this approach is not viable in practice because the coefficients would be contaminated by unavoidable
4916:
1119:
studied eigenvalue problems similar to those of Sturm; the discipline that grew out of their work is now called
25393:
21379:
21232:
19954:
19758:
14531:
14506:
14475:
13632:
8515:
8164:
7674:
5291:
3110:
2539:{\displaystyle \det(A-\lambda I)=(\lambda _{1}-\lambda )(\lambda _{2}-\lambda )\cdots (\lambda _{n}-\lambda ),}
2401:
1420:
859:, then the eigenvalue equation for a linear transformation above can be rewritten as the matrix multiplication
681:
may be negative, in which case the eigenvector reverses direction as part of the scaling, or it may be zero or
590:
15468:, and the roots of a polynomial can be an extremely sensitive function of the coefficients (as exemplified by
25449:
24823:
22171:, le degré de cette équation étant précisément l'order de l'équation différentielle qu'il s'agit d'intégrer."
21714:
20428:
6916:
5920:
is the eigenvalue's algebraic multiplicity. The following are properties of this matrix and its eigenvalues:
4563:
18970:
12094:. As with diagonal matrices, the eigenvalues of triangular matrices are the elements of the main diagonal.
5880:
5837:
25405:
24388:
22129:. See: A. Cayley (1862) "Report on the progress of the solution of certain special problems of dynamics,"
22051:
21961:
20375:
20326:
20143:
19732:
19233:
19213:
19194:
19190:
19173:
19167:
16918:{\displaystyle {\begin{bmatrix}\cos \theta &-\sin \theta \\\sin \theta &\cos \theta \end{bmatrix}}}
16610:
with the LU decomposition results in an algorithm with better convergence than the QR algorithm. For large
4756:
4527:
4350:
4061:
3772:). On the other hand, by definition, any nonzero vector that satisfies this condition is an eigenvector of
22092:
22084:
21661:
19743:
16394:
578:
25320:
24876:
24621:
24471:
23671:
Kublanovskaya, Vera N. (1962), "On some algorithms for the solution of the complete eigenvalue problem",
20874:
19574:
19367:
3171:
terms each corresponding to a distinct eigenvalue and raised to the power of the algebraic multiplicity,
2601:
As a brief example, which is described in more detail in the examples section later, consider the matrix
2266:
20932:
20768:
20665:
17231:
17068:
8115:
8060:
24891:
24526:
24420:
21608:
20593:
20244:
16607:
3030:
1121:
1004:
566:
535:
Eigenvalues and eigenvectors feature prominently in the analysis of linear transformations. The prefix
24232:
21490:
21461:
21432:
21267:
20871:, the infinite-dimensional analog of Hermitian matrices. As in the matrix case, in the equation above
16542:
13394:
6662:
5687:
5612:
5580:
3847:
733:
25439:
25305:
25277:
24914:
24766:
24415:
23561:, Wiley series in mathematical and computational biology, West Sussex, England: John Wiley & Sons
22358:
22118:, which proves that a body has three principal axes of rotation. He then states (on the same page):
21969:
21773:
21138:
21105:
20320:
19331:
19284:
15469:
14832:{\displaystyle \lambda ^{k}-a_{1}\lambda ^{k-1}-a_{2}\lambda ^{k-2}-\cdots -a_{k-1}\lambda -a_{k}=0,}
14497:
inverse. The spectrum of an operator always contains all its eigenvalues but is not limited to them.
14444:
10237:
The roots of the characteristic polynomial are 2, 1, and 11, which are the only three eigenvalues of
7599:
7570:
2371:
2216:
917:
Eigenvalues and eigenvectors give rise to many closely related mathematical concepts, and the prefix
24306:
from
Symbolab (Click on the bottom right button of the 2×12 grid to select a matrix size. Select an
24087:
22659:
16346:
1062:
25350:
24758:
24641:
24244:
24240:
24224:
24144:
22191:
Nachrichten von der
Gesellschaft der Wissenschaften zu Göttingen, Mathematisch-Physikalische Klasse
19727:
19701:
of the World Wide Web graph gives the page ranks as its components. This vector corresponds to the
19268:
19248:
19121:
18419:
18245:
15553:
14549:
8011:
7163:
6629:
5523:
4998:
369:
23164:
20994:
17335:
16575:
16426:
11657:. The eigenvalues of a diagonal matrix are the diagonal elements themselves. Consider the matrix
8399:
5264:
2815:. The eigenvectors corresponding to each eigenvalue can be found by solving for the components of
428:
394:
347:
298:
82:
25444:
25235:
25225:
25195:
25129:
24864:
24804:
24733:
24511:
24381:
22322:
22163:
21984:
purposes. Research related to eigen vision systems determining hand gestures has also been made.
20224:
20001:
19757:
Eigenvalue problems occur naturally in the vibration analysis of mechanical structures with many
19702:
19202:
8273:
6226:{\displaystyle \det(A)=\prod _{i=1}^{n}\lambda _{i}=\lambda _{1}\lambda _{2}\cdots \lambda _{n}.}
3761:
2594:, which may not all have distinct values, are roots of the polynomial and are the eigenvalues of
1052:
266:
20415:
19240:
by the principal components. Principal component analysis of the correlation matrix provides an
18083:
17909:
16189:
16140:
6944:
4645:
3751:{\displaystyle E=\left\{\mathbf {v} :\left(A-\lambda I\right)\mathbf {v} =\mathbf {0} \right\}.}
1167:
The first numerical algorithm for computing eigenvalues and eigenvectors appeared in 1929, when
25454:
25333:
25230:
25210:
25205:
25134:
24859:
24568:
24501:
24491:
24309:
22041:
20611:
20605:
20568:
20506:
19686:
16212:. A similar calculation shows that the corresponding eigenvectors are the nonzero solutions of
15479:
15433:
14512:
13505:
13056:
8289:
7386:
7184:
6728:
5925:
5765:
2342:
956:
513:
23920:
23692:
22288:
22138:
19642:
th principal eigenvector of a graph is defined as either the eigenvector corresponding to the
19516:
17483:{\displaystyle {\begin{aligned}\lambda _{1}&=k_{1}\\\lambda _{2}&=k_{2}\end{aligned}}}
16529:{\displaystyle \lambda ={\frac {\mathbf {v} ^{*}A\mathbf {v} }{\mathbf {v} ^{*}\mathbf {v} }}}
14385:, or equivalently if the direct sum of the eigenspaces associated with all the eigenvalues of
13292:. A widely used class of linear transformations acting on infinite-dimensional spaces are the
10718:
8389:{\displaystyle \mathbf {x} ^{\textsf {T}}H\mathbf {x} /\mathbf {x} ^{\textsf {T}}\mathbf {x} }
5091:
4013:. This can be checked by noting that multiplication of complex matrices by complex numbers is
196:
128:
25360:
25290:
25167:
25091:
25030:
25015:
25010:
24987:
24869:
24583:
24578:
24573:
24506:
24451:
24303:
22193:(News of the Philosophical Society at Göttingen, mathematical-physical section), pp. 49–91.
22103:
22001:
21228:
21217:
20868:
20500:
20220:
20028:
19296:
15620:
with known coefficients. For example, once it is known that 6 is an eigenvalue of the matrix
15193:
14524:
13321:
13293:
13263:
8181:
7456:
7346:
6521:
5192:
4887:
4858:
3953:
3881:
1982:{\displaystyle w_{i}=A_{i1}v_{1}+A_{i2}v_{2}+\cdots +A_{in}v_{n}=\sum _{j=1}^{n}A_{ij}v_{j}.}
1364:
1161:
1138:
1130:
1097:
1015:
1008:
989:
911:
727:
122:
19459:
16215:
13890:
which is the union of the zero vector with the set of all eigenvectors associated with
12179:{\displaystyle A={\begin{bmatrix}1&0&0\\1&2&0\\2&3&3\end{bmatrix}}.}
11739:{\displaystyle A={\begin{bmatrix}1&0&0\\0&2&0\\0&0&3\end{bmatrix}}.}
10510:{\displaystyle A={\begin{bmatrix}0&1&0\\0&0&1\\1&0&0\end{bmatrix}}.}
9818:{\displaystyle A={\begin{bmatrix}2&0&0\\0&3&4\\0&4&9\end{bmatrix}}.}
7532:{\displaystyle A^{\textsf {T}}\mathbf {u} ^{\textsf {T}}=\kappa \mathbf {u} ^{\textsf {T}}.}
6433:
4730:
3025:
The non-real roots of a real polynomial with real coefficients can be grouped into pairs of
25340:
25220:
25215:
25139:
25040:
24593:
24558:
24545:
24436:
24102:
23932:
23868:
23813:
23735:
23650:
23622:
23506:
23408:
23186:
23135:
23109:
22258:
22187:"Grundzüge einer allgemeinen Theorie der linearen Integralgleichungen. (Erste Mitteilung)"
22061:
22021:
21889:
21862:
21835:
21808:
21781:
21581:
21554:
21527:
21209:
21205:
21050:
20659:
20323:. They are associated with eigenvalues interpreted as their energies (increasing downward:
19489:
19089:
15684:
15385:
14520:
13614:
13509:
8442:
8050:
are linearly independent, Q is invertible. Right multiplying both sides of the equation by
6553:
6280:
5500:'s eigenvalues, or equivalently the maximum number of linearly independent eigenvectors of
4447:{\displaystyle {\boldsymbol {v}}_{1},\,\ldots ,\,{\boldsymbol {v}}_{\gamma _{A}(\lambda )}}
4014:
3048:
1200:
1044:
981:
907:
674:
517:
422:
223:
quantities with magnitude and direction, often pictured as arrows. A linear transformation
16018:
15430:
can be determined by finding the roots of the characteristic polynomial. This is easy for
8189:
is the change of basis matrix of the similarity transformation. Essentially, the matrices
7057:
4560:
columns are these eigenvectors, and whose remaining columns can be any orthonormal set of
4051:, or equivalently the maximum number of linearly independent eigenvectors associated with
3056:
8:
25355:
25265:
25187:
25086:
25020:
24977:
24967:
24947:
24771:
24651:
24626:
24476:
23474:
23130:
A First Course In Linear
Algebra: with Optional Introduction to Groups, Rings, and Fields
22262:
20284:
19711:
19337:
16672:
16662:
16569:
14561:
14516:
14471:
8310:
case, eigenvalues can be given a variational characterization. The largest eigenvalue of
6806:
6548:
3131:
1368:
224:
24106:
23936:
23872:
23817:
23791:
23654:
23626:
23510:
23190:
22644:
19263:, the eigenvalues of the correlation matrix determine the Q-methodologist's judgment of
19232:
equal to one). For the covariance or correlation matrix, the eigenvectors correspond to
15505:
320:
25381:
25300:
25240:
25172:
25162:
25101:
25076:
24952:
24909:
24904:
24481:
24368:
23964:
23905:
23884:
23829:
23803:
23522:
23373:
23315:
23297:
23218:
23176:
23128:
22268:
22215:
For the origin and evolution of the terms eigenvalue, characteristic value, etc., see:
22031:
21965:
21236:
20912:
20847:
20827:
20807:
20638:
20537:
20478:
20397:
20276:
20056:
19932:
19908:
19888:
19865:
19665:
19645:
19625:
19554:
19439:
19309:
19272:
19237:
19225:
19155:
18961:
16654:
16448:
16306:
16169:
16120:
16100:
15914:{\displaystyle \left\{{\begin{aligned}4x+y&=6x\\6x+3y&=6y\end{aligned}}\right.}
15583:
15559:
15535:
15529:
15413:
14366:
13230:
10693:
10426:
8313:
8285:
8277:
7648:
7628:
7550:
7436:
7412:
7366:
7285:
7210:
7037:
6832:
6708:
6638:
6611:
6580:
6530:
6413:
6386:
6307:
6260:
6240:
6112:
5931:
5745:
5716:
5661:
5641:
5560:
5503:
5483:
5334:
5172:
4978:
4838:
4818:
4625:
4605:
4507:
3785:
3641:
1184:
1180:
1081:
228:
104:
24276:
24115:
23634:
21950:. The dimension of this vector space is the number of pixels. The eigenvectors of the
16006:{\displaystyle \left\{{\begin{aligned}-2x+y&=0\\6x-3y&=0\end{aligned}}\right.}
11578:, respectively. The two complex eigenvectors also appear in a complex conjugate pair,
25376:
25096:
25081:
25025:
24972:
24679:
24636:
24563:
24456:
24360:
24189:
24171:
23956:
23948:
23888:
23846:
23833:
23777:
23759:
23739:
23717:
23698:
23684:
23658:
23600:
23582:
23547:
23526:
23482:
23457:
23393:
23365:
23319:
23268:
23261:
23248:
23244:
23222:
23210:
23202:
23139:
23113:
23022:
22046:
21951:
21221:
21197:
21185:
21177:
20984:
20419:
20046:
19762:
19221:
19209:
19151:
16618:
16339:
12086:
11653:
8168:
6404:
5477:
3663:
3026:
3003:
1168:
1149:
1085:
244:
19:"Characteristic root" redirects here. For the root of a characteristic equation, see
23968:
23377:
18512:{\displaystyle {\begin{aligned}\gamma _{1}&=1\\\gamma _{2}&=1\end{aligned}}}
18408:{\displaystyle {\begin{aligned}\gamma _{1}&=1\\\gamma _{2}&=1\end{aligned}}}
18338:{\displaystyle {\begin{aligned}\gamma _{1}&=1\\\gamma _{2}&=1\end{aligned}}}
12907:
of the eigenvalue 2 is only 1, because its eigenspace is spanned by just one vector
3394:). The size of each eigenvalue's algebraic multiplicity is related to the dimension
1134:
701:
25310:
25285:
25157:
25005:
24942:
24684:
24588:
24441:
24153:
24120:
24110:
24063:
23940:
23876:
23821:
23680:
23630:
23542:
23514:
23439:
23417:
23355:
23347:
23307:
23198:
23194:
22085:"Du mouvement d'un corps solide quelconque lorsqu'il tourne autour d'un axe mobile"
21977:
21939:
21189:
20371:
20074:
19950:
19698:
19327:
19304:
19241:
19198:
19187:
16622:
16611:
15601:
15577:
14545:
14494:
13881:{\displaystyle E=\left\{\mathbf {v} :T(\mathbf {v} )=\lambda \mathbf {v} \right\},}
13048:
8307:
8269:
6602:
6598:
5948:, defined as the sum of its diagonal elements, is also the sum of all eigenvalues,
3019:
1196:
1116:
1109:
1105:
1101:
1074:
1066:
1032:
574:
556:
216:
181:. It is often important to know these vectors in linear algebra. The corresponding
39:
24132:
22106:
proved that any body has three principal axes of rotation: Johann Andreas Segner,
12090:, while a matrix whose elements below the main diagonal are all zero is called an
8288:
and therefore admits a basis of generalized eigenvectors and a decomposition into
936:
corresponding to the same eigenvalue, together with the zero vector, is called an
723:
eigenvalue equal to one, because the mapping does not change their length either.
25250:
25177:
25106:
24899:
24743:
24536:
24496:
24486:
23977:
23825:
23642:
23281:
Denton, Peter B.; Parke, Stephen J.; Tao, Terence; Zhang, Xining (January 2022).
22668:
22422:
22189:(Fundamentals of a general theory of linear integral equations. (First report)),
21521:
20272:
20053:
are different from the principal compliance modes, which are the eigenvectors of
19280:
19276:
19229:
19217:
19206:
19177:
19067:
16332:
15830:
15465:
14422:
13607:
10762:= 1, any vector with three equal nonzero entries is an eigenvector. For example,
8301:
4347:, consider how the definition of geometric multiplicity implies the existence of
3837:
3011:
2200:
1153:
1126:
1070:
1019:
573:, eigenvalues and eigenvectors have a wide range of applications, for example in
542:
24355:
23165:"Fluctuations and Correlations of Transmission Eigenchannels in Diffusive Media"
20259:
is a key quantity required to determine the rotation of a rigid body around its
12963:{\displaystyle {\begin{bmatrix}0&1&-1&1\end{bmatrix}}^{\textsf {T}}}
8452:
7596:, with the same eigenvalue. Furthermore, since the characteristic polynomial of
25328:
25255:
24962:
24748:
24669:
24404:
23845:, Translated and edited by Richard A. Silverman, New York: Dover Publications,
23751:
23389:
23283:"Eigenvectors from Eigenvalues: A Survey of a Basic Identity in Linear Algebra"
22056:
22026:
21193:
21181:
20601:
20383:
20260:
20240:
20050:
19256:
19066:
A linear transformation that takes a square to a rectangle of the same area (a
15399:
13297:
13023:{\displaystyle {\begin{bmatrix}0&0&0&1\end{bmatrix}}^{\textsf {T}}}
10711:
8331:
6656:
4383:
3022:, which include the rationals, the eigenvalues must also be algebraic numbers.
2231:
1093:
1040:
996:
985:
977:
682:
582:
29:
24068:
24051:
23556:
23163:
Bender, Nicholas; Yamilov, Alexey; Yilmaz, Hasan; Cao, Hui (14 October 2020).
22674:
22440:
18176:{\displaystyle {\begin{aligned}\mu _{1}&=1\\\mu _{2}&=1\end{aligned}}}
18072:{\displaystyle {\begin{aligned}\mu _{1}&=1\\\mu _{2}&=1\end{aligned}}}
18002:{\displaystyle {\begin{aligned}\mu _{1}&=1\\\mu _{2}&=1\end{aligned}}}
12662:
has a characteristic polynomial that is the product of its diagonal elements,
25433:
25116:
25048:
25000:
24781:
24704:
24664:
24631:
24611:
23952:
23470:
23444:
23422:
23206:
22126:
22006:
21201:
20976:
20909:
is understood to be the vector obtained by application of the transformation
20597:
20316:
20312:
19260:
16667:
16614:
15617:
15032:{\displaystyle {\begin{bmatrix}x_{t}&\cdots &x_{t-k+1}\end{bmatrix}}}
13587:
13285:
13284:
remains valid even if the underlying vector space is an infinite-dimensional
13275:
7738:
2421:
1145:
1048:
1011:
realized that the principal axes are the eigenvectors of the inertia matrix.
765:
715:
693:
688:
547:
502:
232:
21942:, processed images of faces can be seen as vectors whose components are the
16625:
to compute eigenvalues and eigenvectors, among several other possibilities.
15456:
matrices, but the difficulty increases rapidly with the size of the matrix.
11637:{\displaystyle \mathbf {v} _{\lambda _{2}}=\mathbf {v} _{\lambda _{3}}^{*}.}
10420:
3960:
is a linear subspace, it is closed under scalar multiplication. That is, if
25058:
25053:
24957:
24714:
24603:
24553:
24446:
24364:
23960:
23214:
22087:(On the movement of any solid body while it rotates around a moving axis),
22016:
21517:
21307:
20531:
20414:
is represented in terms of a differential operator is the time-independent
20300:
19723:
19706:
19300:
19279:). More generally, principal component analysis can be used as a method of
18965:
16603:
14377:
to such a subspace is diagonalizable. Moreover, if the entire vector space
13598:
The concept of eigenvalues and eigenvectors extends naturally to arbitrary
13289:
13269:
12313:. These roots are the diagonal elements as well as the eigenvalues of
11873:. These roots are the diagonal elements as well as the eigenvalues of
7991:
With this in mind, define a diagonal matrix Λ where each diagonal element Λ
4203:{\displaystyle \gamma _{A}(\lambda )=n-\operatorname {rank} (A-\lambda I).}
1372:
1176:
1172:
1152:
by viewing the operators as infinite matrices. He was the first to use the
847:
by 1 matrices. If the linear transformation is expressed in the form of an
252:
24815:
24037:
23567:
23369:
23045:
22089:
Histoire de l'Académie royale des sciences et des belles lettres de Berlin
20227:, but neatly generalize the solution to scalar-valued vibration problems.
20219:
The orthogonality properties of the eigenvectors allows decoupling of the
16390:
this causes it to converge to an eigenvector of the eigenvalue closest to
15182:{\displaystyle x_{t}=c_{1}\lambda _{1}^{t}+\cdots +c_{k}\lambda _{k}^{t}.}
13810:
13280:
The definitions of eigenvalue and eigenvectors of a linear transformation
13262:
th row and column from the original matrix. This identity also extends to
12879:
The roots of this polynomial, and hence the eigenvalues, are 2 and 3. The
10350:{\displaystyle {\begin{bmatrix}0&-2&1\end{bmatrix}}^{\textsf {T}}}
1022:, and generalized it to arbitrary dimensions. Cauchy also coined the term
561:
25260:
24924:
24847:
24694:
24659:
24616:
24461:
24298:
24140:
23944:
23792:"Light fields in complex media: Mesoscopic scattering meets wave control"
20562:
20304:
19953:. Admissible solutions are then a linear combination of solutions to the
19926:
15473:
12083:
A matrix whose elements above the main diagonal are all zero is called a
10408:{\displaystyle {\begin{bmatrix}0&1&2\end{bmatrix}}^{\textsf {T}}}
10290:{\displaystyle {\begin{bmatrix}1&0&0\end{bmatrix}}^{\textsf {T}}}
6106:
3034:
2354:
2235:
1089:
24337:
23311:
22966:
6513:{\textstyle {\frac {1}{\lambda _{1}}},\ldots ,{\frac {1}{\lambda _{n}}}}
4280:{\displaystyle 1\leq \gamma _{A}(\lambda )\leq \mu _{A}(\lambda )\leq n}
537:
455:
is the corresponding eigenvalue. This relationship can be expressed as:
25245:
25124:
24919:
24723:
24466:
23716:, Philadelphia: Society for Industrial and Applied Mathematics (SIAM),
23351:
22111:
22011:
21973:
21954:
associated with a large set of normalized pictures of faces are called
21943:
21225:
20865:
20248:
16683:
14339:
13599:
13309:
12274:{\displaystyle \det(A-\lambda I)=(1-\lambda )(2-\lambda )(3-\lambda ),}
11834:{\displaystyle \det(A-\lambda I)=(1-\lambda )(2-\lambda )(3-\lambda ),}
6524:
of the original, the eigenvalues share the same algebraic multiplicity.
2334:
1000:
906:
by 1 matrix. For a matrix, eigenvalues and eigenvectors can be used to
570:
76:
23519:
10.1002/1096-9837(200012)25:13<1473::AID-ESP158>3.0.CO;2-C
23069:
22853:
20287:
components; the components it does have are the principal components.
16837:{\displaystyle {\begin{bmatrix}k_{1}&0\\0&k_{2}\end{bmatrix}}}
14967:-dimensional system of the first order in the stacked variable vector
13492:
This differential equation can be solved by multiplying both sides by
2874:
In this example, the eigenvectors are any nonzero scalar multiples of
1209:
616:, called an eigenvalue. This condition can be written as the equation
523:
The following section gives a more general viewpoint that also covers
24521:
24124:
23452:
Friedberg, Stephen H.; Insel, Arnold J.; Spence, Lawrence E. (1989),
23360:
21956:
21933:
21925:
21213:
20980:
20236:
19752:
19078:
16296:{\displaystyle {\begin{bmatrix}b&-3b\end{bmatrix}}^{\textsf {T}}}
14683:{\displaystyle x_{t}=a_{1}x_{t-1}+a_{2}x_{t-2}+\cdots +a_{k}x_{t-k}.}
13255:
8887:
Setting the characteristic polynomial equal to zero, it has roots at
7473:
is its associated eigenvalue. Taking the transpose of this equation,
7027:{\displaystyle \{\lambda _{1}+\alpha ,\ldots ,\lambda _{k}+\alpha \}}
2803:
Setting the characteristic polynomial equal to zero, it has roots at
711:
586:
506:
220:
24295:
Eigenvectors and eigenvalues | Essence of linear algebra, chapter 10
23093:
22732:
22730:
22216:
22095:, Euler proves that any body contains a principal axis of rotation:
16090:{\displaystyle {\begin{bmatrix}a&2a\end{bmatrix}}^{\textsf {T}}}
8272:. For defective matrices, the notion of eigenvectors generalizes to
4497:{\displaystyle A{\boldsymbol {v}}_{k}=\lambda {\boldsymbol {v}}_{k}}
3642:
Eigenspaces, geometric multiplicity, and the eigenbasis for matrices
25149:
25068:
24995:
24689:
24243:
external links, and converting useful links where appropriate into
23880:
23808:
23302:
23181:
23002:
22841:
20308:
20280:
19862:
That is, acceleration is proportional to position (i.e., we expect
19694:
19252:
19159:
18825:{\displaystyle \mathbf {u} _{1}={\begin{bmatrix}1\\0\end{bmatrix}}}
15608:
for the roots of a degree 3 polynomial is numerically impractical.
15461:
14275:
is closed under addition and scalar multiplication. The eigenspace
835:
Alternatively, the linear transformation could take the form of an
828:{\displaystyle {\frac {d}{dx}}e^{\lambda x}=\lambda e^{\lambda x}.}
612:
to the eigenvector only scales the eigenvector by the scalar value
248:
24279:– non-technical introduction from PhysLink.com's "Ask the Experts"
23921:"Focusing coherent light through opaque strongly scattering media"
23910:
23533:
Hawkins, T. (1975), "Cauchy and the spectral theory of matrices",
22978:
22939:
22807:
22805:
21369:{\displaystyle \mathbf {v} _{1},\mathbf {v} _{2},\mathbf {v} _{3}}
16713:
16690:
12896:, the order of the characteristic polynomial and the dimension of
2165:{\displaystyle \left(A-\lambda I\right)\mathbf {v} =\mathbf {0} ,}
565:) for 'proper', 'characteristic', 'own'. Originally used to study
24934:
24373:
22727:
21303:
20256:
14500:
8577:{\displaystyle A={\begin{bmatrix}2&1\\1&2\end{bmatrix}}.}
2865:{\displaystyle \left(A-\lambda I\right)\mathbf {v} =\mathbf {0} }
2656:{\displaystyle A={\begin{bmatrix}2&1\\1&2\end{bmatrix}}.}
552:
240:
23749:
23743:
23430:
Francis, J. G. F. (1962), "The QR Transformation, II (part 2)",
22680:
22446:
20295:
15672:{\displaystyle A={\begin{bmatrix}4&1\\6&3\end{bmatrix}}}
2210:
2044:{\displaystyle A\mathbf {v} =\mathbf {w} =\lambda \mathbf {v} ,}
24699:
23481:(3rd ed.), Baltimore, MD: Johns Hopkins University Press,
23406:
Francis, J. G. F. (1961), "The QR Transformation, I (part 1)",
23252:
23158:, Free online book under GNU licence, University of Puget Sound
23057:
22956:
22954:
22802:
22705:
22703:
22701:
22531:
22529:
22452:
22125:
The relevant passage of Segner's work was discussed briefly by
21311:
20555:
20534:, is one of its eigenfunctions corresponding to the eigenvalue
20387:
20252:
19690:
14956:{\displaystyle x_{t-1}=x_{t-1},\ \dots ,\ x_{t-k+1}=x_{t-k+1},}
14542:
are the analogs of eigenvectors and eigenspaces, respectively.
8431:
7274:
The eigenvalue and eigenvector problem can also be defined for
3113:
of the characteristic polynomial, that is, the largest integer
22896:
22375:
22373:
21913:
is then the largest eigenvalue of the next generation matrix.
20394:
An example of an eigenvalue equation where the transformation
16972:{\displaystyle {\begin{bmatrix}1&k\\0&1\end{bmatrix}}}
16769:{\displaystyle {\begin{bmatrix}k&0\\0&k\end{bmatrix}}}
14287:. If that subspace has dimension 1, it is sometimes called an
11651:
Matrices with entries only along the main diagonal are called
4681:, we get a matrix whose top left block is the diagonal matrix
1133:
on general domains towards the end of the 19th century, while
51:
24052:"Eigenvector components of symmetric, graph-related matrices"
23776:(3rd ed.), New York: Springer Science + Business Media,
23335:
22613:
21947:
21921:
20972:
14331:) ≥ 1 because every eigenvalue has at least one eigenvector.
13375:
The functions that satisfy this equation are eigenvectors of
7668:
3014:
or even if they are all integers. However, if the entries of
23033:
22990:
22951:
22698:
22553:
22541:
22526:
19747:
Mode shape of a tuning fork at eigenfrequency 440.09 Hz
12528:
respectively, as well as scalar multiples of these vectors.
12075:
respectively, as well as scalar multiples of these vectors.
8611:
Taking the determinant to find characteristic polynomial of
5159:{\displaystyle \mu _{A}(\lambda )\geq \gamma _{A}(\lambda )}
4340:{\displaystyle \gamma _{A}(\lambda )\leq \mu _{A}(\lambda )}
2571:
may be real but in general is a complex number. The numbers
23094:"Eigenvalue, eigenfunction, eigenvector, and related terms"
22929:
22927:
22925:
22923:
22370:
22217:
Earliest Known Uses of Some of the Words of Mathematics (E)
20212:
This can be reduced to a generalized eigenvalue problem by
16697:
16000:
15908:
12531:
7146:{\displaystyle \{P(\lambda _{1}),\ldots ,P(\lambda _{k})\}}
3836:
by 1 matrices. A property of the nullspace is that it is a
1190:
45:
24035:
23153:
23075:
22908:
22589:
22338:
21964:. They are very useful for expressing any face image as a
21224:, one often represents the Hartree–Fock equation in a non-
13266:, and has been rediscovered many times in the literature.
8276:
and the diagonal matrix of eigenvalues generalizes to the
8237:
whose eigenvalue is the corresponding diagonal element of
5713:
can be written as a linear combination of eigenvectors of
4213:
that an eigenvalue's algebraic multiplicity cannot exceed
22754:
22715:
22601:
20561:
However, in the case where one is interested only in the
10898:
For the complex conjugate pair of imaginary eigenvalues,
10421:
Three-dimensional matrix example with complex eigenvalues
6906:{\displaystyle \{\lambda _{1}+1,\ldots ,\lambda _{k}+1\}}
6601:, then every eigenvalue is real. The same is true of any
6373:{\displaystyle \lambda _{1}^{k},\ldots ,\lambda _{n}^{k}}
48:
22920:
22504:
22502:
22500:
22498:
22114:
p. xxviiii ), Segner derives a third-degree equation in
21605:
are dictated by the nature of the sediment's fabric. If
20203:{\displaystyle \left(\omega ^{2}m+\omega c+k\right)x=0.}
16704:
14401:
can be formed from linearly independent eigenvectors of
13270:
Eigenvalues and eigenfunctions of differential operators
12536:
As in the previous example, the lower triangular matrix
23673:
USSR Computational Mathematics and Mathematical Physics
23333:
23098:
Earliest Known Uses of Some of the Words of Mathematics
23051:
22829:
22817:
22766:
22742:
22686:
22565:
22514:
22473:
22471:
22469:
22467:
20755:{\displaystyle H|\Psi _{E}\rangle =E|\Psi _{E}\rangle }
17321:{\displaystyle \lambda ^{2}-2\cosh(\varphi )\lambda +1}
16465:, then the corresponding eigenvalue can be computed as
15532:
for the roots of a polynomial exist only if the degree
13811:
Eigenspaces, geometric multiplicity, and the eigenbasis
7567:
is the same as the transpose of a right eigenvector of
5737:
5081:{\displaystyle (\xi -\lambda )^{\gamma _{A}(\lambda )}}
3788:
of the zero vector with the set of all eigenvectors of
768:
that are scaled by that differential operator, such as
24018:"Neutrinos Lead to Unexpected Discovery in Basic Math"
23162:
22972:
22390:
22388:
19577:
18917:
18868:
18801:
18735:
18683:
18612:
18563:
16992:
16938:
16857:
16789:
16735:
16260:
16057:
15796:
15761:
15725:
15638:
15087:{\displaystyle \lambda _{1},\,\ldots ,\,\lambda _{k},}
14979:
13399:
13233:
12983:
12920:
12698:
12554:
12485:
12423:
12354:
12115:
12032:
11973:
11914:
11675:
11508:
11417:
11321:
11258:
11208:
11138:
11075:
11025:
10855:
10813:
10777:
10446:
10373:
10312:
10255:
10020:
9949:
9880:
9754:
9646:
9603:
9427:
9384:
9342:
9225:
9179:
9048:
9005:
8969:
8750:
8706:
8664:
8540:
8494:
preserves the direction of purple vectors parallel to
7889:
7773:
6466:
4642:
has full rank and is therefore invertible. Evaluating
2961:
2907:
2717:
2619:
1762:
1698:
1520:
1314:
1294:
1261:
764:, in which case the eigenvectors are functions called
738:
608:
is applied to it, does not change direction. Applying
24312:
23754:; Vetterling, William T.; Flannery, Brian P. (2007),
23456:(2nd ed.), Englewood Cliffs, NJ: Prentice Hall,
22790:
22778:
22625:
22495:
22483:
22400:
22291:
22271:
21892:
21865:
21838:
21811:
21784:
21717:
21664:
21611:
21584:
21557:
21530:
21493:
21464:
21435:
21382:
21323:
21141:
21108:
21073:
21053:
21019:
20997:
20935:
20915:
20877:
20850:
20830:
20810:
20771:
20706:
20668:
20641:
20614:
20571:
20565:
solutions of the Schrödinger equation, one looks for
20540:
20509:
20481:
20431:
20400:
20329:
20152:
20083:
20059:
20031:
20004:
19962:
19935:
19911:
19891:
19868:
19822:
19775:
19668:
19648:
19628:
19557:
19519:
19492:
19462:
19442:
19370:
19340:
19312:
19124:
19092:
19028:
18973:
18839:
18780:
18654:
18534:
18456:
18422:
18352:
18282:
18248:
18198:
18120:
18086:
18016:
17946:
17912:
17862:
17705:
17658:
17497:
17413:
17366:
17338:
17277:
17234:
17220:{\displaystyle \lambda ^{2}-2\cos(\theta )\lambda +1}
17176:
17114:
17071:
16986:
16932:
16851:
16783:
16729:
16578:
16545:
16474:
16451:
16429:
16397:
16349:
16309:
16253:
16218:
16192:
16172:
16143:
16123:
16103:
16050:
16021:
15933:
15838:
15719:
15687:
15681:
we can find its eigenvectors by solving the equation
15626:
15586:
15562:
15538:
15508:
15482:
15436:
15416:
15205:
15103:
15049:
14973:
14852:
14710:
14573:
14109:
13923:
13825:
13733:{\displaystyle T(\mathbf {v} )=\lambda \mathbf {v} .}
13701:
13635:
13517:
13431:
13397:
13329:
13065:
12976:
12913:
12668:
12542:
12326:
12198:
12103:
11886:
11758:
11663:
11584:
11479:
11388:
11199:
11016:
10904:
10768:
10721:
10696:
10529:
10434:
10366:
10305:
10248:
9837:
9742:
9576:
9289:
9152:
9084:
8927:
8621:
8528:
8445:
8402:
8339:
8316:
8118:
8063:
8014:
7874:
7761:
7651:
7631:
7602:
7573:
7553:
7547:), it follows immediately that a left eigenvector of
7479:
7459:
7439:
7415:
7389:
7369:
7349:
7308:
7288:
7239:
7213:
7187:
7089:
7060:
7040:
6976:
6947:
6919:
6855:
6835:
6809:
6796:{\displaystyle \{\lambda _{1},\ldots ,\lambda _{k}\}}
6757:
6731:
6711:
6665:
6641:
6614:
6583:
6556:
6533:
6436:
6416:
6389:
6330:
6310:
6283:
6263:
6243:
6137:
6115:
5956:
5934:
5883:
5840:
5794:
5768:
5748:
5719:
5690:
5664:
5644:
5615:
5583:
5563:
5526:
5506:
5486:
5357:
5337:
5294:
5267:
5221:
5195:
5175:
5114:
5094:
5039:
5001:
4981:
4919:
4890:
4861:
4841:
4821:
4759:
4733:
4687:
4648:
4628:
4608:
4566:
4530:
4510:
4460:
4392:
4353:
4295:
4223:
4144:
4064:
3850:
3832:
is a complex number and the eigenvectors are complex
3685:
3404:
3177:
2880:
2826:
2687:
2607:
2439:
2269:
2123:
2010:
1834:
1514:
1478:
1423:
1385:
1247:
865:
774:
736:
654:{\displaystyle T(\mathbf {v} )=\lambda \mathbf {v} ,}
622:
461:
431:
397:
372:
350:
323:
301:
269:
199:
151:
131:
107:
85:
60:
24039:
Estimation of 3D motion and structure of human faces
23975:
23612:
23451:
23267:(5th ed.), Boston: Prindle, Weber and Schmidt,
23008:
22859:
22643:
Cornell University Department of Mathematics (2016)
22464:
22458:
20279:
tensor is symmetric and so can be decomposed into a
16015:
Both equations reduce to the single linear equation
14548:
is a tensor-multiple of itself and is considered in
13756:
This equation is called the eigenvalue equation for
6803:
are its eigenvalues, then the eigenvalues of matrix
984:. Historically, however, they arose in the study of
24341:Wikiversity uses introductory physics to introduce
24036:Xirouhakis, A.; Votsis, G.; Delopoulus, A. (2004),
23691:Lipschutz, Seymour; Lipson, Marc (12 August 2002).
23572:, Colchester, VT: Online book, St Michael's College
23554:
23280:
23232:
A Practical Guide to the study of Glacial Sediments
23063:
22811:
22577:
22385:
22357:Gilbert Strang. "6: Eigenvalues and Eigenvectors".
21262:
may be too technical for most readers to understand
21204:. The corresponding eigenvalues are interpreted as
21094:{\displaystyle \mathbf {t} ^{\dagger }\mathbf {t} }
21040:{\displaystyle \mathbf {t} ^{\dagger }\mathbf {t} }
19714:. Other methods are also available for clustering.
19697:algorithm. The principal eigenvector of a modified
19247:Principal component analysis is used as a means of
14314:is the dimension of the eigenspace associated with
10241:. These eigenvalues correspond to the eigenvectors
9731:
7265:{\displaystyle A\mathbf {v} =\lambda \mathbf {v} .}
976:Eigenvalues are often introduced in the context of
891:{\displaystyle A\mathbf {v} =\lambda \mathbf {v} ,}
145:, when the linear transformation is applied to it:
42:
24324:
24085:
23756:Numerical Recipes: The Art of Scientific Computing
23260:
23127:
23126:Beauregard, Raymond A.; Fraleigh, John B. (1973),
23125:
22736:
22637:
22313:
22277:
21905:
21878:
21851:
21824:
21797:
21756:
21703:
21650:
21597:
21570:
21543:
21508:
21479:
21450:
21421:
21368:
21160:
21127:
21093:
21059:
21039:
21005:
20956:
20921:
20901:
20856:
20836:
20816:
20792:
20754:
20689:
20647:
20627:
20584:
20546:
20522:
20487:
20464:
20406:
20362:
20202:
20134:
20065:
20037:
20017:
19990:
19941:
19917:
19897:
19874:
19852:
19808:
19674:
19654:
19634:
19614:
19563:
19543:
19505:
19478:
19448:
19424:
19352:
19318:
19142:
19110:
19055:
19010:
18947:
18824:
18765:
18639:
18511:
18441:
18407:
18337:
18267:
18233:
18175:
18105:
18071:
18001:
17931:
17897:
17839:
17690:
17643:
17482:
17398:
17351:
17320:
17262:
17219:
17161:
17099:
17049:
16971:
16917:
16836:
16768:
16586:
16560:
16528:
16457:
16437:
16411:
16380:
16315:
16295:
16239:
16204:
16178:
16155:
16129:
16109:
16089:
16036:
16005:
15913:
15819:
15705:
15671:
15592:
15568:
15544:
15517:
15494:
15448:
15422:
15368:
15181:
15086:
15031:
14955:
14831:
14682:
14222:
14045:
13880:
13732:
13656:
13563:
13482:
13417:
13365:
13246:
13219:
13042:
13022:
12962:
12869:
12654:
12518:
12320:These eigenvalues correspond to the eigenvectors,
12273:
12178:
12065:
11833:
11738:
11636:
11556:
11465:
11368:
11185:
11000:
10888:
10743:
10702:
10682:
10509:
10407:
10349:
10289:
10227:
9817:
9669:
9546:
9251:
9122:
9071:
8877:
8576:
8486:
8410:
8388:
8322:
8268:A matrix that is not diagonalizable is said to be
8149:
8094:
8035:
7980:
7834:
7657:
7637:
7617:
7588:
7559:
7531:
7465:
7445:
7421:
7401:
7375:
7355:
7334:{\displaystyle \mathbf {u} A=\kappa \mathbf {u} ,}
7333:
7294:
7264:
7219:
7199:
7145:
7075:
7046:
7026:
6962:
6933:
6905:
6841:
6821:
6795:
6743:
6717:
6694:
6647:
6620:
6589:
6569:
6539:
6512:
6452:
6422:
6395:
6372:
6316:
6296:
6269:
6249:
6225:
6121:
6093:
5940:
5912:
5869:
5826:
5780:
5754:
5725:
5705:
5670:
5650:
5630:
5598:
5569:
5545:
5512:
5492:
5468:
5343:
5323:
5280:
5253:
5207:
5181:
5158:
5100:
5080:
5025:
4987:
4967:
4905:
4876:
4847:
4827:
4807:
4745:
4719:
4673:
4634:
4614:
4594:
4552:
4516:
4496:
4446:
4375:
4339:
4279:
4202:
4086:
3865:
3750:
3539:
3366:
3033:at least one of the roots is real. Therefore, any
2987:
2864:
2793:
2655:
2538:
2299:
2164:
2043:
1981:
1820:
1500:
1445:
1408:{\displaystyle \mathbf {x} =\lambda \mathbf {y} .}
1407:
1353:
1237:-dimensional vectors that are formed as a list of
890:
827:
756:
653:
484:{\displaystyle A\mathbf {v} =\lambda \mathbf {v} }
483:
439:
405:
383:
358:
332:
309:
283:
205:
174:{\displaystyle T\mathbf {v} =\lambda \mathbf {v} }
173:
137:
113:
93:
24285:– Tutorial and Interactive Program from Revoledu.
24283:Eigen Values and Eigen Vectors Numerical Examples
24227:may not follow Knowledge's policies or guidelines
23597:Mathematical thought from ancient to modern times
20697:. In this notation, the Schrödinger equation is:
19726:is represented by a matrix whose entries are the
19685:The principal eigenvector is used to measure the
19212:are PSD. This orthogonal decomposition is called
18234:{\displaystyle \gamma _{i}=\gamma (\lambda _{i})}
13483:{\displaystyle {\frac {d}{dt}}f(t)=\lambda f(t).}
8426:
8280:. Over an algebraically closed field, any matrix
7737:. The eigenvalues need not be distinct. Define a
5827:{\displaystyle \lambda _{1},\ldots ,\lambda _{n}}
5254:{\displaystyle \lambda _{1},\ldots ,\lambda _{d}}
5088:, which means that the algebraic multiplicity of
4720:{\displaystyle \lambda I_{\gamma _{A}(\lambda )}}
2404:implies that the characteristic polynomial of an
1018:saw how their work could be used to classify the
25431:
24095:Journal of Computational and Applied Mathematics
23919:Vellekoop, I. M.; Mosk, A. P. (15 August 2007).
23558:Mathematical epidemiology of infectious diseases
22652:
22110:( Halle ("Halae"), (Germany): Gebauer, 1755). (
21147:
21114:
21013:. The eigenvectors of the transmission operator
18960:The characteristic equation for a rotation is a
17162:{\displaystyle (\lambda -k_{1})(\lambda -k_{2})}
12669:
12199:
11759:
9842:
8626:
7625:is the same as the characteristic polynomial of
7302:. In this formulation, the defining equation is
6138:
5002:
4944:
4920:
3178:
2688:
2440:
2270:
24086:Golub, Gene F.; van der Vorst, Henk A. (2000),
23790:Rotter, Stefan; Gigan, Sylvain (2 March 2017).
23690:
23334:Diekmann, O; Heesterbeek, JA; Metz, JA (1990),
23259:Burden, Richard L.; Faires, J. Douglas (1993),
22902:
22646:Lower-Level Courses for Freshmen and Sophomores
19073:
16445:is (a good approximation of) an eigenvector of
13784:) is the result of applying the transformation
13304:be a linear differential operator on the space
9687:= 3, as is any scalar multiple of this vector.
9267:= 1, as is any scalar multiple of this vector.
8516:An extended version, showing all four quadrants
8295:
4130:), which relates to the dimension and rank of (
3388:linear terms and this is the same as equation (
1241:scalars, such as the three-dimensional vectors
24289:Introduction to Eigen Vectors and Eigen Values
21458:then is the primary orientation/dip of clast,
20230:
19277:criteria for determining the number of factors
19228:(in which each variable is scaled to have its
19118:with a standard deviation of 3 in roughly the
14701:is found by using its characteristic equation
14501:Associative algebras and representation theory
13386:
8418:that realizes that maximum is an eigenvector.
7157:
1026:(characteristic root), for what is now called
24831:
24389:
23918:
23670:
23555:Heesterbeek, J. A. P.; Diekmann, Odo (2000),
23494:
23469:
23290:Bulletin of the American Mathematical Society
23023:"Endogene Geologie - Ruhr-Universität Bochum"
22984:
22945:
22709:
22619:
22607:
22247:Lemma for linear independence of eigenvectors
20135:{\displaystyle m{\ddot {x}}+c{\dot {x}}+kx=0}
19056:{\displaystyle \cos \theta \pm i\sin \theta }
10169:
10126:
8105:or by instead left multiplying both sides by
3956:of matrix multiplication. Similarly, because
2211:Eigenvalues and the characteristic polynomial
2089:corresponding to that eigenvector. Equation (
1175:. One of the most popular methods today, the
520:, or the language of linear transformations.
24170:, Wellesley, MA: Wellesley-Cambridge Press,
24088:"Eigenvalue Computation in the 20th Century"
23894:
23758:(3rd ed.), Cambridge University Press,
23258:
22933:
22344:
22207:
22197:
21992:recognition systems for speaker adaptation.
21767:
20951:
20896:
20787:
20749:
20725:
20684:
16637:
14532:representation-theoretical concept of weight
14523:. The study of such actions is the field of
12078:
7140:
7090:
7021:
6977:
6900:
6856:
6790:
6758:
5557:The direct sum of the eigenspaces of all of
4602:vectors orthogonal to these eigenvectors of
4032:are not zero, they are also eigenvectors of
1069:have real eigenvalues. This was extended by
75:that has its direction unchanged by a given
24845:
24049:
23903:
23789:
22960:
22835:
22823:
19303:is defined as an eigenvalue of the graph's
17898:{\displaystyle \mu _{i}=\mu (\lambda _{i})}
17691:{\displaystyle \lambda _{1}=\lambda _{2}=1}
17399:{\displaystyle \lambda _{1}=\lambda _{2}=k}
16326:
8257:. It then follows that the eigenvectors of
6407:if and only if every eigenvalue is nonzero.
5788:matrix of complex numbers with eigenvalues
4968:{\displaystyle \det(A-\xi I)=\det(D-\xi I)}
4504:. We can therefore find a (unitary) matrix
3384:then the right-hand side is the product of
3159:) factors the characteristic polynomial of
1501:{\displaystyle A\mathbf {v} =\mathbf {w} ,}
843:matrix, in which case the eigenvectors are
24838:
24824:
24396:
24382:
23714:Matrix analysis and applied linear algebra
23238:
23020:
22658:University of Michigan Mathematics (2016)
22406:
22356:
21200:can be defined by the eigenvectors of the
15829:This matrix equation is equivalent to two
15192:A similar procedure is used for solving a
13917:By definition of a linear transformation,
11646:
8204:be a non-singular square matrix such that
7669:Diagonalization and the eigendecomposition
7665:are associated with the same eigenvalues.
5577:'s eigenvalues is the entire vector space
1456:Now consider the linear transformation of
24361:Numerical solution of eigenvalue problems
24263:Learn how and when to remove this message
24114:
24067:
24045:, National Technical University of Athens
24015:
23995:
23909:
23858:
23807:
23697:. McGraw Hill Professional. p. 111.
23581:, Waltham: Blaisdell Publishing Company,
23546:
23443:
23421:
23359:
23301:
23229:
23180:
23039:
22996:
22796:
21976:, eigenfaces provide a means of applying
21658:, the fabric is said to be isotropic. If
21422:{\displaystyle E_{1}\geq E_{2}\geq E_{3}}
21290:Learn how and when to remove this message
21274:, without removing the technical details.
21242:
20461:
20356:
20349:
20342:
19220:among variables. PCA is performed on the
16647:Eigenvalues of geometric transformations
16405:
16287:
16081:
15576:is the characteristic polynomial of some
15070:
15063:
13014:
12954:
11548:
11457:
11378:Therefore, the other two eigenvectors of
10399:
10341:
10281:
8600:for which the determinant of the matrix (
8590:of this transformation satisfy equation (
8375:
8348:
7609:
7580:
7520:
7500:
7486:
6927:
5693:
5618:
5586:
5324:{\displaystyle \gamma _{A}(\lambda _{i})}
4415:
4408:
3853:
3062:
1446:{\displaystyle \lambda =-{\frac {1}{20}}}
494:There is a direct correspondence between
23895:Trefethen, Lloyd N.; Bau, David (1997),
23640:
23576:
23565:
23383:
23076:Xirouhakis, Votsis & Delopoulus 2004
22847:
22772:
22748:
22692:
22379:
22366:(5 ed.). Wellesley-Cambridge Press.
22238:
21920:
20987:numerous times when traversing a static
20294:
20216:at the cost of solving a larger system.
19742:
19077:
14361:Any subspace spanned by eigenvectors of
14354:operates, and there cannot be more than
12532:Matrix with repeated eigenvalues example
8430:
3884:under addition. That is, if two vectors
3153:distinct eigenvalues. Whereas equation (
1208:
1191:Eigenvalues and eigenvectors of matrices
700:
687:
24151:
24016:Wolchover, Natalie (13 November 2019).
23694:Schaum's Easy Outline of Linear Algebra
23532:
23405:
23091:
22595:
22583:
22508:
22489:
22162:: 827–830, 845–865, 889–907, 931–937.
21757:{\displaystyle E_{1}>E_{2}>E_{3}}
20465:{\displaystyle H\psi _{E}=E\psi _{E}\,}
20315:can be seen as the eigenvectors of the
20290:
13564:{\displaystyle f(t)=f(0)e^{\lambda t},}
6934:{\displaystyle \alpha \in \mathbb {C} }
6430:is invertible, then the eigenvalues of
6129:is the product of all its eigenvalues,
4595:{\displaystyle n-\gamma _{A}(\lambda )}
4484:
4466:
4418:
4395:
3760:On one hand, this set is precisely the
3598:) equals the geometric multiplicity of
3041:
2250:is zero. Therefore, the eigenvalues of
1003:, and discovered the importance of the
921:is applied liberally when naming them:
25432:
24787:Comparison of linear algebra libraries
24183:
24165:
23840:
23729:
23151:
22887:
22883:
22784:
22760:
22477:
22394:
22285:terms it is possible to get away with
22242:
22234:
21711:, the fabric is said to be planar. If
21231:. This particular representation is a
19018:, which is a negative number whenever
19011:{\displaystyle D=-4(\sin \theta )^{2}}
14474:eigenvalues can be generalized to the
14381:can be spanned by the eigenvectors of
12097:Consider the lower triangular matrix,
7997:is the eigenvalue associated with the
6659:, every eigenvalue has absolute value
5913:{\displaystyle \mu _{A}(\lambda _{i})}
5870:{\displaystyle \mu _{A}(\lambda _{i})}
5331:. The total geometric multiplicity of
5261:, where the geometric multiplicity of
4110:is the dimension of the nullspace of (
1108:started by Laplace, by realizing that
24819:
24377:
24188:, Belmont, CA: Thomson, Brooks/Cole,
23771:
23711:
23594:
23498:Earth Surface Processes and Landforms
23329:from the original on 19 January 2022.
23103:
23052:Diekmann, Heesterbeek & Metz 1990
22914:
22871:
22721:
22631:
22571:
22559:
22547:
22535:
22520:
22230:
22151:
21272:make it understandable to non-experts
21171:
20363:{\displaystyle n=1,\,2,\,3,\,\ldots }
19738:
19267:significance (which differs from the
16342:is to instead multiply the vector by
13682:if and only if there exists a scalar
13593:
8253:linearly independent eigenvectors of
7748:linearly independent eigenvectors of
7645:, the left and right eigenvectors of
7541:Comparing this equation to equation (
7429:satisfying this equation is called a
5658:linearly independent eigenvectors of
4808:{\displaystyle (A-\xi I)V=V(D-\xi I)}
4553:{\displaystyle \gamma _{A}(\lambda )}
4376:{\displaystyle \gamma _{A}(\lambda )}
4087:{\displaystyle \gamma _{A}(\lambda )}
4055:, is referred to as the eigenvalue's
3625:), defined in the next section, then
1104:clarified an important aspect in the
25400:
24207:
23021:Busche, Christian; Schiller, Beate.
22417:
22415:
22112:https://books.google.com/books?id=29
21704:{\displaystyle E_{1}=E_{2}>E_{3}}
21246:
19615:{\textstyle 1/{\sqrt {\deg(v_{i})}}}
19158:is more readily visualized than the
18188:
17852:
16606:was designed in 1961. Combining the
16412:{\displaystyle \mu \in \mathbb {C} }
16044:. Therefore, any vector of the form
14555:
14342:. As a consequence, eigenvectors of
13692:
8233:must therefore be an eigenvector of
7227:in the defining equation, equation (
5738:Additional properties of eigenvalues
2430:
2260:
2114:
2001:
512:vector space into itself, given any
25412:
24186:Linear algebra and its applications
24056:Linear Algebra and Its Applications
23241:The New Cassell's German Dictionary
22321:operations, but that does not take
22037:List of numerical-analysis software
21764:, the fabric is said to be linear.
20902:{\displaystyle H|\Psi _{E}\rangle }
20382:. The center of each figure is the
19425:{\displaystyle I-D^{-1/2}AD^{-1/2}}
19326:, or (increasingly) of the graph's
15393:
14534:is an analog of eigenvalues, while
14244:are either zero or eigenvectors of
13621:be a linear transformation mapping
13030:. The total geometric multiplicity
8901:, which are the two eigenvalues of
8608:) equals zero are the eigenvalues.
7034:. More generally, for a polynomial
2811:, which are the two eigenvalues of
2677:, the characteristic polynomial of
2300:{\displaystyle \det(A-\lambda I)=0}
1460:-dimensional vectors defined by an
1096:found the corresponding result for
999:studied the rotational motion of a
505:and linear transformations from an
13:
24403:
24078:
23386:A First Course In Abstract Algebra
23234:, London: Arnold, pp. 103–107
22860:Friedberg, Insel & Spence 1989
20957:{\displaystyle |\Psi _{E}\rangle }
20942:
20887:
20793:{\displaystyle |\Psi _{E}\rangle }
20778:
20740:
20716:
20690:{\displaystyle |\Psi _{E}\rangle }
20675:
19084:multivariate Gaussian distribution
17263:{\displaystyle \ (\lambda -1)^{2}}
17100:{\displaystyle \ (\lambda -k)^{2}}
16703:
16247:, that is, any vector of the form
14693:The solution of this equation for
14463:) may not have an inverse even if
14416:
14283:is therefore a linear subspace of
13366:{\displaystyle Df(t)=\lambda f(t)}
13312:real functions of a real argument
8421:
8150:{\displaystyle Q^{-1}AQ=\Lambda .}
8141:
8095:{\displaystyle A=Q\Lambda Q^{-1},}
8073:
8027:
7691:linearly independent eigenvectors
2327:, the left-hand side of equation (
1144:At the start of the 20th century,
525:infinite-dimensional vector spaces
79:. More precisely, an eigenvector,
21:Characteristic equation (calculus)
14:
25471:
24203:
24155:An introduction to linear algebra
22412:
22206:: ... ) Later on the same page:
21651:{\displaystyle E_{1}=E_{2}=E_{3}}
20967:
20596:functions. Since this space is a
19809:{\displaystyle m{\ddot {x}}+kx=0}
19216:(PCA) in statistics. PCA studies
16597:
15094:for use in the solution equation
13772:corresponding to the eigenvector
13391:Consider the derivative operator
12188:The characteristic polynomial of
11748:The characteristic polynomial of
10417:or any nonzero multiple thereof.
9827:The characteristic polynomial of
9570:solves this equation. Therefore,
9146:solves this equation. Therefore,
8451:
8245:must be linearly independent for
4106:), the geometric multiplicity of
2361:, except that its term of degree
1999:are scalar multiples, that is if
1221:, not changing its direction, so
25411:
25399:
25388:
25387:
25375:
24800:
24799:
24777:Basic Linear Algebra Subprograms
24535:
24367:, Jack Dongarra, Axel Ruhe, and
24336:
24212:
23732:Linear Algebra and Matrix Theory
23155:A first course in linear algebra
23009:Knox-Robinson & Gardoll 1998
21509:{\displaystyle \mathbf {v} _{3}}
21496:
21480:{\displaystyle \mathbf {v} _{2}}
21467:
21451:{\displaystyle \mathbf {v} _{1}}
21438:
21356:
21341:
21326:
21251:
21087:
21076:
21033:
21022:
20999:
20266:
19991:{\displaystyle kx=\omega ^{2}mx}
19853:{\displaystyle m{\ddot {x}}=-kx}
19717:
18895:
18846:
18783:
18713:
18661:
18590:
18541:
16712:
16696:
16689:
16682:
16580:
16561:{\displaystyle \mathbf {v} ^{*}}
16548:
16519:
16508:
16500:
16486:
16431:
14206:
14182:
14158:
14150:
14129:
14121:
14029:
14005:
13981:
13964:
13943:
13935:
13866:
13852:
13838:
13723:
13709:
13602:on arbitrary vector spaces. Let
13418:{\displaystyle {\tfrac {d}{dt}}}
12460:
12398:
12329:
12007:
11948:
11889:
11609:
11587:
11482:
11391:
9732:Three-dimensional matrix example
9714:associated with the eigenvalues
9579:
9314:
9155:
8945:
8504:= and blue vectors parallel to
8404:
8382:
8369:
8358:
8342:
7957:
7928:
7904:
7811:
7792:
7778:
7514:
7494:
7324:
7310:
7255:
7244:
6695:{\displaystyle |\lambda _{i}|=1}
5706:{\displaystyle \mathbb {C} ^{n}}
5631:{\displaystyle \mathbb {C} ^{n}}
5599:{\displaystyle \mathbb {C} ^{n}}
4043:The dimension of the eigenspace
3952:. This can be checked using the
3866:{\displaystyle \mathbb {C} ^{n}}
3736:
3728:
3698:
2937:
2883:
2858:
2850:
2369:. This polynomial is called the
2325:Leibniz formula for determinants
2155:
2147:
2112:) can be stated equivalently as
2034:
2023:
2015:
1491:
1483:
1398:
1387:
1302:
1249:
1179:, was proposed independently by
1129:studied the first eigenvalue of
1058:Théorie analytique de la chaleur
948:associated with that eigenvalue.
881:
870:
757:{\displaystyle {\tfrac {d}{dx}}}
644:
630:
477:
466:
433:
399:
377:
352:
303:
167:
156:
87:
38:
25296:Computational complexity theory
24675:Seven-dimensional cross product
23340:Journal of Mathematical Biology
23064:Heesterbeek & Diekmann 2000
23014:
22877:
22865:
22251:
22245:, Theorem EDELI on p. 469; and
22229:For a proof of this lemma, see
22223:
22176:
22145:
22074:
21778:The basic reproduction number (
21216:procedure, called in this case
21161:{\displaystyle \tau _{\min }=0}
21128:{\displaystyle \tau _{\max }=1}
19689:of its vertices. An example is
19255:, such as those encountered in
16632:
16621:is one example of an efficient
15611:
15552:is 4 or less. According to the
13043:Eigenvector-eigenvalue identity
12457:
12395:
12004:
11945:
10966:
10934:
9123:{\displaystyle 1v_{1}+1v_{2}=0}
7618:{\displaystyle A^{\textsf {T}}}
7589:{\displaystyle A^{\textsf {T}}}
6628:is not only Hermitian but also
2934:
2416:, being a polynomial of degree
1300:
1292:
1084:proved that the eigenvalues of
1073:in 1855 to what are now called
963:, then this basis is called an
932:The set of all eigenvectors of
710:The example here, based on the
604:is a nonzero vector that, when
24304:Matrix Eigenvectors Calculator
24168:Introduction to linear algebra
23239:Betteridge, Harold T. (1965),
23199:10.1103/physrevlett.125.165901
22737:Beauregard & Fraleigh 1973
22360:Introduction to Linear Algebra
22350:
22308:
22295:
21233:generalized eigenvalue problem
20937:
20882:
20773:
20735:
20711:
20670:
19955:generalized eigenvalue problem
19607:
19594:
19486:equal to the degree of vertex
19137:
19125:
19105:
19093:
19070:) has reciprocal eigenvalues.
18999:
18986:
18228:
18215:
17892:
17879:
17306:
17300:
17251:
17238:
17205:
17199:
17156:
17137:
17134:
17115:
17088:
17075:
16381:{\displaystyle (A-\mu I)^{-1}}
16366:
16350:
16303:, for any nonzero real number
16097:, for any nonzero real number
15405:
15379:
14507:Weight (representation theory)
14210:
14199:
14186:
14175:
14162:
14146:
14133:
14117:
14033:
14025:
14009:
13998:
13985:
13977:
13968:
13960:
13947:
13931:
13856:
13848:
13713:
13705:
13645:
13606:be any vector space over some
13590:article gives other examples.
13542:
13536:
13527:
13521:
13474:
13468:
13456:
13450:
13360:
13354:
13342:
13336:
13316:. The eigenvalue equation for
13207:
13181:
13158:
13155:
13142:
13116:
13089:
13067:
12855:
12842:
12833:
12820:
12687:
12672:
12265:
12253:
12250:
12238:
12235:
12223:
12217:
12202:
11825:
11813:
11810:
11798:
11795:
11783:
11777:
11762:
10158:
10146:
10143:
10131:
10121:
10109:
9860:
9845:
9309:
9294:
8940:
8928:
8865:
8853:
8850:
8838:
8644:
8629:
8427:Two-dimensional matrix example
8249:to be invertible, there exist
7865:by its associated eigenvalue,
7683:form a basis, or equivalently
7675:Eigendecomposition of a matrix
7137:
7124:
7109:
7096:
7070:
7064:
6682:
6667:
6147:
6141:
5969:
5963:
5907:
5894:
5864:
5851:
5423:
5410:
5318:
5305:
5153:
5147:
5131:
5125:
5073:
5067:
5053:
5040:
5020:
5005:
4962:
4947:
4938:
4923:
4802:
4787:
4775:
4760:
4712:
4706:
4589:
4583:
4547:
4541:
4439:
4433:
4370:
4364:
4334:
4328:
4312:
4306:
4268:
4262:
4246:
4240:
4194:
4179:
4161:
4155:
4081:
4075:
3646:Given a particular eigenvalue
3442:
3429:
3356:
3343:
3329:
3309:
3301:
3288:
3274:
3254:
3249:
3236:
3222:
3202:
3196:
3181:
2706:
2691:
2530:
2511:
2505:
2486:
2483:
2464:
2458:
2443:
2402:fundamental theorem of algebra
2288:
2273:
1217:acts by stretching the vector
1043:used the work of Lagrange and
634:
626:
101:, of a linear transformation,
1:
24116:10.1016/S0377-0427(00)00413-1
23635:10.1016/S0098-3004(97)00122-2
21916:
21306:, especially in the study of
19143:{\displaystyle (0.878,0.478)}
18442:{\displaystyle \gamma _{1}=1}
18268:{\displaystyle \gamma _{1}=2}
16186:above has another eigenvalue
14476:spectrum of a linear operator
14447:, and therefore its inverse (
14350:of the vector space on which
13799:is the product of the scalar
13666:We say that a nonzero vector
8196:Conversely, suppose a matrix
8036:{\displaystyle AQ=Q\Lambda .}
5546:{\displaystyle \gamma _{A}=n}
5026:{\displaystyle \det(D-\xi I)}
4975:. But from the definition of
2997:If the entries of the matrix
2077:of the linear transformation
1363:These vectors are said to be
384:{\displaystyle A\mathbf {v} }
258:
25460:Singular value decomposition
24517:Eigenvalues and eigenvectors
24344:Eigenvalues and eigenvectors
24297:– A visual explanation with
23826:10.1103/RevModPhys.89.015005
23685:10.1016/0041-5553(63)90168-X
23548:10.1016/0315-0860(75)90032-4
23230:Benn, D.; Evans, D. (2004),
22423:"Eigenvector and Eigenvalue"
22332:
22052:Quadratic eigenvalue problem
21962:principal component analysis
21006:{\displaystyle \mathbf {t} }
20144:quadratic eigenvalue problem
19285:structural equation modeling
19214:principal component analysis
19174:Positive semidefinite matrix
19168:Principal component analysis
19074:Principal component analysis
17352:{\displaystyle \lambda _{i}}
16587:{\displaystyle \mathbf {v} }
16438:{\displaystyle \mathbf {v} }
15410:The eigenvalues of a matrix
8411:{\displaystyle \mathbf {x} }
8330:is the maximum value of the
8296:Variational characterization
8261:form a basis if and only if
8179:to the diagonal matrix Λ or
7714:with associated eigenvalues
7679:Suppose the eigenvectors of
6849:is the identity matrix) are
5678:; such a basis is called an
5281:{\displaystyle \lambda _{i}}
3880:is a linear subspace, it is
3764:or nullspace of the matrix (
951:If a set of eigenvectors of
569:of the rotational motion of
447:is called an eigenvector of
440:{\displaystyle \mathbf {v} }
406:{\displaystyle \mathbf {v} }
359:{\displaystyle \mathbf {v} }
310:{\displaystyle \mathbf {v} }
94:{\displaystyle \mathbf {v} }
16:Concepts from linear algebra
7:
24291:– lecture from Khan Academy
23996:Weisstein, Eric W. (n.d.).
23615:Computers & Geosciences
23599:, Oxford University Press,
22903:Lipschutz & Lipson 2002
21995:
21928:as examples of eigenvectors
20844:represents the eigenvalue.
20231:Tensor of moment of inertia
20018:{\displaystyle \omega ^{2}}
19882:to be sinusoidal in time).
19236:and the eigenvalues to the
14389:is the entire vector space
14087:associated with eigenvalue
13746:
13387:Derivative operator example
9279:
8917:
8592:
8215:. Left multiplying both by
7543:
7229:
7158:Left and right eigenvectors
6304:, for any positive integer
6277:; i.e., the eigenvalues of
3675:
3390:
3155:
3109:) of the eigenvalue is its
3006:even if all the entries of
2552:
2381:
2329:
2313:
2223:
2178:
2108:
2091:
2057:
1148:studied the eigenvalues of
1014:In the early 19th century,
600:of a linear transformation
596:In essence, an eigenvector
530:
284:{\displaystyle n{\times }n}
123:scaled by a constant factor
10:
25476:
25346:Films about mathematicians
24356:Computation of Eigenvalues
24161:, Brigham Young University
23841:Shilov, Georgi E. (1977),
23734:(2nd ed.), New York:
23384:Fraleigh, John B. (1976),
23152:Beezer, Robert A. (2006),
23108:(5th ed.), New York:
23084:
22137:: 184–252; see especially
22108:Specimen theoriae turbinum
21931:
21771:
20239:, the eigenvectors of the
19761:. The eigenvalues are the
19750:
19456:is a diagonal matrix with
19171:
19165:
18106:{\displaystyle \mu _{1}=2}
17932:{\displaystyle \mu _{1}=2}
16608:Householder transformation
16330:
16205:{\displaystyle \lambda =1}
16156:{\displaystyle \lambda =6}
15618:system of linear equations
15383:
14504:
14481:as the set of all scalars
14420:
13273:
13051:, the norm squared of the
8435:The transformation matrix
8299:
7672:
7161:
7054:the eigenvalues of matrix
6963:{\displaystyle \alpha I+A}
5877:times in this list, where
5834:. Each eigenvalue appears
5480:of all the eigenspaces of
4674:{\displaystyle D:=V^{T}AV}
4098:is also the nullspace of (
3031:intermediate value theorem
2665:Taking the determinant of
2349:, the order of the matrix
2258:that satisfy the equation
2214:
1194:
971:
193:is the multiplying factor
18:
25369:
25319:
25276:
25186:
25148:
25115:
25067:
25039:
24986:
24933:
24915:Philosophy of mathematics
24890:
24855:
24795:
24757:
24713:
24650:
24602:
24544:
24533:
24429:
24411:
24349:
24325:{\displaystyle n\times n}
24152:Kuttler, Kenneth (2017),
24069:10.1016/j.laa.2024.03.035
23796:Reviews of Modern Physics
23388:(2nd ed.), Reading:
23106:Elementary Linear Algebra
23096:, in Miller, Jeff (ed.),
22985:Graham & Midgley 2000
22946:Vellekoop & Mosk 2007
22710:Golub & Van Loan 1996
22671:. Accessed on 2016-03-27.
22649:. Accessed on 2016-03-27.
22620:Golub & Van Loan 1996
22233:, Theorem 8.2 on p. 186;
21987:Similar to this concept,
21774:Basic reproduction number
21768:Basic reproduction number
20628:{\displaystyle \psi _{E}}
20585:{\displaystyle \psi _{E}}
20523:{\displaystyle \psi _{E}}
20321:angular momentum operator
20317:hydrogen atom Hamiltonian
19332:discrete Laplace operator
19290:
18522:
18186:
17850:
17331:
17060:
16720:
16678:
16671:
16666:
16661:
16658:
16653:
16638:Geometric transformations
15495:{\displaystyle n\times n}
15449:{\displaystyle 2\times 2}
14846: – 1 equations
14373:, and the restriction of
13657:{\displaystyle T:V\to V.}
13425:with eigenvalue equation
12079:Triangular matrix example
10427:cyclic permutation matrix
8169:similarity transformation
7402:{\displaystyle 1\times n}
7200:{\displaystyle n\times n}
6744:{\displaystyle n\times n}
5781:{\displaystyle n\times n}
3800:equals the nullspace of (
2372:characteristic polynomial
2357:depend on the entries of
2337:function of the variable
2227:) has a nonzero solution
2217:Characteristic polynomial
25351:Recreational mathematics
24184:Strang, Gilbert (2006),
24166:Strang, Gilbert (1993),
24145:University of Nottingham
24050:Van Mieghem, P. (2024).
23897:Numerical Linear Algebra
23730:Nering, Evar D. (1970),
23649:(2nd Revised ed.),
23577:Herstein, I. N. (1964),
22934:Trefethen & Bau 1997
22892:Lemma for the eigenspace
22459:Wolfram.com: Eigenvector
22345:Burden & Faires 1993
22314:{\displaystyle O(n^{4})}
22169:équation caractéristique
22067:
21968:of some of them. In the
21960:; this is an example of
20241:moment of inertia tensor
19733:Perron–Frobenius theorem
19728:transition probabilities
19544:{\displaystyle D^{-1/2}}
19269:statistical significance
19249:dimensionality reduction
16327:Simple iterative methods
15926:
15922:
14550:Langlands correspondence
14485:for which the operator (
13379:and are commonly called
10755:For the real eigenvalue
10744:{\displaystyle i^{2}=-1}
9556:Any nonzero vector with
9132:Any nonzero vector with
8274:generalized eigenvectors
8211:is some diagonal matrix
7164:left and right (algebra)
5476:is the dimension of the
5101:{\displaystyle \lambda }
4289:To prove the inequality
3844:is a linear subspace of
1055:in his famous 1822 book
206:{\displaystyle \lambda }
138:{\displaystyle \lambda }
25236:Mathematical statistics
25226:Mathematical psychology
25196:Engineering mathematics
25130:Algebraic number theory
24363:Edited by Zhaojun Bai,
23774:Advanced linear algebra
23712:Meyer, Carl D. (2000),
23169:Physical Review Letters
22961:Rotter & Gigan 2017
22323:combinatorial explosion
21180:, and in particular in
20225:finite element analysis
20038:{\displaystyle \omega }
19703:stationary distribution
19362:combinatorial Laplacian
16117:, is an eigenvector of
13264:diagonalizable matrices
13258:formed by removing the
12903:On the other hand, the
12092:upper triangular matrix
11647:Diagonal matrix example
8290:generalized eigenspaces
8241:. Since the columns of
8200:is diagonalizable. Let
8046:Because the columns of
7466:{\displaystyle \kappa }
7409:matrix. Any row vector
7356:{\displaystyle \kappa }
6237:The eigenvalues of the
5208:{\displaystyle d\leq n}
4906:{\displaystyle D-\xi I}
4877:{\displaystyle A-\xi I}
3876:Because the eigenspace
3673:that satisfy equation (
3076:be an eigenvalue of an
2387:characteristic equation
1375:, if there is a scalar
1112:can cause instability.
1098:skew-symmetric matrices
1053:separation of variables
1033:characteristic equation
1030:; his term survives in
929:of that transformation.
25382:Mathematics portal
25231:Mathematical sociology
25211:Mathematical economics
25206:Mathematical chemistry
25135:Analytic number theory
25016:Differential equations
24502:Row and column vectors
24326:
24277:What are Eigen Values?
23772:Roman, Steven (2008),
23595:Kline, Morris (1972),
23566:Hefferon, Jim (2001),
23445:10.1093/comjnl/4.4.332
23423:10.1093/comjnl/4.3.265
23104:Anton, Howard (1987),
23092:Aldrich, John (2006),
23027:www.ruhr-uni-bochum.de
22315:
22279:
22208:
22198:
22042:Nonlinear eigenproblem
21929:
21907:
21886:has passed. The value
21880:
21853:
21826:
21799:
21758:
21705:
21652:
21599:
21572:
21545:
21510:
21481:
21452:
21423:
21370:
21243:Geology and glaciology
21162:
21129:
21095:
21061:
21041:
21007:
20958:
20923:
20903:
20858:
20838:
20818:
20794:
20756:
20691:
20649:
20629:
20604:, one can introduce a
20586:
20548:
20524:
20489:
20466:
20408:
20391:
20364:
20221:differential equations
20214:algebraic manipulation
20204:
20136:
20067:
20039:
20025:is the eigenvalue and
20019:
19992:
19943:
19919:
19899:
19876:
19854:
19810:
19748:
19676:
19656:
19636:
19616:
19565:
19545:
19507:
19480:
19479:{\displaystyle D_{ii}}
19450:
19432:(sometimes called the
19426:
19360:(sometimes called the
19354:
19320:
19251:in the study of large
19163:
19144:
19112:
19057:
19012:
18949:
18826:
18767:
18641:
18513:
18443:
18409:
18339:
18269:
18235:
18177:
18107:
18073:
18003:
17933:
17899:
17841:
17692:
17645:
17484:
17400:
17353:
17322:
17264:
17221:
17163:
17101:
17051:
16973:
16919:
16838:
16770:
16708:
16588:
16562:
16530:
16459:
16439:
16413:
16382:
16317:
16297:
16241:
16240:{\displaystyle 3x+y=0}
16206:
16180:
16157:
16131:
16111:
16091:
16038:
16007:
15915:
15821:
15707:
15673:
15594:
15570:
15546:
15519:
15496:
15470:Wilkinson's polynomial
15450:
15424:
15370:
15183:
15088:
15033:
14957:
14833:
14684:
14513:algebra representation
14467:is not an eigenvalue.
14409:admits an eigenbasis,
14358:distinct eigenvalues.
14296:geometric multiplicity
14224:
14047:
13882:
13734:
13658:
13600:linear transformations
13575:= 0 the eigenfunction
13565:
13484:
13419:
13367:
13294:differential operators
13248:
13221:
13024:
12964:
12905:geometric multiplicity
12881:algebraic multiplicity
12871:
12656:
12520:
12275:
12180:
12067:
11835:
11740:
11638:
11558:
11467:
11370:
11187:
11002:
10890:
10745:
10704:
10684:
10511:
10409:
10351:
10291:
10229:
9819:
9671:
9548:
9253:
9124:
9073:
8879:
8578:
8519:
8488:
8412:
8390:
8324:
8151:
8096:
8037:
7982:
7861:scales each column of
7836:
7744:whose columns are the
7659:
7639:
7619:
7590:
7561:
7533:
7467:
7447:
7423:
7403:
7377:
7357:
7335:
7296:
7266:
7221:
7201:
7147:
7077:
7048:
7028:
6964:
6935:
6907:
6843:
6823:
6797:
6745:
6719:
6696:
6649:
6622:
6591:
6571:
6541:
6514:
6454:
6453:{\displaystyle A^{-1}}
6424:
6397:
6374:
6318:
6298:
6271:
6251:
6227:
6173:
6123:
6095:
6032:
5995:
5942:
5914:
5871:
5828:
5782:
5756:
5727:
5707:
5672:
5652:
5632:
5600:
5571:
5547:
5514:
5494:
5470:
5399:
5345:
5325:
5282:
5255:
5209:
5183:
5160:
5102:
5082:
5027:
4989:
4969:
4907:
4878:
4849:
4829:
4809:
4753:on both sides, we get
4747:
4746:{\displaystyle -\xi V}
4721:
4675:
4636:
4616:
4596:
4554:
4518:
4498:
4448:
4377:
4341:
4281:
4204:
4088:
4057:geometric multiplicity
3867:
3752:
3541:
3495:
3368:
3111:multiplicity as a root
3090:algebraic multiplicity
3063:Algebraic multiplicity
2989:
2866:
2795:
2657:
2540:
2345:of this polynomial is
2301:
2166:
2045:
1983:
1952:
1822:
1502:
1447:
1409:
1355:
1230:
1122:Sturm–Liouville theory
1080:Around the same time,
1063:Charles-François Sturm
1024:racine caractéristique
990:differential equations
898:where the eigenvector
892:
829:
758:
707:
698:
655:
591:matrix diagonalization
485:
441:
407:
385:
360:
334:
311:
285:
217:Geometrically, vectors
207:
175:
139:
115:
95:
25361:Mathematics education
25291:Theory of computation
25011:Hypercomplex analysis
24507:Row and column spaces
24452:Scalar multiplication
24327:
24002:mathworld.wolfram.com
23982:mathworld.wolfram.com
23647:New York: McGraw-Hill
23040:Benn & Evans 2004
22997:Sneed & Folk 1958
22987:, pp. 1473–1477.
22948:, pp. 2309–2311.
22661:Math Course Catalogue
22316:
22280:
22185:David Hilbert (1904)
22104:Johann Andreas Segner
22002:Antieigenvalue theory
21924:
21908:
21906:{\displaystyle R_{0}}
21881:
21879:{\displaystyle t_{G}}
21854:
21852:{\displaystyle t_{G}}
21827:
21825:{\displaystyle R_{0}}
21800:
21798:{\displaystyle R_{0}}
21759:
21706:
21653:
21600:
21598:{\displaystyle E_{3}}
21573:
21571:{\displaystyle E_{2}}
21546:
21544:{\displaystyle E_{1}}
21511:
21487:is the secondary and
21482:
21453:
21424:
21376:by their eigenvalues
21371:
21218:self-consistent field
21206:ionization potentials
21163:
21130:
21096:
21062:
21060:{\displaystyle \tau }
21042:
21008:
20959:
20924:
20904:
20869:self-adjoint operator
20859:
20839:
20819:
20795:
20757:
20692:
20650:
20630:
20587:
20554:, interpreted as its
20549:
20525:
20501:differential operator
20490:
20467:
20409:
20365:
20298:
20205:
20142:leads to a so-called
20137:
20068:
20040:
20020:
19993:
19944:
19920:
19900:
19877:
19855:
19811:
19746:
19677:
19657:
19637:
19617:
19571:th diagonal entry is
19566:
19546:
19508:
19506:{\displaystyle v_{i}}
19481:
19451:
19427:
19355:
19321:
19299:, an eigenvalue of a
19297:spectral graph theory
19203:multivariate analysis
19191:positive semidefinite
19145:
19113:
19111:{\displaystyle (1,3)}
19081:
19058:
19013:
18950:
18827:
18768:
18642:
18514:
18444:
18410:
18340:
18270:
18236:
18178:
18108:
18074:
18004:
17934:
17900:
17842:
17693:
17646:
17485:
17401:
17354:
17323:
17265:
17222:
17164:
17102:
17052:
16974:
16920:
16839:
16771:
16707:
16589:
16563:
16531:
16460:
16440:
16414:
16383:
16318:
16298:
16242:
16207:
16181:
16158:
16132:
16112:
16092:
16039:
16008:
15916:
15822:
15708:
15706:{\displaystyle Av=6v}
15674:
15595:
15571:
15547:
15520:
15497:
15451:
15425:
15371:
15194:differential equation
15184:
15089:
15043:characteristic roots
15034:
14958:
14834:
14685:
14525:representation theory
14435:, then the operator (
14225:
14048:
13910:associated with
13883:
13735:
13659:
13566:
13485:
13420:
13368:
13322:differential equation
13249:
13222:
13025:
12965:
12872:
12657:
12521:
12276:
12181:
12068:
11836:
11741:
11639:
11559:
11468:
11371:
11188:
11003:
10891:
10746:
10705:
10685:
10512:
10410:
10352:
10292:
10230:
9820:
9679:is an eigenvector of
9672:
9549:
9259:is an eigenvector of
9254:
9125:
9074:
8880:
8596:), and the values of
8579:
8489:
8487:{\displaystyle \left}
8434:
8413:
8391:
8325:
8152:
8097:
8038:
7983:
7849:is an eigenvector of
7845:Since each column of
7837:
7660:
7640:
7620:
7591:
7562:
7534:
7468:
7448:
7424:
7404:
7378:
7358:
7336:
7297:
7267:
7222:
7202:
7148:
7078:
7049:
7029:
6965:
6941:, the eigenvalues of
6936:
6908:
6844:
6824:
6798:
6746:
6720:
6697:
6650:
6623:
6592:
6577:, or equivalently if
6572:
6570:{\displaystyle A^{*}}
6542:
6522:reciprocal polynomial
6515:
6455:
6425:
6398:
6375:
6319:
6299:
6297:{\displaystyle A^{k}}
6272:
6252:
6228:
6153:
6124:
6096:
6012:
5975:
5943:
5915:
5872:
5829:
5783:
5757:
5728:
5708:
5673:
5653:
5633:
5601:
5572:
5548:
5515:
5495:
5471:
5379:
5346:
5326:
5283:
5256:
5215:distinct eigenvalues
5210:
5184:
5161:
5103:
5083:
5028:
4990:
4970:
4908:
4879:
4850:
4830:
4810:
4748:
4722:
4676:
4637:
4617:
4597:
4555:
4519:
4499:
4449:
4378:
4342:
4282:
4205:
4089:
3974:is a complex number,
3954:distributive property
3868:
3753:
3636:semisimple eigenvalue
3542:
3475:
3369:
2990:
2867:
2796:
2658:
2541:
2302:
2167:
2081:and the scale factor
2046:
1984:
1932:
1828:where, for each row,
1823:
1503:
1448:
1410:
1356:
1225:is an eigenvector of
1212:
1162:Hermann von Helmholtz
1016:Augustin-Louis Cauchy
1009:Joseph-Louis Lagrange
995:In the 18th century,
893:
830:
759:
728:differential operator
704:
691:
656:
486:
442:
408:
386:
361:
335:
312:
295:and a nonzero vector
286:
208:
176:
140:
116:
96:
77:linear transformation
69:characteristic vector
25450:Mathematical physics
25341:Informal mathematics
25221:Mathematical physics
25216:Mathematical finance
25201:Mathematical biology
25140:Diophantine geometry
24642:Gram–Schmidt process
24594:Gaussian elimination
24310:
24233:improve this article
24131:Hill, Roger (2009).
23945:10.1364/OL.32.002309
23535:Historia Mathematica
23475:Van Loan, Charles F.
23432:The Computer Journal
23409:The Computer Journal
23136:Houghton Mifflin Co.
22848:Korn & Korn 2000
22724:, pp. 305, 307.
22382:, pp. 228, 229.
22289:
22269:
22259:Gaussian elimination
22062:Spectrum of a matrix
22022:Eigenvalue algorithm
21890:
21863:
21836:
21809:
21782:
21715:
21662:
21609:
21582:
21555:
21528:
21491:
21462:
21433:
21380:
21321:
21139:
21106:
21071:
21051:
21017:
20995:
20933:
20913:
20875:
20848:
20828:
20808:
20769:
20704:
20666:
20639:
20612:
20600:with a well-defined
20592:within the space of
20569:
20538:
20507:
20499:, is a second-order
20479:
20429:
20416:Schrödinger equation
20398:
20327:
20303:associated with the
20291:Schrödinger equation
20150:
20081:
20073:alone. Furthermore,
20057:
20029:
20002:
19960:
19933:
19909:
19889:
19866:
19820:
19773:
19666:
19646:
19626:
19575:
19555:
19517:
19490:
19460:
19440:
19434:normalized Laplacian
19368:
19338:
19310:
19234:principal components
19122:
19090:
19026:
18971:
18837:
18778:
18652:
18532:
18526:All nonzero vectors
18454:
18420:
18350:
18280:
18246:
18196:
18118:
18084:
18014:
17944:
17910:
17860:
17703:
17656:
17495:
17411:
17364:
17336:
17275:
17232:
17174:
17112:
17069:
16984:
16930:
16849:
16781:
16727:
16576:
16543:
16472:
16449:
16427:
16395:
16347:
16307:
16251:
16216:
16190:
16170:
16141:
16121:
16101:
16048:
16037:{\displaystyle y=2x}
16019:
15931:
15836:
15717:
15685:
15624:
15584:
15560:
15554:Abel–Ruffini theorem
15536:
15525:different products.
15506:
15480:
15434:
15414:
15386:Eigenvalue algorithm
15203:
15101:
15047:
14971:
14850:
14708:
14571:
14562:difference equations
14470:For this reason, in
14431:is an eigenvalue of
14107:
14083:are eigenvectors of
13921:
13904:characteristic space
13823:
13815:Given an eigenvalue
13699:
13633:
13515:
13510:exponential function
13508:. Its solution, the
13429:
13395:
13327:
13231:
13063:
12974:
12911:
12666:
12540:
12324:
12283:which has the roots
12196:
12101:
11884:
11843:which has the roots
11756:
11661:
11582:
11477:
11386:
11382:are complex and are
11197:
11014:
10902:
10766:
10719:
10694:
10527:
10432:
10364:
10303:
10246:
9835:
9740:
9736:Consider the matrix
9710:are eigenvectors of
9574:
9287:
9150:
9082:
8925:
8619:
8526:
8522:Consider the matrix
8443:
8400:
8337:
8314:
8116:
8061:
8012:
7872:
7853:, right multiplying
7759:
7649:
7629:
7600:
7571:
7551:
7477:
7457:
7437:
7413:
7387:
7367:
7347:
7306:
7286:
7237:
7211:
7185:
7087:
7076:{\displaystyle P(A)}
7058:
7038:
6974:
6945:
6917:
6853:
6833:
6807:
6755:
6729:
6709:
6663:
6639:
6612:
6581:
6554:
6531:
6464:
6434:
6414:
6387:
6328:
6308:
6281:
6261:
6241:
6135:
6113:
5954:
5932:
5881:
5838:
5792:
5766:
5746:
5717:
5688:
5662:
5642:
5613:
5581:
5561:
5524:
5504:
5484:
5355:
5335:
5292:
5265:
5219:
5193:
5173:
5112:
5092:
5037:
4999:
4979:
4917:
4888:
4859:
4839:
4819:
4757:
4731:
4685:
4646:
4626:
4606:
4564:
4528:
4508:
4458:
4390:
4351:
4293:
4221:
4142:
4062:
3848:
3818:characteristic space
3683:
3402:
3175:
3163:into the product of
3042:Spectrum of a matrix
2878:
2824:
2685:
2605:
2437:
2424:into the product of
2267:
2207:is the zero vector.
2121:
2008:
1832:
1512:
1476:
1421:
1383:
1245:
1201:Matrix (mathematics)
1045:Pierre-Simon Laplace
942:characteristic space
908:decompose the matrix
863:
772:
734:
620:
541:is adopted from the
459:
429:
395:
370:
348:
321:
299:
267:
197:
187:characteristic value
149:
129:
105:
83:
25356:Mathematics and art
25266:Operations research
25021:Functional analysis
24772:Numerical stability
24652:Multilinear algebra
24627:Inner product space
24477:Linear independence
24245:footnote references
24107:2000JCoAM.123...35G
23976:Weisstein, Eric W.
23937:2007OptL...32.2309V
23873:1958JG.....66..114S
23818:2017RvMP...89a5005R
23750:Press, William H.;
23655:1968mhse.book.....K
23627:1998CG.....24..243K
23511:2000ESPL...25.1473G
23479:Matrix computations
23191:2020PhRvL.125p5901B
23054:, pp. 365–382.
23042:, pp. 103–107.
22999:, pp. 114–150.
22763:, pp. 115–116.
22598:, pp. 265–271.
22574:, p. 1063, p..
22263:formal power series
20376:probability density
20045:is the (imaginary)
19763:natural frequencies
19712:spectral clustering
19353:{\displaystyle D-A}
19210:covariance matrices
16673:Hyperbolic rotation
16648:
16570:conjugate transpose
15502:matrix is a sum of
15175:
15141:
14517:associative algebra
14472:functional analysis
14413:is diagonalizable.
14334:The eigenspaces of
14310:) of an eigenvalue
13819:, consider the set
11630:
10981:
10949:
10641:
8265:is diagonalizable.
6822:{\displaystyle I+A}
6549:conjugate transpose
6369:
6345:
5638:can be formed from
4118:), also called the
2097:eigenvalue equation
1141:a few years later.
1086:orthogonal matrices
663:eigenvalue equation
661:referred to as the
191:characteristic root
25301:Numerical analysis
24910:Mathematical logic
24905:Information theory
24482:Linear combination
24369:Henk van der Vorst
24322:
23861:Journal of Geology
23752:Teukolsky, Saul A.
23641:Korn, Granino A.;
23352:10.1007/BF00178324
23263:Numerical Analysis
23245:Funk & Wagnall
22973:Bender et al. 2020
22850:, Section 14.3.5a.
22812:Denton et al. 2022
22667:2015-11-01 at the
22608:Kublanovskaya 1962
22427:www.mathsisfun.com
22311:
22275:
22032:Jordan normal form
21970:facial recognition
21966:linear combination
21930:
21903:
21876:
21849:
21822:
21795:
21754:
21701:
21648:
21595:
21568:
21541:
21506:
21477:
21448:
21419:
21366:
21237:Roothaan equations
21198:molecular orbitals
21172:Molecular orbitals
21158:
21125:
21091:
21057:
21037:
21003:
20954:
20919:
20899:
20854:
20834:
20814:
20790:
20752:
20687:
20645:
20625:
20582:
20544:
20520:
20485:
20462:
20404:
20392:
20360:
20319:as well as of the
20200:
20132:
20063:
20035:
20015:
19988:
19939:
19915:
19895:
19872:
19850:
19806:
19759:degrees of freedom
19749:
19739:Vibration analysis
19672:
19652:
19632:
19612:
19561:
19541:
19503:
19476:
19446:
19422:
19350:
19334:, which is either
19316:
19273:hypothesis testing
19238:variance explained
19226:correlation matrix
19184:eigendecomposition
19164:
19156:standard deviation
19140:
19108:
19053:
19008:
18962:quadratic equation
18945:
18943:
18935:
18883:
18822:
18816:
18763:
18761:
18753:
18701:
18637:
18635:
18627:
18578:
18509:
18507:
18439:
18405:
18403:
18335:
18333:
18265:
18231:
18173:
18171:
18103:
18069:
18067:
17999:
17997:
17929:
17895:
17837:
17835:
17688:
17641:
17639:
17480:
17478:
17396:
17349:
17318:
17260:
17217:
17159:
17097:
17047:
17041:
16969:
16963:
16915:
16909:
16834:
16828:
16766:
16760:
16709:
16646:
16584:
16558:
16526:
16455:
16435:
16409:
16378:
16313:
16293:
16279:
16237:
16202:
16176:
16153:
16127:
16107:
16087:
16073:
16034:
16003:
15998:
15911:
15906:
15817:
15811:
15776:
15750:
15703:
15669:
15663:
15590:
15566:
15542:
15530:algebraic formulas
15518:{\displaystyle n!}
15515:
15492:
15446:
15420:
15366:
15179:
15161:
15127:
15084:
15029:
15023:
14953:
14829:
14680:
14393:, then a basis of
14367:invariant subspace
14220:
14218:
14043:
14041:
13878:
13730:
13654:
13594:General definition
13561:
13480:
13415:
13413:
13363:
13247:{\textstyle M_{j}}
13244:
13217:
13179:
13114:
13020:
13006:
12960:
12946:
12867:
12811:
12652:
12643:
12516:
12507:
12448:
12386:
12271:
12176:
12167:
12063:
12054:
11995:
11936:
11831:
11736:
11727:
11634:
11607:
11554:
11540:
11463:
11449:
11366:
11357:
11294:
11244:
11183:
11174:
11111:
11061:
10998:
10967:
10935:
10886:
10877:
10835:
10799:
10741:
10700:
10680:
10678:
10627:
10523:, whose roots are
10507:
10498:
10405:
10391:
10347:
10333:
10287:
10273:
10225:
10223:
10090:
10001:
9932:
9815:
9806:
9690:Thus, the vectors
9667:
9661:
9632:
9544:
9542:
9442:
9413:
9373:
9249:
9243:
9211:
9120:
9069:
9063:
9034:
8994:
8875:
8873:
8787:
8731:
8689:
8574:
8565:
8520:
8484:
8478:
8477:
8408:
8386:
8320:
8286:Jordan normal form
8278:Jordan normal form
8165:eigendecomposition
8147:
8092:
8033:
7978:
7969:
7832:
7823:
7655:
7635:
7615:
7586:
7557:
7529:
7463:
7443:
7419:
7399:
7373:
7353:
7331:
7292:
7262:
7217:
7197:
7143:
7073:
7044:
7024:
6960:
6931:
6903:
6839:
6819:
6793:
6741:
6715:
6692:
6645:
6618:
6587:
6567:
6537:
6510:
6450:
6420:
6393:
6370:
6355:
6331:
6314:
6294:
6267:
6247:
6223:
6119:
6091:
5938:
5910:
5867:
5824:
5778:
5752:
5723:
5703:
5668:
5648:
5628:
5596:
5567:
5543:
5510:
5490:
5466:
5464:
5341:
5321:
5278:
5251:
5205:
5179:
5156:
5098:
5078:
5033:contains a factor
5023:
4985:
4965:
4903:
4874:
4855:. In other words,
4845:
4825:
4805:
4743:
4717:
4671:
4632:
4612:
4592:
4550:
4514:
4494:
4444:
4373:
4337:
4277:
4200:
4084:
3892:belong to the set
3863:
3748:
3669:to be all vectors
3537:
3535:
3364:
3027:complex conjugates
3004:irrational numbers
2985:
2976:
2925:
2862:
2791:
2754:
2653:
2644:
2536:
2297:
2162:
2041:
1991:If it occurs that
1979:
1818:
1812:
1748:
1687:
1498:
1443:
1405:
1367:of each other, or
1351:
1342:
1298:
1286:
1231:
1185:Vera Kublanovskaya
1181:John G. F. Francis
1150:integral operators
1139:Poisson's equation
1131:Laplace's equation
1110:defective matrices
1082:Francesco Brioschi
1075:Hermitian matrices
1067:symmetric matrices
888:
825:
754:
752:
708:
699:
651:
587:facial recognition
579:vibration analysis
575:stability analysis
481:
437:
403:
381:
356:
333:{\displaystyle n.}
330:
307:
281:
203:
171:
135:
111:
91:
25427:
25426:
25026:Harmonic analysis
24813:
24812:
24680:Geometric algebra
24637:Kronecker product
24472:Linear projection
24457:Vector projection
24273:
24272:
24265:
24133:"λ – Eigenvalues"
23931:(16): 2309–2311.
23783:978-0-387-72828-5
23723:978-0-89871-454-8
23579:Topics In Algebra
23505:(13): 1473–1477,
23488:978-0-8018-5414-9
23312:10.1090/bull/1722
22975:, p. 165901.
22681:Press et al. 2007
22447:Press et al. 2007
22278:{\displaystyle n}
22047:Normal eigenvalue
21952:covariance matrix
21300:
21299:
21292:
21222:quantum chemistry
21210:Koopmans' theorem
21186:molecular physics
21178:quantum mechanics
20989:disordered system
20922:{\displaystyle H}
20857:{\displaystyle H}
20837:{\displaystyle E}
20817:{\displaystyle H}
20648:{\displaystyle H}
20594:square integrable
20547:{\displaystyle E}
20488:{\displaystyle H}
20420:quantum mechanics
20407:{\displaystyle T}
20114:
20096:
20066:{\displaystyle k}
20047:angular frequency
19942:{\displaystyle k}
19918:{\displaystyle m}
19898:{\displaystyle n}
19875:{\displaystyle x}
19835:
19788:
19675:{\displaystyle k}
19655:{\displaystyle k}
19635:{\displaystyle k}
19610:
19564:{\displaystyle i}
19449:{\displaystyle D}
19319:{\displaystyle A}
19222:covariance matrix
19152:covariance matrix
18958:
18957:
17237:
17074:
16619:Lanczos algorithm
16524:
16458:{\displaystyle A}
16316:{\displaystyle b}
16289:
16179:{\displaystyle A}
16130:{\displaystyle A}
16110:{\displaystyle a}
16083:
15602:numerical methods
15593:{\displaystyle n}
15569:{\displaystyle n}
15545:{\displaystyle n}
15423:{\displaystyle A}
15342:
15303:
15238:
14902:
14893:
14556:Dynamic equations
13760:, and the scalar
13754:
13753:
13583:) is a constant.
13445:
13412:
13212:
13164:
13105:
13016:
12956:
12382:
12087:triangular matrix
11654:diagonal matrices
11564:with eigenvalues
11550:
11459:
10703:{\displaystyle i}
10674:
10670:
10656:
10604:
10600:
10586:
10401:
10343:
10283:
9683:corresponding to
9263:corresponding to
8377:
8350:
8323:{\displaystyle H}
8229:. Each column of
7658:{\displaystyle A}
7638:{\displaystyle A}
7611:
7582:
7560:{\displaystyle A}
7522:
7502:
7488:
7446:{\displaystyle A}
7422:{\displaystyle u}
7376:{\displaystyle u}
7295:{\displaystyle A}
7220:{\displaystyle A}
7171:right eigenvector
7047:{\displaystyle P}
6842:{\displaystyle I}
6718:{\displaystyle A}
6648:{\displaystyle A}
6630:positive-definite
6621:{\displaystyle A}
6590:{\displaystyle A}
6540:{\displaystyle A}
6508:
6482:
6423:{\displaystyle A}
6396:{\displaystyle A}
6317:{\displaystyle k}
6270:{\displaystyle A}
6250:{\displaystyle k}
6122:{\displaystyle A}
5941:{\displaystyle A}
5755:{\displaystyle A}
5726:{\displaystyle A}
5671:{\displaystyle A}
5651:{\displaystyle n}
5570:{\displaystyle A}
5513:{\displaystyle A}
5493:{\displaystyle A}
5344:{\displaystyle A}
5182:{\displaystyle A}
4988:{\displaystyle D}
4848:{\displaystyle V}
4828:{\displaystyle I}
4635:{\displaystyle V}
4615:{\displaystyle A}
4517:{\displaystyle V}
3578:simple eigenvalue
3137:Suppose a matrix
3134:that polynomial.
3020:algebraic numbers
2560:
2559:
2321:
2320:
2186:
2185:
2065:
2064:
1441:
1297:
1169:Richard von Mises
1115:In the meantime,
959:of the domain of
788:
751:
245:quantum mechanics
114:{\displaystyle T}
25467:
25440:Abstract algebra
25415:
25414:
25403:
25402:
25391:
25390:
25380:
25379:
25311:Computer algebra
25286:Computer science
25006:Complex analysis
24840:
24833:
24826:
24817:
24816:
24803:
24802:
24685:Exterior algebra
24622:Hadamard product
24539:
24527:Linear equations
24398:
24391:
24384:
24375:
24374:
24340:
24331:
24329:
24328:
24323:
24268:
24261:
24257:
24254:
24248:
24216:
24215:
24208:
24198:
24180:
24162:
24160:
24148:
24127:
24118:
24092:
24073:
24071:
24046:
24044:
24032:
24030:
24028:
24012:
24010:
24008:
23992:
23990:
23988:
23972:
23915:
23913:
23900:
23891:
23855:
23837:
23811:
23786:
23768:
23746:
23726:
23708:
23687:
23667:
23643:Korn, Theresa M.
23637:
23609:
23591:
23573:
23562:
23551:
23550:
23529:
23491:
23466:
23448:
23447:
23426:
23425:
23402:
23380:
23363:
23330:
23328:
23305:
23287:
23277:
23266:
23255:
23235:
23226:
23184:
23159:
23148:
23133:
23122:
23100:
23079:
23073:
23067:
23061:
23055:
23049:
23043:
23037:
23031:
23030:
23018:
23012:
23006:
23000:
22994:
22988:
22982:
22976:
22970:
22964:
22963:, p. 15005.
22958:
22949:
22943:
22937:
22931:
22918:
22912:
22906:
22900:
22894:
22881:
22875:
22869:
22863:
22857:
22851:
22845:
22839:
22836:Van Mieghem 2024
22833:
22827:
22824:Van Mieghem 2014
22821:
22815:
22809:
22800:
22794:
22788:
22782:
22776:
22770:
22764:
22758:
22752:
22746:
22740:
22734:
22725:
22719:
22713:
22707:
22696:
22690:
22684:
22678:
22672:
22656:
22650:
22641:
22635:
22629:
22623:
22617:
22611:
22605:
22599:
22593:
22587:
22581:
22575:
22569:
22563:
22557:
22551:
22545:
22539:
22533:
22524:
22518:
22512:
22506:
22493:
22487:
22481:
22475:
22462:
22456:
22450:
22444:
22438:
22437:
22435:
22433:
22419:
22410:
22404:
22398:
22392:
22383:
22377:
22368:
22367:
22365:
22354:
22348:
22342:
22326:
22320:
22318:
22317:
22312:
22307:
22306:
22284:
22282:
22281:
22276:
22255:
22249:
22227:
22221:
22211:
22201:
22180:
22174:
22149:
22143:
22091:, pp. 176–227.
22078:
21978:data compression
21940:image processing
21912:
21910:
21909:
21904:
21902:
21901:
21885:
21883:
21882:
21877:
21875:
21874:
21858:
21856:
21855:
21850:
21848:
21847:
21831:
21829:
21828:
21823:
21821:
21820:
21804:
21802:
21801:
21796:
21794:
21793:
21763:
21761:
21760:
21755:
21753:
21752:
21740:
21739:
21727:
21726:
21710:
21708:
21707:
21702:
21700:
21699:
21687:
21686:
21674:
21673:
21657:
21655:
21654:
21649:
21647:
21646:
21634:
21633:
21621:
21620:
21604:
21602:
21601:
21596:
21594:
21593:
21577:
21575:
21574:
21569:
21567:
21566:
21550:
21548:
21547:
21542:
21540:
21539:
21515:
21513:
21512:
21507:
21505:
21504:
21499:
21486:
21484:
21483:
21478:
21476:
21475:
21470:
21457:
21455:
21454:
21449:
21447:
21446:
21441:
21428:
21426:
21425:
21420:
21418:
21417:
21405:
21404:
21392:
21391:
21375:
21373:
21372:
21367:
21365:
21364:
21359:
21350:
21349:
21344:
21335:
21334:
21329:
21295:
21288:
21284:
21281:
21275:
21255:
21254:
21247:
21167:
21165:
21164:
21159:
21151:
21150:
21134:
21132:
21131:
21126:
21118:
21117:
21100:
21098:
21097:
21092:
21090:
21085:
21084:
21079:
21066:
21064:
21063:
21058:
21046:
21044:
21043:
21038:
21036:
21031:
21030:
21025:
21012:
21010:
21009:
21004:
21002:
20963:
20961:
20960:
20955:
20950:
20949:
20940:
20928:
20926:
20925:
20920:
20908:
20906:
20905:
20900:
20895:
20894:
20885:
20863:
20861:
20860:
20855:
20843:
20841:
20840:
20835:
20823:
20821:
20820:
20815:
20799:
20797:
20796:
20791:
20786:
20785:
20776:
20761:
20759:
20758:
20753:
20748:
20747:
20738:
20724:
20723:
20714:
20696:
20694:
20693:
20688:
20683:
20682:
20673:
20660:bra–ket notation
20654:
20652:
20651:
20646:
20634:
20632:
20631:
20626:
20624:
20623:
20591:
20589:
20588:
20583:
20581:
20580:
20553:
20551:
20550:
20545:
20529:
20527:
20526:
20521:
20519:
20518:
20494:
20492:
20491:
20486:
20471:
20469:
20468:
20463:
20460:
20459:
20444:
20443:
20413:
20411:
20410:
20405:
20372:angular momentum
20369:
20367:
20366:
20361:
20209:
20207:
20206:
20201:
20190:
20186:
20167:
20166:
20141:
20139:
20138:
20133:
20116:
20115:
20107:
20098:
20097:
20089:
20075:damped vibration
20072:
20070:
20069:
20064:
20049:. The principal
20044:
20042:
20041:
20036:
20024:
20022:
20021:
20016:
20014:
20013:
19997:
19995:
19994:
19989:
19981:
19980:
19951:stiffness matrix
19948:
19946:
19945:
19940:
19924:
19922:
19921:
19916:
19904:
19902:
19901:
19896:
19881:
19879:
19878:
19873:
19859:
19857:
19856:
19851:
19837:
19836:
19828:
19815:
19813:
19812:
19807:
19790:
19789:
19781:
19767:eigenfrequencies
19699:adjacency matrix
19681:
19679:
19678:
19673:
19661:
19659:
19658:
19653:
19641:
19639:
19638:
19633:
19621:
19619:
19618:
19613:
19611:
19606:
19605:
19587:
19585:
19570:
19568:
19567:
19562:
19550:
19548:
19547:
19542:
19540:
19539:
19535:
19512:
19510:
19509:
19504:
19502:
19501:
19485:
19483:
19482:
19477:
19475:
19474:
19455:
19453:
19452:
19447:
19431:
19429:
19428:
19423:
19421:
19420:
19416:
19397:
19396:
19392:
19359:
19357:
19356:
19351:
19328:Laplacian matrix
19325:
19323:
19322:
19317:
19305:adjacency matrix
19242:orthogonal basis
19218:linear relations
19199:orthogonal basis
19149:
19147:
19146:
19141:
19117:
19115:
19114:
19109:
19062:
19060:
19059:
19054:
19021:
19017:
19015:
19014:
19009:
19007:
19006:
18954:
18952:
18951:
18946:
18944:
18940:
18939:
18904:
18903:
18898:
18888:
18887:
18855:
18854:
18849:
18831:
18829:
18828:
18823:
18821:
18820:
18792:
18791:
18786:
18772:
18770:
18769:
18764:
18762:
18758:
18757:
18722:
18721:
18716:
18706:
18705:
18670:
18669:
18664:
18646:
18644:
18643:
18638:
18636:
18632:
18631:
18599:
18598:
18593:
18583:
18582:
18550:
18549:
18544:
18518:
18516:
18515:
18510:
18508:
18494:
18493:
18470:
18469:
18448:
18446:
18445:
18440:
18432:
18431:
18414:
18412:
18411:
18406:
18404:
18390:
18389:
18366:
18365:
18344:
18342:
18341:
18336:
18334:
18320:
18319:
18296:
18295:
18274:
18272:
18271:
18266:
18258:
18257:
18240:
18238:
18237:
18232:
18227:
18226:
18208:
18207:
18190:
18182:
18180:
18179:
18174:
18172:
18158:
18157:
18134:
18133:
18112:
18110:
18109:
18104:
18096:
18095:
18078:
18076:
18075:
18070:
18068:
18054:
18053:
18030:
18029:
18008:
18006:
18005:
18000:
17998:
17984:
17983:
17960:
17959:
17938:
17936:
17935:
17930:
17922:
17921:
17904:
17902:
17901:
17896:
17891:
17890:
17872:
17871:
17854:
17846:
17844:
17843:
17838:
17836:
17805:
17801:
17800:
17781:
17780:
17740:
17736:
17735:
17719:
17718:
17697:
17695:
17694:
17689:
17681:
17680:
17668:
17667:
17650:
17648:
17647:
17642:
17640:
17606:
17602:
17601:
17579:
17578:
17535:
17531:
17530:
17511:
17510:
17489:
17487:
17486:
17481:
17479:
17475:
17474:
17458:
17457:
17444:
17443:
17427:
17426:
17405:
17403:
17402:
17397:
17389:
17388:
17376:
17375:
17358:
17356:
17355:
17350:
17348:
17347:
17327:
17325:
17324:
17319:
17287:
17286:
17269:
17267:
17266:
17261:
17259:
17258:
17235:
17226:
17224:
17223:
17218:
17186:
17185:
17168:
17166:
17165:
17160:
17155:
17154:
17133:
17132:
17106:
17104:
17103:
17098:
17096:
17095:
17072:
17056:
17054:
17053:
17048:
17046:
17045:
16978:
16976:
16975:
16970:
16968:
16967:
16924:
16922:
16921:
16916:
16914:
16913:
16843:
16841:
16840:
16835:
16833:
16832:
16825:
16824:
16801:
16800:
16775:
16773:
16772:
16767:
16765:
16764:
16716:
16700:
16693:
16686:
16668:Horizontal shear
16659:Unequal scaling
16649:
16645:
16623:iterative method
16593:
16591:
16590:
16585:
16583:
16567:
16565:
16564:
16559:
16557:
16556:
16551:
16535:
16533:
16532:
16527:
16525:
16523:
16522:
16517:
16516:
16511:
16504:
16503:
16495:
16494:
16489:
16482:
16464:
16462:
16461:
16456:
16444:
16442:
16441:
16436:
16434:
16420:
16418:
16416:
16415:
16410:
16408:
16389:
16387:
16385:
16384:
16379:
16377:
16376:
16322:
16320:
16319:
16314:
16302:
16300:
16299:
16294:
16292:
16291:
16290:
16284:
16283:
16246:
16244:
16243:
16238:
16211:
16209:
16208:
16203:
16185:
16183:
16182:
16177:
16162:
16160:
16159:
16154:
16137:with eigenvalue
16136:
16134:
16133:
16128:
16116:
16114:
16113:
16108:
16096:
16094:
16093:
16088:
16086:
16085:
16084:
16078:
16077:
16043:
16041:
16040:
16035:
16012:
16010:
16009:
16004:
16002:
15999:
15927:
15923:
15920:
15918:
15917:
15912:
15910:
15907:
15831:linear equations
15826:
15824:
15823:
15818:
15816:
15815:
15781:
15780:
15755:
15754:
15712:
15710:
15709:
15704:
15678:
15676:
15675:
15670:
15668:
15667:
15599:
15597:
15596:
15591:
15578:companion matrix
15575:
15573:
15572:
15567:
15551:
15549:
15548:
15543:
15524:
15522:
15521:
15516:
15501:
15499:
15498:
15493:
15466:round-off errors
15455:
15453:
15452:
15447:
15429:
15427:
15426:
15421:
15394:Classical method
15375:
15373:
15372:
15367:
15356:
15355:
15343:
15341:
15333:
15325:
15323:
15322:
15304:
15302:
15301:
15300:
15281:
15277:
15276:
15260:
15258:
15257:
15239:
15237:
15236:
15235:
15222:
15218:
15217:
15207:
15188:
15186:
15185:
15180:
15174:
15169:
15160:
15159:
15140:
15135:
15126:
15125:
15113:
15112:
15093:
15091:
15090:
15085:
15080:
15079:
15059:
15058:
15038:
15036:
15035:
15030:
15028:
15027:
15020:
15019:
14991:
14990:
14962:
14960:
14959:
14954:
14949:
14948:
14924:
14923:
14900:
14891:
14887:
14886:
14868:
14867:
14838:
14836:
14835:
14830:
14819:
14818:
14803:
14802:
14778:
14777:
14762:
14761:
14749:
14748:
14733:
14732:
14720:
14719:
14689:
14687:
14686:
14681:
14676:
14675:
14660:
14659:
14641:
14640:
14625:
14624:
14612:
14611:
14596:
14595:
14583:
14582:
14546:Hecke eigensheaf
14279:associated with
14248:associated with
14229:
14227:
14226:
14221:
14219:
14209:
14185:
14161:
14153:
14132:
14124:
14075:. Therefore, if
14052:
14050:
14049:
14044:
14042:
14032:
14008:
13984:
13967:
13946:
13938:
13887:
13885:
13884:
13879:
13874:
13870:
13869:
13855:
13841:
13748:
13739:
13737:
13736:
13731:
13726:
13712:
13693:
13663:
13661:
13660:
13655:
13570:
13568:
13567:
13562:
13557:
13556:
13489:
13487:
13486:
13481:
13446:
13444:
13433:
13424:
13422:
13421:
13416:
13414:
13411:
13400:
13372:
13370:
13369:
13364:
13253:
13251:
13250:
13245:
13243:
13242:
13226:
13224:
13223:
13218:
13213:
13211:
13210:
13206:
13205:
13193:
13192:
13178:
13162:
13161:
13154:
13153:
13141:
13140:
13128:
13127:
13113:
13103:
13098:
13097:
13092:
13086:
13085:
13070:
13049:Hermitian matrix
13029:
13027:
13026:
13021:
13019:
13018:
13017:
13011:
13010:
12969:
12967:
12966:
12961:
12959:
12958:
12957:
12951:
12950:
12876:
12874:
12873:
12868:
12863:
12862:
12841:
12840:
12816:
12815:
12661:
12659:
12658:
12653:
12648:
12647:
12525:
12523:
12522:
12517:
12512:
12511:
12476:
12475:
12474:
12473:
12463:
12453:
12452:
12414:
12413:
12412:
12411:
12401:
12391:
12390:
12383:
12375:
12345:
12344:
12343:
12342:
12332:
12312:
12302:
12292:
12280:
12278:
12277:
12272:
12185:
12183:
12182:
12177:
12172:
12171:
12072:
12070:
12069:
12064:
12059:
12058:
12023:
12022:
12021:
12020:
12010:
12000:
11999:
11964:
11963:
11962:
11961:
11951:
11941:
11940:
11905:
11904:
11903:
11902:
11892:
11872:
11862:
11852:
11840:
11838:
11837:
11832:
11745:
11743:
11742:
11737:
11732:
11731:
11643:
11641:
11640:
11635:
11629:
11624:
11623:
11622:
11612:
11603:
11602:
11601:
11600:
11590:
11563:
11561:
11560:
11555:
11553:
11552:
11551:
11545:
11544:
11537:
11536:
11525:
11524:
11498:
11497:
11496:
11495:
11485:
11472:
11470:
11469:
11464:
11462:
11461:
11460:
11454:
11453:
11446:
11445:
11434:
11433:
11407:
11406:
11405:
11404:
11394:
11375:
11373:
11372:
11367:
11362:
11361:
11354:
11353:
11340:
11339:
11312:
11311:
11299:
11298:
11284:
11283:
11270:
11269:
11249:
11248:
11241:
11240:
11227:
11226:
11192:
11190:
11189:
11184:
11179:
11178:
11171:
11170:
11157:
11156:
11129:
11128:
11116:
11115:
11101:
11100:
11087:
11086:
11066:
11065:
11058:
11057:
11044:
11043:
11007:
11005:
11004:
10999:
10994:
10993:
10980:
10975:
10962:
10961:
10948:
10943:
10924:
10923:
10914:
10913:
10895:
10893:
10892:
10887:
10882:
10881:
10840:
10839:
10804:
10803:
10752:
10750:
10748:
10747:
10742:
10731:
10730:
10709:
10707:
10706:
10701:
10689:
10687:
10686:
10681:
10679:
10675:
10666:
10665:
10657:
10649:
10640:
10635:
10619:
10618:
10605:
10596:
10595:
10587:
10579:
10567:
10566:
10543:
10542:
10516:
10514:
10513:
10508:
10503:
10502:
10416:
10414:
10412:
10411:
10406:
10404:
10403:
10402:
10396:
10395:
10358:
10356:
10354:
10353:
10348:
10346:
10345:
10344:
10338:
10337:
10298:
10296:
10294:
10293:
10288:
10286:
10285:
10284:
10278:
10277:
10234:
10232:
10231:
10226:
10224:
10205:
10204:
10189:
10188:
10173:
10172:
10130:
10129:
10102:
10095:
10094:
10011:
10007:
10006:
10005:
9937:
9936:
9824:
9822:
9821:
9816:
9811:
9810:
9728:, respectively.
9727:
9720:
9676:
9674:
9673:
9668:
9666:
9665:
9637:
9636:
9629:
9628:
9615:
9614:
9594:
9593:
9582:
9553:
9551:
9550:
9545:
9543:
9529:
9528:
9513:
9512:
9483:
9482:
9467:
9466:
9447:
9446:
9418:
9417:
9410:
9409:
9396:
9395:
9378:
9377:
9329:
9328:
9317:
9276:
9258:
9256:
9255:
9250:
9248:
9247:
9216:
9215:
9208:
9207:
9191:
9190:
9170:
9169:
9158:
9129:
9127:
9126:
9121:
9113:
9112:
9097:
9096:
9078:
9076:
9075:
9070:
9068:
9067:
9039:
9038:
9031:
9030:
9017:
9016:
8999:
8998:
8960:
8959:
8948:
8914:
8900:
8893:
8884:
8882:
8881:
8876:
8874:
8831:
8827:
8826:
8796:
8792:
8791:
8741:
8737:
8736:
8735:
8694:
8693:
8583:
8581:
8580:
8575:
8570:
8569:
8493:
8491:
8490:
8485:
8483:
8479:
8417:
8415:
8414:
8409:
8407:
8395:
8393:
8392:
8387:
8385:
8380:
8379:
8378:
8372:
8366:
8361:
8353:
8352:
8351:
8345:
8329:
8327:
8326:
8321:
8228:
8171:. Such a matrix
8156:
8154:
8153:
8148:
8131:
8130:
8101:
8099:
8098:
8093:
8088:
8087:
8042:
8040:
8039:
8034:
7987:
7985:
7984:
7979:
7974:
7973:
7966:
7965:
7960:
7954:
7953:
7937:
7936:
7931:
7925:
7924:
7913:
7912:
7907:
7901:
7900:
7841:
7839:
7838:
7833:
7828:
7827:
7820:
7819:
7814:
7801:
7800:
7795:
7787:
7786:
7781:
7664:
7662:
7661:
7656:
7644:
7642:
7641:
7636:
7624:
7622:
7621:
7616:
7614:
7613:
7612:
7595:
7593:
7592:
7587:
7585:
7584:
7583:
7566:
7564:
7563:
7558:
7538:
7536:
7535:
7530:
7525:
7524:
7523:
7517:
7505:
7504:
7503:
7497:
7491:
7490:
7489:
7472:
7470:
7469:
7464:
7452:
7450:
7449:
7444:
7431:left eigenvector
7428:
7426:
7425:
7420:
7408:
7406:
7405:
7400:
7382:
7380:
7379:
7374:
7363:is a scalar and
7362:
7360:
7359:
7354:
7340:
7338:
7337:
7332:
7327:
7313:
7301:
7299:
7298:
7293:
7282:multiply matrix
7271:
7269:
7268:
7263:
7258:
7247:
7226:
7224:
7223:
7218:
7206:
7204:
7203:
7198:
7152:
7150:
7149:
7144:
7136:
7135:
7108:
7107:
7082:
7080:
7079:
7074:
7053:
7051:
7050:
7045:
7033:
7031:
7030:
7025:
7014:
7013:
6989:
6988:
6969:
6967:
6966:
6961:
6940:
6938:
6937:
6932:
6930:
6912:
6910:
6909:
6904:
6893:
6892:
6868:
6867:
6848:
6846:
6845:
6840:
6828:
6826:
6825:
6820:
6802:
6800:
6799:
6794:
6789:
6788:
6770:
6769:
6750:
6748:
6747:
6742:
6724:
6722:
6721:
6716:
6701:
6699:
6698:
6693:
6685:
6680:
6679:
6670:
6654:
6652:
6651:
6646:
6627:
6625:
6624:
6619:
6596:
6594:
6593:
6588:
6576:
6574:
6573:
6568:
6566:
6565:
6547:is equal to its
6546:
6544:
6543:
6538:
6519:
6517:
6516:
6511:
6509:
6507:
6506:
6494:
6483:
6481:
6480:
6468:
6459:
6457:
6456:
6451:
6449:
6448:
6429:
6427:
6426:
6421:
6402:
6400:
6399:
6394:
6379:
6377:
6376:
6371:
6368:
6363:
6344:
6339:
6323:
6321:
6320:
6315:
6303:
6301:
6300:
6295:
6293:
6292:
6276:
6274:
6273:
6268:
6256:
6254:
6253:
6248:
6232:
6230:
6229:
6224:
6219:
6218:
6206:
6205:
6196:
6195:
6183:
6182:
6172:
6167:
6128:
6126:
6125:
6120:
6100:
6098:
6097:
6092:
6087:
6086:
6068:
6067:
6055:
6054:
6042:
6041:
6031:
6026:
6008:
6007:
5994:
5989:
5947:
5945:
5944:
5939:
5919:
5917:
5916:
5911:
5906:
5905:
5893:
5892:
5876:
5874:
5873:
5868:
5863:
5862:
5850:
5849:
5833:
5831:
5830:
5825:
5823:
5822:
5804:
5803:
5787:
5785:
5784:
5779:
5762:be an arbitrary
5761:
5759:
5758:
5753:
5732:
5730:
5729:
5724:
5712:
5710:
5709:
5704:
5702:
5701:
5696:
5677:
5675:
5674:
5669:
5657:
5655:
5654:
5649:
5637:
5635:
5634:
5629:
5627:
5626:
5621:
5605:
5603:
5602:
5597:
5595:
5594:
5589:
5576:
5574:
5573:
5568:
5552:
5550:
5549:
5544:
5536:
5535:
5519:
5517:
5516:
5511:
5499:
5497:
5496:
5491:
5475:
5473:
5472:
5467:
5465:
5452:
5451:
5422:
5421:
5409:
5408:
5398:
5393:
5371:
5370:
5350:
5348:
5347:
5342:
5330:
5328:
5327:
5322:
5317:
5316:
5304:
5303:
5287:
5285:
5284:
5279:
5277:
5276:
5260:
5258:
5257:
5252:
5250:
5249:
5231:
5230:
5214:
5212:
5211:
5206:
5188:
5186:
5185:
5180:
5165:
5163:
5162:
5157:
5146:
5145:
5124:
5123:
5107:
5105:
5104:
5099:
5087:
5085:
5084:
5079:
5077:
5076:
5066:
5065:
5032:
5030:
5029:
5024:
4994:
4992:
4991:
4986:
4974:
4972:
4971:
4966:
4912:
4910:
4909:
4904:
4883:
4881:
4880:
4875:
4854:
4852:
4851:
4846:
4834:
4832:
4831:
4826:
4814:
4812:
4811:
4806:
4752:
4750:
4749:
4744:
4726:
4724:
4723:
4718:
4716:
4715:
4705:
4704:
4680:
4678:
4677:
4672:
4664:
4663:
4641:
4639:
4638:
4633:
4621:
4619:
4618:
4613:
4601:
4599:
4598:
4593:
4582:
4581:
4559:
4557:
4556:
4551:
4540:
4539:
4523:
4521:
4520:
4515:
4503:
4501:
4500:
4495:
4493:
4492:
4487:
4475:
4474:
4469:
4453:
4451:
4450:
4445:
4443:
4442:
4432:
4431:
4421:
4404:
4403:
4398:
4382:
4380:
4379:
4374:
4363:
4362:
4346:
4344:
4343:
4338:
4327:
4326:
4305:
4304:
4286:
4284:
4283:
4278:
4261:
4260:
4239:
4238:
4209:
4207:
4206:
4201:
4154:
4153:
4093:
4091:
4090:
4085:
4074:
4073:
4047:associated with
4036:associated with
4012:
3988:or equivalently
3987:
3969:
3951:
3925:or equivalently
3924:
3909:
3872:
3870:
3869:
3864:
3862:
3861:
3856:
3824:associated with
3792:associated with
3776:associated with
3757:
3755:
3754:
3749:
3744:
3740:
3739:
3731:
3726:
3722:
3701:
3634:is said to be a
3576:is said to be a
3546:
3544:
3543:
3538:
3536:
3523:
3519:
3518:
3505:
3504:
3494:
3489:
3467:
3466:
3441:
3440:
3428:
3427:
3373:
3371:
3370:
3365:
3360:
3359:
3355:
3354:
3342:
3341:
3321:
3320:
3305:
3304:
3300:
3299:
3287:
3286:
3266:
3265:
3253:
3252:
3248:
3247:
3235:
3234:
3214:
3213:
3012:rational numbers
2994:
2992:
2991:
2986:
2981:
2980:
2952:
2951:
2940:
2930:
2929:
2898:
2897:
2886:
2873:
2871:
2869:
2868:
2863:
2861:
2853:
2848:
2844:
2819:in the equation
2810:
2806:
2800:
2798:
2797:
2792:
2787:
2786:
2759:
2758:
2676:
2662:
2660:
2659:
2654:
2649:
2648:
2554:
2545:
2543:
2542:
2537:
2523:
2522:
2498:
2497:
2476:
2475:
2431:
2391:secular equation
2385:) is called the
2315:
2306:
2304:
2303:
2298:
2261:
2249:
2199:
2195:
2191:
2180:
2171:
2169:
2168:
2163:
2158:
2150:
2145:
2141:
2115:
2102:
2084:
2080:
2072:
2059:
2050:
2048:
2047:
2042:
2037:
2026:
2018:
2002:
1998:
1994:
1988:
1986:
1985:
1980:
1975:
1974:
1965:
1964:
1951:
1946:
1928:
1927:
1918:
1917:
1896:
1895:
1886:
1885:
1870:
1869:
1860:
1859:
1844:
1843:
1827:
1825:
1824:
1819:
1817:
1816:
1809:
1808:
1788:
1787:
1774:
1773:
1753:
1752:
1745:
1744:
1724:
1723:
1710:
1709:
1692:
1691:
1684:
1683:
1664:
1663:
1649:
1648:
1610:
1609:
1590:
1589:
1578:
1577:
1564:
1563:
1544:
1543:
1532:
1531:
1507:
1505:
1504:
1499:
1494:
1486:
1471:
1467:
1463:
1459:
1452:
1450:
1449:
1444:
1442:
1434:
1414:
1412:
1411:
1406:
1401:
1390:
1378:
1365:scalar multiples
1360:
1358:
1357:
1352:
1347:
1346:
1305:
1299:
1295:
1291:
1290:
1252:
1240:
1236:
1197:Euclidean vector
1117:Joseph Liouville
1106:stability theory
1102:Karl Weierstrass
1020:quadric surfaces
910:—for example by
897:
895:
894:
889:
884:
873:
834:
832:
831:
826:
821:
820:
802:
801:
789:
787:
776:
763:
761:
760:
755:
753:
750:
739:
660:
658:
657:
652:
647:
633:
490:
488:
487:
482:
480:
469:
454:
450:
446:
444:
443:
438:
436:
420:
416:
412:
410:
409:
404:
402:
391:) simply scales
390:
388:
387:
382:
380:
365:
363:
362:
357:
355:
343:
339:
337:
336:
331:
316:
314:
313:
308:
306:
294:
290:
288:
287:
282:
277:
212:
210:
209:
204:
180:
178:
177:
172:
170:
159:
144:
142:
141:
136:
120:
118:
117:
112:
100:
98:
97:
92:
90:
63:
58:
57:
54:
53:
50:
47:
44:
25475:
25474:
25470:
25469:
25468:
25466:
25465:
25464:
25430:
25429:
25428:
25423:
25374:
25365:
25315:
25272:
25251:Systems science
25182:
25178:Homotopy theory
25144:
25111:
25063:
25035:
24982:
24929:
24900:Category theory
24886:
24851:
24844:
24814:
24809:
24791:
24753:
24709:
24646:
24598:
24540:
24531:
24497:Change of basis
24487:Multilinear map
24425:
24407:
24402:
24352:
24311:
24308:
24307:
24269:
24258:
24252:
24249:
24230:
24221:This article's
24217:
24213:
24206:
24201:
24196:
24178:
24158:
24130:
24090:
24081:
24079:Further reading
24076:
24042:
24026:
24024:
24022:Quanta Magazine
24006:
24004:
23986:
23984:
23853:
23784:
23766:
23724:
23705:
23704:978-007139880-0
23665:
23607:
23589:
23489:
23464:
23429:
23400:
23326:
23285:
23275:
23146:
23120:
23087:
23082:
23074:
23070:
23062:
23058:
23050:
23046:
23038:
23034:
23019:
23015:
23007:
23003:
22995:
22991:
22983:
22979:
22971:
22967:
22959:
22952:
22944:
22940:
22932:
22921:
22913:
22909:
22901:
22897:
22886:, p. 107;
22882:
22878:
22870:
22866:
22858:
22854:
22846:
22842:
22834:
22830:
22822:
22818:
22810:
22803:
22795:
22791:
22783:
22779:
22771:
22767:
22759:
22755:
22747:
22743:
22735:
22728:
22720:
22716:
22708:
22699:
22691:
22687:
22679:
22675:
22669:Wayback Machine
22657:
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22642:
22638:
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22626:
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22590:
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22578:
22570:
22566:
22558:
22554:
22546:
22542:
22534:
22527:
22519:
22515:
22507:
22496:
22488:
22484:
22476:
22465:
22457:
22453:
22445:
22441:
22431:
22429:
22421:
22420:
22413:
22407:Betteridge 1965
22405:
22401:
22393:
22386:
22378:
22371:
22363:
22355:
22351:
22343:
22339:
22335:
22330:
22329:
22302:
22298:
22290:
22287:
22286:
22270:
22267:
22266:
22256:
22252:
22228:
22224:
22181:
22177:
22150:
22146:
22079:
22075:
22070:
21998:
21936:
21919:
21897:
21893:
21891:
21888:
21887:
21870:
21866:
21864:
21861:
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21843:
21839:
21837:
21834:
21833:
21816:
21812:
21810:
21807:
21806:
21789:
21785:
21783:
21780:
21779:
21776:
21770:
21748:
21744:
21735:
21731:
21722:
21718:
21716:
21713:
21712:
21695:
21691:
21682:
21678:
21669:
21665:
21663:
21660:
21659:
21642:
21638:
21629:
21625:
21616:
21612:
21610:
21607:
21606:
21589:
21585:
21583:
21580:
21579:
21562:
21558:
21556:
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21535:
21531:
21529:
21526:
21525:
21500:
21495:
21494:
21492:
21489:
21488:
21471:
21466:
21465:
21463:
21460:
21459:
21442:
21437:
21436:
21434:
21431:
21430:
21413:
21409:
21400:
21396:
21387:
21383:
21381:
21378:
21377:
21360:
21355:
21354:
21345:
21340:
21339:
21330:
21325:
21324:
21322:
21319:
21318:
21296:
21285:
21279:
21276:
21268:help improve it
21265:
21256:
21252:
21245:
21174:
21146:
21142:
21140:
21137:
21136:
21113:
21109:
21107:
21104:
21103:
21086:
21080:
21075:
21074:
21072:
21069:
21068:
21052:
21049:
21048:
21032:
21026:
21021:
21020:
21018:
21015:
21014:
20998:
20996:
20993:
20992:
20970:
20945:
20941:
20936:
20934:
20931:
20930:
20914:
20911:
20910:
20890:
20886:
20881:
20876:
20873:
20872:
20849:
20846:
20845:
20829:
20826:
20825:
20809:
20806:
20805:
20781:
20777:
20772:
20770:
20767:
20766:
20743:
20739:
20734:
20719:
20715:
20710:
20705:
20702:
20701:
20678:
20674:
20669:
20667:
20664:
20663:
20640:
20637:
20636:
20619:
20615:
20613:
20610:
20609:
20576:
20572:
20570:
20567:
20566:
20539:
20536:
20535:
20514:
20510:
20508:
20505:
20504:
20480:
20477:
20476:
20455:
20451:
20439:
20435:
20430:
20427:
20426:
20399:
20396:
20395:
20378:for a position
20328:
20325:
20324:
20293:
20273:solid mechanics
20269:
20233:
20162:
20158:
20157:
20153:
20151:
20148:
20147:
20106:
20105:
20088:
20087:
20082:
20079:
20078:
20058:
20055:
20054:
20051:vibration modes
20030:
20027:
20026:
20009:
20005:
20003:
20000:
19999:
19976:
19972:
19961:
19958:
19957:
19934:
19931:
19930:
19910:
19907:
19906:
19890:
19887:
19886:
19867:
19864:
19863:
19827:
19826:
19821:
19818:
19817:
19780:
19779:
19774:
19771:
19770:
19755:
19741:
19720:
19667:
19664:
19663:
19647:
19644:
19643:
19627:
19624:
19623:
19601:
19597:
19586:
19581:
19576:
19573:
19572:
19556:
19553:
19552:
19531:
19524:
19520:
19518:
19515:
19514:
19497:
19493:
19491:
19488:
19487:
19467:
19463:
19461:
19458:
19457:
19441:
19438:
19437:
19412:
19405:
19401:
19388:
19381:
19377:
19369:
19366:
19365:
19339:
19336:
19335:
19311:
19308:
19307:
19293:
19281:factor analysis
19230:sample variance
19180:
19178:Factor analysis
19170:
19123:
19120:
19119:
19091:
19088:
19087:
19076:
19068:squeeze mapping
19027:
19024:
19023:
19019:
19002:
18998:
18972:
18969:
18968:
18942:
18941:
18934:
18933:
18924:
18923:
18913:
18912:
18905:
18899:
18894:
18893:
18890:
18889:
18882:
18881:
18875:
18874:
18864:
18863:
18856:
18850:
18845:
18844:
18840:
18838:
18835:
18834:
18815:
18814:
18808:
18807:
18797:
18796:
18787:
18782:
18781:
18779:
18776:
18775:
18760:
18759:
18752:
18751:
18742:
18741:
18731:
18730:
18723:
18717:
18712:
18711:
18708:
18707:
18700:
18699:
18690:
18689:
18679:
18678:
18671:
18665:
18660:
18659:
18655:
18653:
18650:
18649:
18634:
18633:
18626:
18625:
18619:
18618:
18608:
18607:
18600:
18594:
18589:
18588:
18585:
18584:
18577:
18576:
18570:
18569:
18559:
18558:
18551:
18545:
18540:
18539:
18535:
18533:
18530:
18529:
18506:
18505:
18495:
18489:
18485:
18482:
18481:
18471:
18465:
18461:
18457:
18455:
18452:
18451:
18427:
18423:
18421:
18418:
18417:
18402:
18401:
18391:
18385:
18381:
18378:
18377:
18367:
18361:
18357:
18353:
18351:
18348:
18347:
18332:
18331:
18321:
18315:
18311:
18308:
18307:
18297:
18291:
18287:
18283:
18281:
18278:
18277:
18253:
18249:
18247:
18244:
18243:
18222:
18218:
18203:
18199:
18197:
18194:
18193:
18192:
18170:
18169:
18159:
18153:
18149:
18146:
18145:
18135:
18129:
18125:
18121:
18119:
18116:
18115:
18091:
18087:
18085:
18082:
18081:
18066:
18065:
18055:
18049:
18045:
18042:
18041:
18031:
18025:
18021:
18017:
18015:
18012:
18011:
17996:
17995:
17985:
17979:
17975:
17972:
17971:
17961:
17955:
17951:
17947:
17945:
17942:
17941:
17917:
17913:
17911:
17908:
17907:
17886:
17882:
17867:
17863:
17861:
17858:
17857:
17856:
17834:
17833:
17803:
17802:
17793:
17789:
17782:
17776:
17772:
17769:
17768:
17738:
17737:
17731:
17727:
17720:
17714:
17710:
17706:
17704:
17701:
17700:
17676:
17672:
17663:
17659:
17657:
17654:
17653:
17638:
17637:
17604:
17603:
17591:
17587:
17580:
17574:
17570:
17567:
17566:
17533:
17532:
17523:
17519:
17512:
17506:
17502:
17498:
17496:
17493:
17492:
17477:
17476:
17470:
17466:
17459:
17453:
17449:
17446:
17445:
17439:
17435:
17428:
17422:
17418:
17414:
17412:
17409:
17408:
17384:
17380:
17371:
17367:
17365:
17362:
17361:
17343:
17339:
17337:
17334:
17333:
17282:
17278:
17276:
17273:
17272:
17254:
17250:
17233:
17230:
17229:
17181:
17177:
17175:
17172:
17171:
17150:
17146:
17128:
17124:
17113:
17110:
17109:
17091:
17087:
17070:
17067:
17066:
17062:
17040:
17039:
17028:
17016:
17015:
17004:
16988:
16987:
16985:
16982:
16981:
16962:
16961:
16956:
16950:
16949:
16944:
16934:
16933:
16931:
16928:
16927:
16908:
16907:
16896:
16884:
16883:
16869:
16853:
16852:
16850:
16847:
16846:
16827:
16826:
16820:
16816:
16814:
16808:
16807:
16802:
16796:
16792:
16785:
16784:
16782:
16779:
16778:
16759:
16758:
16753:
16747:
16746:
16741:
16731:
16730:
16728:
16725:
16724:
16640:
16635:
16615:sparse matrices
16600:
16579:
16577:
16574:
16573:
16552:
16547:
16546:
16544:
16541:
16540:
16518:
16512:
16507:
16506:
16505:
16499:
16490:
16485:
16484:
16483:
16481:
16473:
16470:
16469:
16450:
16447:
16446:
16430:
16428:
16425:
16424:
16404:
16396:
16393:
16392:
16391:
16369:
16365:
16348:
16345:
16344:
16343:
16335:
16333:Power iteration
16329:
16308:
16305:
16304:
16286:
16285:
16278:
16277:
16266:
16256:
16255:
16254:
16252:
16249:
16248:
16217:
16214:
16213:
16191:
16188:
16187:
16171:
16168:
16167:
16142:
16139:
16138:
16122:
16119:
16118:
16102:
16099:
16098:
16080:
16079:
16072:
16071:
16063:
16053:
16052:
16051:
16049:
16046:
16045:
16020:
16017:
16016:
15997:
15996:
15986:
15968:
15967:
15957:
15938:
15934:
15932:
15929:
15928:
15925:
15921:
15905:
15904:
15891:
15873:
15872:
15859:
15843:
15839:
15837:
15834:
15833:
15810:
15809:
15803:
15802:
15792:
15791:
15775:
15774:
15768:
15767:
15757:
15756:
15749:
15748:
15743:
15737:
15736:
15731:
15721:
15720:
15718:
15715:
15714:
15686:
15683:
15682:
15662:
15661:
15656:
15650:
15649:
15644:
15634:
15633:
15625:
15622:
15621:
15614:
15585:
15582:
15581:
15561:
15558:
15557:
15537:
15534:
15533:
15507:
15504:
15503:
15481:
15478:
15477:
15476:, which for an
15435:
15432:
15431:
15415:
15412:
15411:
15408:
15396:
15388:
15382:
15351:
15347:
15334:
15326:
15324:
15318:
15314:
15290:
15286:
15282:
15266:
15262:
15261:
15259:
15247:
15243:
15231:
15227:
15223:
15213:
15209:
15208:
15206:
15204:
15201:
15200:
15170:
15165:
15155:
15151:
15136:
15131:
15121:
15117:
15108:
15104:
15102:
15099:
15098:
15075:
15071:
15054:
15050:
15048:
15045:
15044:
15022:
15021:
15003:
14999:
14997:
14992:
14986:
14982:
14975:
14974:
14972:
14969:
14968:
14932:
14928:
14907:
14903:
14876:
14872:
14857:
14853:
14851:
14848:
14847:
14814:
14810:
14792:
14788:
14767:
14763:
14757:
14753:
14738:
14734:
14728:
14724:
14715:
14711:
14709:
14706:
14705:
14665:
14661:
14655:
14651:
14630:
14626:
14620:
14616:
14601:
14597:
14591:
14587:
14578:
14574:
14572:
14569:
14568:
14558:
14509:
14503:
14425:
14423:Spectral theory
14419:
14417:Spectral theory
14326:
14305:
14217:
14216:
14205:
14189:
14181:
14169:
14168:
14157:
14149:
14136:
14128:
14120:
14110:
14108:
14105:
14104:
14040:
14039:
14028:
14012:
14004:
13992:
13991:
13980:
13963:
13950:
13942:
13934:
13924:
13922:
13919:
13918:
13865:
13851:
13837:
13836:
13832:
13824:
13821:
13820:
13813:
13722:
13708:
13700:
13697:
13696:
13634:
13631:
13630:
13596:
13549:
13545:
13516:
13513:
13512:
13437:
13432:
13430:
13427:
13426:
13404:
13398:
13396:
13393:
13392:
13389:
13328:
13325:
13324:
13298:function spaces
13278:
13272:
13238:
13234:
13232:
13229:
13228:
13201:
13197:
13188:
13184:
13180:
13168:
13163:
13149:
13145:
13136:
13132:
13123:
13119:
13115:
13109:
13104:
13102:
13093:
13088:
13087:
13075:
13071:
13066:
13064:
13061:
13060:
13045:
13038:
13013:
13012:
13005:
13004:
12999:
12994:
12989:
12979:
12978:
12977:
12975:
12972:
12971:
12953:
12952:
12945:
12944:
12939:
12931:
12926:
12916:
12915:
12914:
12912:
12909:
12908:
12891:
12858:
12854:
12836:
12832:
12810:
12809:
12798:
12793:
12788:
12782:
12781:
12776:
12765:
12760:
12754:
12753:
12748:
12743:
12732:
12726:
12725:
12720:
12715:
12710:
12694:
12693:
12667:
12664:
12663:
12642:
12641:
12636:
12631:
12626:
12620:
12619:
12614:
12609:
12604:
12598:
12597:
12592:
12587:
12582:
12576:
12575:
12570:
12565:
12560:
12550:
12549:
12541:
12538:
12537:
12534:
12506:
12505:
12499:
12498:
12492:
12491:
12481:
12480:
12469:
12465:
12464:
12459:
12458:
12447:
12446:
12437:
12436:
12430:
12429:
12419:
12418:
12407:
12403:
12402:
12397:
12396:
12385:
12384:
12374:
12371:
12370:
12361:
12360:
12350:
12349:
12338:
12334:
12333:
12328:
12327:
12325:
12322:
12321:
12310:
12304:
12300:
12294:
12290:
12284:
12197:
12194:
12193:
12166:
12165:
12160:
12155:
12149:
12148:
12143:
12138:
12132:
12131:
12126:
12121:
12111:
12110:
12102:
12099:
12098:
12081:
12053:
12052:
12046:
12045:
12039:
12038:
12028:
12027:
12016:
12012:
12011:
12006:
12005:
11994:
11993:
11987:
11986:
11980:
11979:
11969:
11968:
11957:
11953:
11952:
11947:
11946:
11935:
11934:
11928:
11927:
11921:
11920:
11910:
11909:
11898:
11894:
11893:
11888:
11887:
11885:
11882:
11881:
11870:
11864:
11860:
11854:
11850:
11844:
11757:
11754:
11753:
11726:
11725:
11720:
11715:
11709:
11708:
11703:
11698:
11692:
11691:
11686:
11681:
11671:
11670:
11662:
11659:
11658:
11649:
11625:
11618:
11614:
11613:
11608:
11596:
11592:
11591:
11586:
11585:
11583:
11580:
11579:
11577:
11570:
11547:
11546:
11539:
11538:
11532:
11528:
11526:
11520:
11516:
11514:
11504:
11503:
11502:
11491:
11487:
11486:
11481:
11480:
11478:
11475:
11474:
11456:
11455:
11448:
11447:
11441:
11437:
11435:
11429:
11425:
11423:
11413:
11412:
11411:
11400:
11396:
11395:
11390:
11389:
11387:
11384:
11383:
11356:
11355:
11349:
11345:
11342:
11341:
11335:
11331:
11328:
11327:
11317:
11316:
11307:
11303:
11293:
11292:
11286:
11285:
11279:
11275:
11272:
11271:
11265:
11261:
11254:
11253:
11243:
11242:
11236:
11232:
11229:
11228:
11222:
11218:
11215:
11214:
11204:
11203:
11198:
11195:
11194:
11173:
11172:
11166:
11162:
11159:
11158:
11152:
11148:
11145:
11144:
11134:
11133:
11124:
11120:
11110:
11109:
11103:
11102:
11096:
11092:
11089:
11088:
11082:
11078:
11071:
11070:
11060:
11059:
11053:
11049:
11046:
11045:
11039:
11035:
11032:
11031:
11021:
11020:
11015:
11012:
11011:
10989:
10985:
10976:
10971:
10957:
10953:
10944:
10939:
10919:
10915:
10909:
10905:
10903:
10900:
10899:
10876:
10875:
10869:
10868:
10862:
10861:
10851:
10850:
10834:
10833:
10827:
10826:
10820:
10819:
10809:
10808:
10798:
10797:
10791:
10790:
10784:
10783:
10773:
10772:
10767:
10764:
10763:
10761:
10726:
10722:
10720:
10717:
10716:
10715:
10695:
10692:
10691:
10677:
10676:
10664:
10648:
10636:
10631:
10620:
10614:
10610:
10607:
10606:
10594:
10578:
10568:
10562:
10558:
10555:
10554:
10544:
10538:
10534:
10530:
10528:
10525:
10524:
10497:
10496:
10491:
10486:
10480:
10479:
10474:
10469:
10463:
10462:
10457:
10452:
10442:
10441:
10433:
10430:
10429:
10423:
10398:
10397:
10390:
10389:
10384:
10379:
10369:
10368:
10367:
10365:
10362:
10361:
10360:
10340:
10339:
10332:
10331:
10326:
10318:
10308:
10307:
10306:
10304:
10301:
10300:
10299:
10280:
10279:
10272:
10271:
10266:
10261:
10251:
10250:
10249:
10247:
10244:
10243:
10242:
10222:
10221:
10200:
10196:
10184:
10180:
10168:
10167:
10125:
10124:
10100:
10099:
10089:
10088:
10077:
10072:
10066:
10065:
10060:
10049:
10043:
10042:
10037:
10032:
10016:
10015:
10000:
9999:
9994:
9989:
9983:
9982:
9977:
9972:
9966:
9965:
9960:
9955:
9945:
9944:
9931:
9930:
9925:
9920:
9914:
9913:
9908:
9903:
9897:
9896:
9891:
9886:
9876:
9875:
9874:
9870:
9863:
9838:
9836:
9833:
9832:
9805:
9804:
9799:
9794:
9788:
9787:
9782:
9777:
9771:
9770:
9765:
9760:
9750:
9749:
9741:
9738:
9737:
9734:
9722:
9715:
9709:
9699:
9660:
9659:
9653:
9652:
9642:
9641:
9631:
9630:
9624:
9620:
9617:
9616:
9610:
9606:
9599:
9598:
9583:
9578:
9577:
9575:
9572:
9571:
9569:
9562:
9541:
9540:
9530:
9524:
9520:
9508:
9504:
9498:
9497:
9484:
9478:
9474:
9462:
9458:
9449:
9448:
9441:
9440:
9434:
9433:
9423:
9422:
9412:
9411:
9405:
9401:
9398:
9397:
9391:
9387:
9380:
9379:
9372:
9371:
9363:
9357:
9356:
9351:
9338:
9337:
9330:
9318:
9313:
9312:
9290:
9288:
9285:
9284:
9271:
9242:
9241:
9232:
9231:
9221:
9220:
9210:
9209:
9203:
9199:
9193:
9192:
9186:
9182:
9175:
9174:
9159:
9154:
9153:
9151:
9148:
9147:
9145:
9138:
9108:
9104:
9092:
9088:
9083:
9080:
9079:
9062:
9061:
9055:
9054:
9044:
9043:
9033:
9032:
9026:
9022:
9019:
9018:
9012:
9008:
9001:
9000:
8993:
8992:
8987:
8981:
8980:
8975:
8965:
8964:
8949:
8944:
8943:
8926:
8923:
8922:
8909:
8895:
8888:
8872:
8871:
8829:
8828:
8822:
8818:
8794:
8793:
8786:
8785:
8774:
8768:
8767:
8762:
8746:
8745:
8730:
8729:
8724:
8718:
8717:
8712:
8702:
8701:
8688:
8687:
8682:
8676:
8675:
8670:
8660:
8659:
8658:
8654:
8647:
8622:
8620:
8617:
8616:
8564:
8563:
8558:
8552:
8551:
8546:
8536:
8535:
8527:
8524:
8523:
8513:
8503:
8476:
8475:
8470:
8464:
8463:
8458:
8450:
8446:
8444:
8441:
8440:
8429:
8424:
8422:Matrix examples
8403:
8401:
8398:
8397:
8381:
8374:
8373:
8368:
8367:
8362:
8357:
8347:
8346:
8341:
8340:
8338:
8335:
8334:
8315:
8312:
8311:
8304:
8302:Min-max theorem
8298:
8220:
8123:
8119:
8117:
8114:
8113:
8080:
8076:
8062:
8059:
8058:
8013:
8010:
8009:
7996:
7968:
7967:
7961:
7956:
7955:
7949:
7945:
7943:
7938:
7932:
7927:
7926:
7920:
7916:
7914:
7908:
7903:
7902:
7896:
7892:
7885:
7884:
7873:
7870:
7869:
7822:
7821:
7815:
7810:
7809:
7807:
7802:
7796:
7791:
7790:
7788:
7782:
7777:
7776:
7769:
7768:
7760:
7757:
7756:
7736:
7727:
7720:
7713:
7704:
7697:
7677:
7671:
7650:
7647:
7646:
7630:
7627:
7626:
7608:
7607:
7603:
7601:
7598:
7597:
7579:
7578:
7574:
7572:
7569:
7568:
7552:
7549:
7548:
7519:
7518:
7513:
7512:
7499:
7498:
7493:
7492:
7485:
7484:
7480:
7478:
7475:
7474:
7458:
7455:
7454:
7438:
7435:
7434:
7414:
7411:
7410:
7388:
7385:
7384:
7368:
7365:
7364:
7348:
7345:
7344:
7323:
7309:
7307:
7304:
7303:
7287:
7284:
7283:
7254:
7243:
7238:
7235:
7234:
7212:
7209:
7208:
7186:
7183:
7182:
7181:multiplies the
7166:
7160:
7131:
7127:
7103:
7099:
7088:
7085:
7084:
7059:
7056:
7055:
7039:
7036:
7035:
7009:
7005:
6984:
6980:
6975:
6972:
6971:
6946:
6943:
6942:
6926:
6918:
6915:
6914:
6913:. Moreover, if
6888:
6884:
6863:
6859:
6854:
6851:
6850:
6834:
6831:
6830:
6808:
6805:
6804:
6784:
6780:
6765:
6761:
6756:
6753:
6752:
6730:
6727:
6726:
6710:
6707:
6706:
6681:
6675:
6671:
6666:
6664:
6661:
6660:
6640:
6637:
6636:
6613:
6610:
6609:
6582:
6579:
6578:
6561:
6557:
6555:
6552:
6551:
6532:
6529:
6528:
6502:
6498:
6493:
6476:
6472:
6467:
6465:
6462:
6461:
6441:
6437:
6435:
6432:
6431:
6415:
6412:
6411:
6388:
6385:
6384:
6364:
6359:
6340:
6335:
6329:
6326:
6325:
6309:
6306:
6305:
6288:
6284:
6282:
6279:
6278:
6262:
6259:
6258:
6242:
6239:
6238:
6214:
6210:
6201:
6197:
6191:
6187:
6178:
6174:
6168:
6157:
6136:
6133:
6132:
6114:
6111:
6110:
6082:
6078:
6063:
6059:
6050:
6046:
6037:
6033:
6027:
6016:
6000:
5996:
5990:
5979:
5955:
5952:
5951:
5933:
5930:
5929:
5901:
5897:
5888:
5884:
5882:
5879:
5878:
5858:
5854:
5845:
5841:
5839:
5836:
5835:
5818:
5814:
5799:
5795:
5793:
5790:
5789:
5767:
5764:
5763:
5747:
5744:
5743:
5740:
5718:
5715:
5714:
5697:
5692:
5691:
5689:
5686:
5685:
5663:
5660:
5659:
5643:
5640:
5639:
5622:
5617:
5616:
5614:
5611:
5610:
5590:
5585:
5584:
5582:
5579:
5578:
5562:
5559:
5558:
5531:
5527:
5525:
5522:
5521:
5505:
5502:
5501:
5485:
5482:
5481:
5463:
5462:
5447:
5443:
5436:
5430:
5429:
5417:
5413:
5404:
5400:
5394:
5383:
5372:
5366:
5362:
5358:
5356:
5353:
5352:
5336:
5333:
5332:
5312:
5308:
5299:
5295:
5293:
5290:
5289:
5272:
5268:
5266:
5263:
5262:
5245:
5241:
5226:
5222:
5220:
5217:
5216:
5194:
5191:
5190:
5174:
5171:
5170:
5141:
5137:
5119:
5115:
5113:
5110:
5109:
5093:
5090:
5089:
5061:
5057:
5056:
5052:
5038:
5035:
5034:
5000:
4997:
4996:
4995:, we know that
4980:
4977:
4976:
4918:
4915:
4914:
4889:
4886:
4885:
4860:
4857:
4856:
4840:
4837:
4836:
4820:
4817:
4816:
4758:
4755:
4754:
4732:
4729:
4728:
4700:
4696:
4695:
4691:
4686:
4683:
4682:
4659:
4655:
4647:
4644:
4643:
4627:
4624:
4623:
4607:
4604:
4603:
4577:
4573:
4565:
4562:
4561:
4535:
4531:
4529:
4526:
4525:
4509:
4506:
4505:
4488:
4483:
4482:
4470:
4465:
4464:
4459:
4456:
4455:
4427:
4423:
4422:
4417:
4416:
4399:
4394:
4393:
4391:
4388:
4387:
4358:
4354:
4352:
4349:
4348:
4322:
4318:
4300:
4296:
4294:
4291:
4290:
4256:
4252:
4234:
4230:
4222:
4219:
4218:
4149:
4145:
4143:
4140:
4139:
4069:
4065:
4063:
4060:
4059:
3989:
3975:
3961:
3926:
3911:
3897:
3857:
3852:
3851:
3849:
3846:
3845:
3838:linear subspace
3735:
3727:
3709:
3705:
3697:
3696:
3692:
3684:
3681:
3680:
3644:
3633:
3624:
3615:
3606:
3597:
3588:
3575:
3566:
3557:
3534:
3533:
3514:
3510:
3506:
3500:
3496:
3490:
3479:
3468:
3462:
3458:
3455:
3454:
3436:
3432:
3423:
3419:
3412:
3405:
3403:
3400:
3399:
3350:
3346:
3337:
3333:
3332:
3328:
3316:
3312:
3295:
3291:
3282:
3278:
3277:
3273:
3261:
3257:
3243:
3239:
3230:
3226:
3225:
3221:
3209:
3205:
3176:
3173:
3172:
3129:
3108:
3099:
3075:
3065:
3059:of the matrix.
3057:spectral radius
3044:
2975:
2974:
2968:
2967:
2957:
2956:
2941:
2936:
2935:
2924:
2923:
2914:
2913:
2903:
2902:
2887:
2882:
2881:
2879:
2876:
2875:
2857:
2849:
2831:
2827:
2825:
2822:
2821:
2820:
2808:
2804:
2782:
2778:
2753:
2752:
2741:
2735:
2734:
2729:
2713:
2712:
2686:
2683:
2682:
2666:
2643:
2642:
2637:
2631:
2630:
2625:
2615:
2614:
2606:
2603:
2602:
2593:
2584:
2577:
2570:
2518:
2514:
2493:
2489:
2471:
2467:
2438:
2435:
2434:
2268:
2265:
2264:
2239:
2219:
2213:
2201:identity matrix
2197:
2193:
2189:
2154:
2146:
2128:
2124:
2122:
2119:
2118:
2100:
2099:for the matrix
2082:
2078:
2068:
2033:
2022:
2014:
2009:
2006:
2005:
1996:
1992:
1970:
1966:
1957:
1953:
1947:
1936:
1923:
1919:
1910:
1906:
1891:
1887:
1878:
1874:
1865:
1861:
1852:
1848:
1839:
1835:
1833:
1830:
1829:
1811:
1810:
1804:
1800:
1797:
1796:
1790:
1789:
1783:
1779:
1776:
1775:
1769:
1765:
1758:
1757:
1747:
1746:
1740:
1736:
1733:
1732:
1726:
1725:
1719:
1715:
1712:
1711:
1705:
1701:
1694:
1693:
1686:
1685:
1676:
1672:
1670:
1665:
1656:
1652:
1650:
1641:
1637:
1634:
1633:
1628:
1623:
1618:
1612:
1611:
1602:
1598:
1596:
1591:
1585:
1581:
1579:
1573:
1569:
1566:
1565:
1556:
1552:
1550:
1545:
1539:
1535:
1533:
1527:
1523:
1516:
1515:
1513:
1510:
1509:
1490:
1482:
1477:
1474:
1473:
1469:
1465:
1461:
1457:
1433:
1422:
1419:
1418:
1397:
1386:
1384:
1381:
1380:
1376:
1341:
1340:
1331:
1330:
1324:
1323:
1310:
1309:
1301:
1293:
1285:
1284:
1278:
1277:
1268:
1267:
1257:
1256:
1248:
1246:
1243:
1242:
1238:
1234:
1203:
1193:
1071:Charles Hermite
986:quadratic forms
974:
880:
869:
864:
861:
860:
813:
809:
794:
790:
780:
775:
773:
770:
769:
743:
737:
735:
732:
731:
677:. For example,
643:
629:
621:
618:
617:
583:atomic orbitals
533:
503:square matrices
476:
465:
460:
457:
456:
452:
448:
432:
430:
427:
426:
418:
414:
413:by a factor of
398:
396:
393:
392:
376:
371:
368:
367:
351:
349:
346:
345:
341:
340:If multiplying
322:
319:
318:
302:
300:
297:
296:
292:
273:
268:
265:
264:
261:
255:of the system.
198:
195:
194:
166:
155:
150:
147:
146:
130:
127:
126:
106:
103:
102:
86:
84:
81:
80:
61:
41:
37:
24:
17:
12:
11:
5:
25473:
25463:
25462:
25457:
25452:
25447:
25445:Linear algebra
25442:
25425:
25424:
25422:
25421:
25409:
25397:
25385:
25370:
25367:
25366:
25364:
25363:
25358:
25353:
25348:
25343:
25338:
25337:
25336:
25329:Mathematicians
25325:
25323:
25321:Related topics
25317:
25316:
25314:
25313:
25308:
25303:
25298:
25293:
25288:
25282:
25280:
25274:
25273:
25271:
25270:
25269:
25268:
25263:
25258:
25256:Control theory
25248:
25243:
25238:
25233:
25228:
25223:
25218:
25213:
25208:
25203:
25198:
25192:
25190:
25184:
25183:
25181:
25180:
25175:
25170:
25165:
25160:
25154:
25152:
25146:
25145:
25143:
25142:
25137:
25132:
25127:
25121:
25119:
25113:
25112:
25110:
25109:
25104:
25099:
25094:
25089:
25084:
25079:
25073:
25071:
25065:
25064:
25062:
25061:
25056:
25051:
25045:
25043:
25037:
25036:
25034:
25033:
25031:Measure theory
25028:
25023:
25018:
25013:
25008:
25003:
24998:
24992:
24990:
24984:
24983:
24981:
24980:
24975:
24970:
24965:
24960:
24955:
24950:
24945:
24939:
24937:
24931:
24930:
24928:
24927:
24922:
24917:
24912:
24907:
24902:
24896:
24894:
24888:
24887:
24885:
24884:
24879:
24874:
24873:
24872:
24867:
24856:
24853:
24852:
24843:
24842:
24835:
24828:
24820:
24811:
24810:
24808:
24807:
24796:
24793:
24792:
24790:
24789:
24784:
24779:
24774:
24769:
24767:Floating-point
24763:
24761:
24755:
24754:
24752:
24751:
24749:Tensor product
24746:
24741:
24736:
24734:Function space
24731:
24726:
24720:
24718:
24711:
24710:
24708:
24707:
24702:
24697:
24692:
24687:
24682:
24677:
24672:
24670:Triple product
24667:
24662:
24656:
24654:
24648:
24647:
24645:
24644:
24639:
24634:
24629:
24624:
24619:
24614:
24608:
24606:
24600:
24599:
24597:
24596:
24591:
24586:
24584:Transformation
24581:
24576:
24574:Multiplication
24571:
24566:
24561:
24556:
24550:
24548:
24542:
24541:
24534:
24532:
24530:
24529:
24524:
24519:
24514:
24509:
24504:
24499:
24494:
24489:
24484:
24479:
24474:
24469:
24464:
24459:
24454:
24449:
24444:
24439:
24433:
24431:
24430:Basic concepts
24427:
24426:
24424:
24423:
24418:
24412:
24409:
24408:
24405:Linear algebra
24401:
24400:
24393:
24386:
24378:
24372:
24371:
24358:
24351:
24348:
24334:
24333:
24321:
24318:
24315:
24301:
24292:
24286:
24280:
24271:
24270:
24225:external links
24220:
24218:
24211:
24205:
24204:External links
24202:
24200:
24199:
24194:
24181:
24176:
24163:
24149:
24128:
24101:(1–2): 35–65,
24082:
24080:
24077:
24075:
24074:
24047:
24033:
24013:
23993:
23973:
23925:Optics Letters
23916:
23901:
23892:
23881:10.1086/626490
23867:(2): 114–150,
23856:
23851:
23843:Linear algebra
23838:
23787:
23782:
23769:
23765:978-0521880688
23764:
23747:
23727:
23722:
23709:
23703:
23688:
23679:(3): 637–657,
23668:
23663:
23638:
23610:
23605:
23592:
23588:978-1114541016
23587:
23574:
23569:Linear Algebra
23563:
23552:
23530:
23492:
23487:
23471:Golub, Gene H.
23467:
23462:
23454:Linear algebra
23449:
23438:(4): 332–345,
23427:
23416:(3): 265–271,
23403:
23398:
23390:Addison-Wesley
23381:
23346:(4): 365–382,
23331:
23278:
23273:
23256:
23236:
23227:
23175:(16): 165901.
23160:
23149:
23144:
23123:
23118:
23101:
23088:
23086:
23083:
23081:
23080:
23068:
23056:
23044:
23032:
23013:
23011:, p. 243.
23001:
22989:
22977:
22965:
22950:
22938:
22919:
22907:
22905:, p. 111.
22895:
22890:, p. 109
22876:
22864:
22852:
22840:
22828:
22816:
22801:
22797:Wolchover 2019
22789:
22787:, p. 116.
22777:
22775:, p. 290.
22765:
22753:
22751:, p. 272.
22741:
22739:, p. 307.
22726:
22714:
22712:, p. 316.
22697:
22695:, p. 358.
22685:
22673:
22651:
22636:
22624:
22612:
22600:
22588:
22576:
22564:
22562:, pp. 706–707.
22552:
22550:, pp. 715–716.
22540:
22538:, pp. 807–808.
22525:
22513:
22494:
22482:
22480:, p. 107.
22463:
22451:
22449:, p. 536.
22439:
22411:
22399:
22384:
22369:
22349:
22347:, p. 401.
22336:
22334:
22331:
22328:
22327:
22310:
22305:
22301:
22297:
22294:
22274:
22250:
22222:
22220:
22219:
22213:
22204:eigenfunctions
22175:
22156:Comptes rendus
22144:
22142:
22141:
22123:
22100:
22072:
22071:
22069:
22066:
22065:
22064:
22059:
22057:Singular value
22054:
22049:
22044:
22039:
22034:
22029:
22027:Quantum states
22024:
22019:
22014:
22009:
22004:
21997:
21994:
21982:identification
21932:Main article:
21918:
21915:
21900:
21896:
21873:
21869:
21846:
21842:
21819:
21815:
21792:
21788:
21772:Main article:
21769:
21766:
21751:
21747:
21743:
21738:
21734:
21730:
21725:
21721:
21698:
21694:
21690:
21685:
21681:
21677:
21672:
21668:
21645:
21641:
21637:
21632:
21628:
21624:
21619:
21615:
21592:
21588:
21565:
21561:
21538:
21534:
21503:
21498:
21474:
21469:
21445:
21440:
21416:
21412:
21408:
21403:
21399:
21395:
21390:
21386:
21363:
21358:
21353:
21348:
21343:
21338:
21333:
21328:
21298:
21297:
21259:
21257:
21250:
21244:
21241:
21173:
21170:
21157:
21154:
21149:
21145:
21124:
21121:
21116:
21112:
21089:
21083:
21078:
21056:
21035:
21029:
21024:
21001:
20977:acoustic waves
20969:
20968:Wave transport
20966:
20953:
20948:
20944:
20939:
20918:
20898:
20893:
20889:
20884:
20880:
20853:
20833:
20813:
20789:
20784:
20780:
20775:
20763:
20762:
20751:
20746:
20742:
20737:
20733:
20730:
20727:
20722:
20718:
20713:
20709:
20686:
20681:
20677:
20672:
20644:
20622:
20618:
20602:scalar product
20579:
20575:
20543:
20517:
20513:
20484:
20473:
20472:
20458:
20454:
20450:
20447:
20442:
20438:
20434:
20403:
20384:atomic nucleus
20359:
20355:
20352:
20348:
20345:
20341:
20338:
20335:
20332:
20292:
20289:
20268:
20265:
20261:center of mass
20245:principal axes
20232:
20229:
20199:
20196:
20193:
20189:
20185:
20182:
20179:
20176:
20173:
20170:
20165:
20161:
20156:
20131:
20128:
20125:
20122:
20119:
20113:
20110:
20104:
20101:
20095:
20092:
20086:
20077:, governed by
20062:
20034:
20012:
20008:
19987:
19984:
19979:
19975:
19971:
19968:
19965:
19938:
19914:
19894:
19871:
19849:
19846:
19843:
19840:
19834:
19831:
19825:
19805:
19802:
19799:
19796:
19793:
19787:
19784:
19778:
19751:Main article:
19740:
19737:
19719:
19716:
19671:
19662:th largest or
19651:
19631:
19609:
19604:
19600:
19596:
19593:
19590:
19584:
19580:
19560:
19538:
19534:
19530:
19527:
19523:
19500:
19496:
19473:
19470:
19466:
19445:
19419:
19415:
19411:
19408:
19404:
19400:
19395:
19391:
19387:
19384:
19380:
19376:
19373:
19349:
19346:
19343:
19315:
19292:
19289:
19257:bioinformatics
19166:Main article:
19139:
19136:
19133:
19130:
19127:
19107:
19104:
19101:
19098:
19095:
19075:
19072:
19052:
19049:
19046:
19043:
19040:
19037:
19034:
19031:
19005:
19001:
18997:
18994:
18991:
18988:
18985:
18982:
18979:
18976:
18956:
18955:
18938:
18932:
18929:
18926:
18925:
18922:
18919:
18918:
18916:
18911:
18908:
18906:
18902:
18897:
18892:
18891:
18886:
18880:
18877:
18876:
18873:
18870:
18869:
18867:
18862:
18859:
18857:
18853:
18848:
18843:
18842:
18832:
18819:
18813:
18810:
18809:
18806:
18803:
18802:
18800:
18795:
18790:
18785:
18773:
18756:
18750:
18747:
18744:
18743:
18740:
18737:
18736:
18734:
18729:
18726:
18724:
18720:
18715:
18710:
18709:
18704:
18698:
18695:
18692:
18691:
18688:
18685:
18684:
18682:
18677:
18674:
18672:
18668:
18663:
18658:
18657:
18647:
18630:
18624:
18621:
18620:
18617:
18614:
18613:
18611:
18606:
18603:
18601:
18597:
18592:
18587:
18586:
18581:
18575:
18572:
18571:
18568:
18565:
18564:
18562:
18557:
18554:
18552:
18548:
18543:
18538:
18537:
18527:
18524:
18520:
18519:
18504:
18501:
18498:
18496:
18492:
18488:
18484:
18483:
18480:
18477:
18474:
18472:
18468:
18464:
18460:
18459:
18449:
18438:
18435:
18430:
18426:
18415:
18400:
18397:
18394:
18392:
18388:
18384:
18380:
18379:
18376:
18373:
18370:
18368:
18364:
18360:
18356:
18355:
18345:
18330:
18327:
18324:
18322:
18318:
18314:
18310:
18309:
18306:
18303:
18300:
18298:
18294:
18290:
18286:
18285:
18275:
18264:
18261:
18256:
18252:
18241:
18230:
18225:
18221:
18217:
18214:
18211:
18206:
18202:
18184:
18183:
18168:
18165:
18162:
18160:
18156:
18152:
18148:
18147:
18144:
18141:
18138:
18136:
18132:
18128:
18124:
18123:
18113:
18102:
18099:
18094:
18090:
18079:
18064:
18061:
18058:
18056:
18052:
18048:
18044:
18043:
18040:
18037:
18034:
18032:
18028:
18024:
18020:
18019:
18009:
17994:
17991:
17988:
17986:
17982:
17978:
17974:
17973:
17970:
17967:
17964:
17962:
17958:
17954:
17950:
17949:
17939:
17928:
17925:
17920:
17916:
17905:
17894:
17889:
17885:
17881:
17878:
17875:
17870:
17866:
17848:
17847:
17832:
17829:
17826:
17823:
17820:
17817:
17814:
17811:
17808:
17806:
17804:
17799:
17796:
17792:
17788:
17785:
17783:
17779:
17775:
17771:
17770:
17767:
17764:
17761:
17758:
17755:
17752:
17749:
17746:
17743:
17741:
17739:
17734:
17730:
17726:
17723:
17721:
17717:
17713:
17709:
17708:
17698:
17687:
17684:
17679:
17675:
17671:
17666:
17662:
17651:
17636:
17633:
17630:
17627:
17624:
17621:
17618:
17615:
17612:
17609:
17607:
17605:
17600:
17597:
17594:
17590:
17586:
17583:
17581:
17577:
17573:
17569:
17568:
17565:
17562:
17559:
17556:
17553:
17550:
17547:
17544:
17541:
17538:
17536:
17534:
17529:
17526:
17522:
17518:
17515:
17513:
17509:
17505:
17501:
17500:
17490:
17473:
17469:
17465:
17462:
17460:
17456:
17452:
17448:
17447:
17442:
17438:
17434:
17431:
17429:
17425:
17421:
17417:
17416:
17406:
17395:
17392:
17387:
17383:
17379:
17374:
17370:
17359:
17346:
17342:
17329:
17328:
17317:
17314:
17311:
17308:
17305:
17302:
17299:
17296:
17293:
17290:
17285:
17281:
17270:
17257:
17253:
17249:
17246:
17243:
17240:
17227:
17216:
17213:
17210:
17207:
17204:
17201:
17198:
17195:
17192:
17189:
17184:
17180:
17169:
17158:
17153:
17149:
17145:
17142:
17139:
17136:
17131:
17127:
17123:
17120:
17117:
17107:
17094:
17090:
17086:
17083:
17080:
17077:
17064:
17061:Characteristic
17058:
17057:
17044:
17038:
17035:
17032:
17029:
17027:
17024:
17021:
17018:
17017:
17014:
17011:
17008:
17005:
17003:
17000:
16997:
16994:
16993:
16991:
16979:
16966:
16960:
16957:
16955:
16952:
16951:
16948:
16945:
16943:
16940:
16939:
16937:
16925:
16912:
16906:
16903:
16900:
16897:
16895:
16892:
16889:
16886:
16885:
16882:
16879:
16876:
16873:
16870:
16868:
16865:
16862:
16859:
16858:
16856:
16844:
16831:
16823:
16819:
16815:
16813:
16810:
16809:
16806:
16803:
16799:
16795:
16791:
16790:
16788:
16776:
16763:
16757:
16754:
16752:
16749:
16748:
16745:
16742:
16740:
16737:
16736:
16734:
16722:
16718:
16717:
16710:
16701:
16694:
16687:
16680:
16676:
16675:
16670:
16665:
16660:
16657:
16652:
16639:
16636:
16634:
16631:
16599:
16598:Modern methods
16596:
16582:
16555:
16550:
16537:
16536:
16521:
16515:
16510:
16502:
16498:
16493:
16488:
16480:
16477:
16454:
16433:
16407:
16403:
16400:
16375:
16372:
16368:
16364:
16361:
16358:
16355:
16352:
16331:Main article:
16328:
16325:
16312:
16282:
16276:
16273:
16270:
16267:
16265:
16262:
16261:
16259:
16236:
16233:
16230:
16227:
16224:
16221:
16201:
16198:
16195:
16175:
16152:
16149:
16146:
16126:
16106:
16076:
16070:
16067:
16064:
16062:
16059:
16058:
16056:
16033:
16030:
16027:
16024:
16001:
15995:
15992:
15989:
15987:
15985:
15982:
15979:
15976:
15973:
15970:
15969:
15966:
15963:
15960:
15958:
15956:
15953:
15950:
15947:
15944:
15941:
15940:
15937:
15909:
15903:
15900:
15897:
15894:
15892:
15890:
15887:
15884:
15881:
15878:
15875:
15874:
15871:
15868:
15865:
15862:
15860:
15858:
15855:
15852:
15849:
15846:
15845:
15842:
15814:
15808:
15805:
15804:
15801:
15798:
15797:
15795:
15790:
15787:
15784:
15779:
15773:
15770:
15769:
15766:
15763:
15762:
15760:
15753:
15747:
15744:
15742:
15739:
15738:
15735:
15732:
15730:
15727:
15726:
15724:
15702:
15699:
15696:
15693:
15690:
15666:
15660:
15657:
15655:
15652:
15651:
15648:
15645:
15643:
15640:
15639:
15637:
15632:
15629:
15613:
15610:
15589:
15565:
15541:
15514:
15511:
15491:
15488:
15485:
15445:
15442:
15439:
15419:
15407:
15404:
15400:floating-point
15395:
15392:
15384:Main article:
15381:
15378:
15377:
15376:
15365:
15362:
15359:
15354:
15350:
15346:
15340:
15337:
15332:
15329:
15321:
15317:
15313:
15310:
15307:
15299:
15296:
15293:
15289:
15285:
15280:
15275:
15272:
15269:
15265:
15256:
15253:
15250:
15246:
15242:
15234:
15230:
15226:
15221:
15216:
15212:
15190:
15189:
15178:
15173:
15168:
15164:
15158:
15154:
15150:
15147:
15144:
15139:
15134:
15130:
15124:
15120:
15116:
15111:
15107:
15083:
15078:
15074:
15069:
15066:
15062:
15057:
15053:
15026:
15018:
15015:
15012:
15009:
15006:
15002:
14998:
14996:
14993:
14989:
14985:
14981:
14980:
14978:
14952:
14947:
14944:
14941:
14938:
14935:
14931:
14927:
14922:
14919:
14916:
14913:
14910:
14906:
14899:
14896:
14890:
14885:
14882:
14879:
14875:
14871:
14866:
14863:
14860:
14856:
14840:
14839:
14828:
14825:
14822:
14817:
14813:
14809:
14806:
14801:
14798:
14795:
14791:
14787:
14784:
14781:
14776:
14773:
14770:
14766:
14760:
14756:
14752:
14747:
14744:
14741:
14737:
14731:
14727:
14723:
14718:
14714:
14691:
14690:
14679:
14674:
14671:
14668:
14664:
14658:
14654:
14650:
14647:
14644:
14639:
14636:
14633:
14629:
14623:
14619:
14615:
14610:
14607:
14604:
14600:
14594:
14590:
14586:
14581:
14577:
14564:have the form
14557:
14554:
14536:weight vectors
14505:Main article:
14502:
14499:
14421:Main article:
14418:
14415:
14338:always form a
14322:
14301:
14215:
14212:
14208:
14204:
14201:
14198:
14195:
14192:
14190:
14188:
14184:
14180:
14177:
14174:
14171:
14170:
14167:
14164:
14160:
14156:
14152:
14148:
14145:
14142:
14139:
14137:
14135:
14131:
14127:
14123:
14119:
14116:
14113:
14112:
14038:
14035:
14031:
14027:
14024:
14021:
14018:
14015:
14013:
14011:
14007:
14003:
14000:
13997:
13994:
13993:
13990:
13987:
13983:
13979:
13976:
13973:
13970:
13966:
13962:
13959:
13956:
13953:
13951:
13949:
13945:
13941:
13937:
13933:
13930:
13927:
13926:
13898:is called the
13877:
13873:
13868:
13864:
13861:
13858:
13854:
13850:
13847:
13844:
13840:
13835:
13831:
13828:
13812:
13809:
13788:to the vector
13752:
13751:
13742:
13740:
13729:
13725:
13721:
13718:
13715:
13711:
13707:
13704:
13653:
13650:
13647:
13644:
13641:
13638:
13595:
13592:
13560:
13555:
13552:
13548:
13544:
13541:
13538:
13535:
13532:
13529:
13526:
13523:
13520:
13479:
13476:
13473:
13470:
13467:
13464:
13461:
13458:
13455:
13452:
13449:
13443:
13440:
13436:
13410:
13407:
13403:
13388:
13385:
13381:eigenfunctions
13362:
13359:
13356:
13353:
13350:
13347:
13344:
13341:
13338:
13335:
13332:
13310:differentiable
13308:of infinitely
13274:Main article:
13271:
13268:
13241:
13237:
13216:
13209:
13204:
13200:
13196:
13191:
13187:
13183:
13177:
13174:
13171:
13167:
13160:
13157:
13152:
13148:
13144:
13139:
13135:
13131:
13126:
13122:
13118:
13112:
13108:
13101:
13096:
13091:
13084:
13081:
13078:
13074:
13069:
13044:
13041:
13034:
13009:
13003:
13000:
12998:
12995:
12993:
12990:
12988:
12985:
12984:
12982:
12949:
12943:
12940:
12938:
12935:
12932:
12930:
12927:
12925:
12922:
12921:
12919:
12887:
12866:
12861:
12857:
12853:
12850:
12847:
12844:
12839:
12835:
12831:
12828:
12825:
12822:
12819:
12814:
12808:
12805:
12802:
12799:
12797:
12794:
12792:
12789:
12787:
12784:
12783:
12780:
12777:
12775:
12772:
12769:
12766:
12764:
12761:
12759:
12756:
12755:
12752:
12749:
12747:
12744:
12742:
12739:
12736:
12733:
12731:
12728:
12727:
12724:
12721:
12719:
12716:
12714:
12711:
12709:
12706:
12703:
12700:
12699:
12697:
12692:
12689:
12686:
12683:
12680:
12677:
12674:
12671:
12651:
12646:
12640:
12637:
12635:
12632:
12630:
12627:
12625:
12622:
12621:
12618:
12615:
12613:
12610:
12608:
12605:
12603:
12600:
12599:
12596:
12593:
12591:
12588:
12586:
12583:
12581:
12578:
12577:
12574:
12571:
12569:
12566:
12564:
12561:
12559:
12556:
12555:
12553:
12548:
12545:
12533:
12530:
12515:
12510:
12504:
12501:
12500:
12497:
12494:
12493:
12490:
12487:
12486:
12484:
12479:
12472:
12468:
12462:
12456:
12451:
12445:
12442:
12439:
12438:
12435:
12432:
12431:
12428:
12425:
12424:
12422:
12417:
12410:
12406:
12400:
12394:
12389:
12381:
12378:
12373:
12372:
12369:
12366:
12363:
12362:
12359:
12356:
12355:
12353:
12348:
12341:
12337:
12331:
12308:
12298:
12288:
12270:
12267:
12264:
12261:
12258:
12255:
12252:
12249:
12246:
12243:
12240:
12237:
12234:
12231:
12228:
12225:
12222:
12219:
12216:
12213:
12210:
12207:
12204:
12201:
12175:
12170:
12164:
12161:
12159:
12156:
12154:
12151:
12150:
12147:
12144:
12142:
12139:
12137:
12134:
12133:
12130:
12127:
12125:
12122:
12120:
12117:
12116:
12114:
12109:
12106:
12080:
12077:
12062:
12057:
12051:
12048:
12047:
12044:
12041:
12040:
12037:
12034:
12033:
12031:
12026:
12019:
12015:
12009:
12003:
11998:
11992:
11989:
11988:
11985:
11982:
11981:
11978:
11975:
11974:
11972:
11967:
11960:
11956:
11950:
11944:
11939:
11933:
11930:
11929:
11926:
11923:
11922:
11919:
11916:
11915:
11913:
11908:
11901:
11897:
11891:
11868:
11858:
11848:
11830:
11827:
11824:
11821:
11818:
11815:
11812:
11809:
11806:
11803:
11800:
11797:
11794:
11791:
11788:
11785:
11782:
11779:
11776:
11773:
11770:
11767:
11764:
11761:
11735:
11730:
11724:
11721:
11719:
11716:
11714:
11711:
11710:
11707:
11704:
11702:
11699:
11697:
11694:
11693:
11690:
11687:
11685:
11682:
11680:
11677:
11676:
11674:
11669:
11666:
11648:
11645:
11633:
11628:
11621:
11617:
11611:
11606:
11599:
11595:
11589:
11575:
11568:
11543:
11535:
11531:
11527:
11523:
11519:
11515:
11513:
11510:
11509:
11507:
11501:
11494:
11490:
11484:
11452:
11444:
11440:
11436:
11432:
11428:
11424:
11422:
11419:
11418:
11416:
11410:
11403:
11399:
11393:
11365:
11360:
11352:
11348:
11344:
11343:
11338:
11334:
11330:
11329:
11326:
11323:
11322:
11320:
11315:
11310:
11306:
11302:
11297:
11291:
11288:
11287:
11282:
11278:
11274:
11273:
11268:
11264:
11260:
11259:
11257:
11252:
11247:
11239:
11235:
11231:
11230:
11225:
11221:
11217:
11216:
11213:
11210:
11209:
11207:
11202:
11182:
11177:
11169:
11165:
11161:
11160:
11155:
11151:
11147:
11146:
11143:
11140:
11139:
11137:
11132:
11127:
11123:
11119:
11114:
11108:
11105:
11104:
11099:
11095:
11091:
11090:
11085:
11081:
11077:
11076:
11074:
11069:
11064:
11056:
11052:
11048:
11047:
11042:
11038:
11034:
11033:
11030:
11027:
11026:
11024:
11019:
10997:
10992:
10988:
10984:
10979:
10974:
10970:
10965:
10960:
10956:
10952:
10947:
10942:
10938:
10933:
10930:
10927:
10922:
10918:
10912:
10908:
10885:
10880:
10874:
10871:
10870:
10867:
10864:
10863:
10860:
10857:
10856:
10854:
10849:
10846:
10843:
10838:
10832:
10829:
10828:
10825:
10822:
10821:
10818:
10815:
10814:
10812:
10807:
10802:
10796:
10793:
10792:
10789:
10786:
10785:
10782:
10779:
10778:
10776:
10771:
10759:
10740:
10737:
10734:
10729:
10725:
10712:imaginary unit
10699:
10673:
10669:
10663:
10660:
10655:
10652:
10647:
10644:
10639:
10634:
10630:
10626:
10623:
10621:
10617:
10613:
10609:
10608:
10603:
10599:
10593:
10590:
10585:
10582:
10577:
10574:
10571:
10569:
10565:
10561:
10557:
10556:
10553:
10550:
10547:
10545:
10541:
10537:
10533:
10532:
10506:
10501:
10495:
10492:
10490:
10487:
10485:
10482:
10481:
10478:
10475:
10473:
10470:
10468:
10465:
10464:
10461:
10458:
10456:
10453:
10451:
10448:
10447:
10445:
10440:
10437:
10422:
10419:
10394:
10388:
10385:
10383:
10380:
10378:
10375:
10374:
10372:
10336:
10330:
10327:
10325:
10322:
10319:
10317:
10314:
10313:
10311:
10276:
10270:
10267:
10265:
10262:
10260:
10257:
10256:
10254:
10220:
10217:
10214:
10211:
10208:
10203:
10199:
10195:
10192:
10187:
10183:
10179:
10176:
10171:
10166:
10163:
10160:
10157:
10154:
10151:
10148:
10145:
10142:
10139:
10136:
10133:
10128:
10123:
10120:
10117:
10114:
10111:
10108:
10105:
10103:
10101:
10098:
10093:
10087:
10084:
10081:
10078:
10076:
10073:
10071:
10068:
10067:
10064:
10061:
10059:
10056:
10053:
10050:
10048:
10045:
10044:
10041:
10038:
10036:
10033:
10031:
10028:
10025:
10022:
10021:
10019:
10014:
10010:
10004:
9998:
9995:
9993:
9990:
9988:
9985:
9984:
9981:
9978:
9976:
9973:
9971:
9968:
9967:
9964:
9961:
9959:
9956:
9954:
9951:
9950:
9948:
9943:
9940:
9935:
9929:
9926:
9924:
9921:
9919:
9916:
9915:
9912:
9909:
9907:
9904:
9902:
9899:
9898:
9895:
9892:
9890:
9887:
9885:
9882:
9881:
9879:
9873:
9869:
9866:
9864:
9862:
9859:
9856:
9853:
9850:
9847:
9844:
9841:
9840:
9814:
9809:
9803:
9800:
9798:
9795:
9793:
9790:
9789:
9786:
9783:
9781:
9778:
9776:
9773:
9772:
9769:
9766:
9764:
9761:
9759:
9756:
9755:
9753:
9748:
9745:
9733:
9730:
9704:
9694:
9664:
9658:
9655:
9654:
9651:
9648:
9647:
9645:
9640:
9635:
9627:
9623:
9619:
9618:
9613:
9609:
9605:
9604:
9602:
9597:
9592:
9589:
9586:
9581:
9567:
9560:
9539:
9536:
9533:
9531:
9527:
9523:
9519:
9516:
9511:
9507:
9503:
9500:
9499:
9496:
9493:
9490:
9487:
9485:
9481:
9477:
9473:
9470:
9465:
9461:
9457:
9454:
9451:
9450:
9445:
9439:
9436:
9435:
9432:
9429:
9428:
9426:
9421:
9416:
9408:
9404:
9400:
9399:
9394:
9390:
9386:
9385:
9383:
9376:
9370:
9367:
9364:
9362:
9359:
9358:
9355:
9352:
9350:
9347:
9344:
9343:
9341:
9336:
9333:
9331:
9327:
9324:
9321:
9316:
9311:
9308:
9305:
9302:
9299:
9296:
9293:
9292:
9246:
9240:
9237:
9234:
9233:
9230:
9227:
9226:
9224:
9219:
9214:
9206:
9202:
9198:
9195:
9194:
9189:
9185:
9181:
9180:
9178:
9173:
9168:
9165:
9162:
9157:
9143:
9136:
9119:
9116:
9111:
9107:
9103:
9100:
9095:
9091:
9087:
9066:
9060:
9057:
9056:
9053:
9050:
9049:
9047:
9042:
9037:
9029:
9025:
9021:
9020:
9015:
9011:
9007:
9006:
9004:
8997:
8991:
8988:
8986:
8983:
8982:
8979:
8976:
8974:
8971:
8970:
8968:
8963:
8958:
8955:
8952:
8947:
8942:
8939:
8936:
8933:
8930:
8870:
8867:
8864:
8861:
8858:
8855:
8852:
8849:
8846:
8843:
8840:
8837:
8834:
8832:
8830:
8825:
8821:
8817:
8814:
8811:
8808:
8805:
8802:
8799:
8797:
8795:
8790:
8784:
8781:
8778:
8775:
8773:
8770:
8769:
8766:
8763:
8761:
8758:
8755:
8752:
8751:
8749:
8744:
8740:
8734:
8728:
8725:
8723:
8720:
8719:
8716:
8713:
8711:
8708:
8707:
8705:
8700:
8697:
8692:
8686:
8683:
8681:
8678:
8677:
8674:
8671:
8669:
8666:
8665:
8663:
8657:
8653:
8650:
8648:
8646:
8643:
8640:
8637:
8634:
8631:
8628:
8625:
8624:
8573:
8568:
8562:
8559:
8557:
8554:
8553:
8550:
8547:
8545:
8542:
8541:
8539:
8534:
8531:
8508:
8498:
8482:
8474:
8471:
8469:
8466:
8465:
8462:
8459:
8457:
8454:
8453:
8449:
8428:
8425:
8423:
8420:
8406:
8384:
8371:
8365:
8360:
8356:
8344:
8332:quadratic form
8319:
8300:Main article:
8297:
8294:
8182:diagonalizable
8175:is said to be
8158:
8157:
8146:
8143:
8140:
8137:
8134:
8129:
8126:
8122:
8103:
8102:
8091:
8086:
8083:
8079:
8075:
8072:
8069:
8066:
8044:
8043:
8032:
8029:
8026:
8023:
8020:
8017:
7992:
7989:
7988:
7977:
7972:
7964:
7959:
7952:
7948:
7944:
7942:
7939:
7935:
7930:
7923:
7919:
7915:
7911:
7906:
7899:
7895:
7891:
7890:
7888:
7883:
7880:
7877:
7843:
7842:
7831:
7826:
7818:
7813:
7808:
7806:
7803:
7799:
7794:
7789:
7785:
7780:
7775:
7774:
7772:
7767:
7764:
7732:
7725:
7718:
7709:
7702:
7695:
7673:Main article:
7670:
7667:
7654:
7634:
7606:
7577:
7556:
7528:
7516:
7511:
7508:
7496:
7483:
7462:
7442:
7418:
7398:
7395:
7392:
7372:
7352:
7330:
7326:
7322:
7319:
7316:
7312:
7291:
7261:
7257:
7253:
7250:
7246:
7242:
7216:
7196:
7193:
7190:
7159:
7156:
7155:
7154:
7142:
7139:
7134:
7130:
7126:
7123:
7120:
7117:
7114:
7111:
7106:
7102:
7098:
7095:
7092:
7072:
7069:
7066:
7063:
7043:
7023:
7020:
7017:
7012:
7008:
7004:
7001:
6998:
6995:
6992:
6987:
6983:
6979:
6959:
6956:
6953:
6950:
6929:
6925:
6922:
6902:
6899:
6896:
6891:
6887:
6883:
6880:
6877:
6874:
6871:
6866:
6862:
6858:
6838:
6818:
6815:
6812:
6792:
6787:
6783:
6779:
6776:
6773:
6768:
6764:
6760:
6740:
6737:
6734:
6714:
6703:
6691:
6688:
6684:
6678:
6674:
6669:
6644:
6633:
6617:
6606:
6586:
6564:
6560:
6536:
6525:
6505:
6501:
6497:
6492:
6489:
6486:
6479:
6475:
6471:
6447:
6444:
6440:
6419:
6408:
6392:
6381:
6367:
6362:
6358:
6354:
6351:
6348:
6343:
6338:
6334:
6313:
6291:
6287:
6266:
6246:
6235:
6234:
6233:
6222:
6217:
6213:
6209:
6204:
6200:
6194:
6190:
6186:
6181:
6177:
6171:
6166:
6163:
6160:
6156:
6152:
6149:
6146:
6143:
6140:
6118:
6103:
6102:
6101:
6090:
6085:
6081:
6077:
6074:
6071:
6066:
6062:
6058:
6053:
6049:
6045:
6040:
6036:
6030:
6025:
6022:
6019:
6015:
6011:
6006:
6003:
5999:
5993:
5988:
5985:
5982:
5978:
5974:
5971:
5968:
5965:
5962:
5959:
5937:
5909:
5904:
5900:
5896:
5891:
5887:
5866:
5861:
5857:
5853:
5848:
5844:
5821:
5817:
5813:
5810:
5807:
5802:
5798:
5777:
5774:
5771:
5751:
5739:
5736:
5735:
5734:
5722:
5700:
5695:
5684:Any vector in
5682:
5667:
5647:
5625:
5620:
5607:
5593:
5588:
5566:
5542:
5539:
5534:
5530:
5509:
5489:
5461:
5458:
5455:
5450:
5446:
5442:
5439:
5437:
5435:
5432:
5431:
5428:
5425:
5420:
5416:
5412:
5407:
5403:
5397:
5392:
5389:
5386:
5382:
5378:
5375:
5373:
5369:
5365:
5361:
5360:
5340:
5320:
5315:
5311:
5307:
5302:
5298:
5275:
5271:
5248:
5244:
5240:
5237:
5234:
5229:
5225:
5204:
5201:
5198:
5178:
5155:
5152:
5149:
5144:
5140:
5136:
5133:
5130:
5127:
5122:
5118:
5097:
5075:
5072:
5069:
5064:
5060:
5055:
5051:
5048:
5045:
5042:
5022:
5019:
5016:
5013:
5010:
5007:
5004:
4984:
4964:
4961:
4958:
4955:
4952:
4949:
4946:
4943:
4940:
4937:
4934:
4931:
4928:
4925:
4922:
4902:
4899:
4896:
4893:
4884:is similar to
4873:
4870:
4867:
4864:
4844:
4835:commutes with
4824:
4804:
4801:
4798:
4795:
4792:
4789:
4786:
4783:
4780:
4777:
4774:
4771:
4768:
4765:
4762:
4742:
4739:
4736:
4714:
4711:
4708:
4703:
4699:
4694:
4690:
4670:
4667:
4662:
4658:
4654:
4651:
4631:
4611:
4591:
4588:
4585:
4580:
4576:
4572:
4569:
4549:
4546:
4543:
4538:
4534:
4513:
4491:
4486:
4481:
4478:
4473:
4468:
4463:
4441:
4438:
4435:
4430:
4426:
4420:
4414:
4411:
4407:
4402:
4397:
4372:
4369:
4366:
4361:
4357:
4336:
4333:
4330:
4325:
4321:
4317:
4314:
4311:
4308:
4303:
4299:
4276:
4273:
4270:
4267:
4264:
4259:
4255:
4251:
4248:
4245:
4242:
4237:
4233:
4229:
4226:
4199:
4196:
4193:
4190:
4187:
4184:
4181:
4178:
4175:
4172:
4169:
4166:
4163:
4160:
4157:
4152:
4148:
4083:
4080:
4077:
4072:
4068:
3860:
3855:
3812:is called the
3780:. So, the set
3747:
3743:
3738:
3734:
3730:
3725:
3721:
3718:
3715:
3712:
3708:
3704:
3700:
3695:
3691:
3688:
3643:
3640:
3629:
3620:
3611:
3602:
3593:
3584:
3571:
3562:
3553:
3532:
3529:
3526:
3522:
3517:
3513:
3509:
3503:
3499:
3493:
3488:
3485:
3482:
3478:
3474:
3471:
3469:
3465:
3461:
3457:
3456:
3453:
3450:
3447:
3444:
3439:
3435:
3431:
3426:
3422:
3418:
3415:
3413:
3411:
3408:
3407:
3363:
3358:
3353:
3349:
3345:
3340:
3336:
3331:
3327:
3324:
3319:
3315:
3311:
3308:
3303:
3298:
3294:
3290:
3285:
3281:
3276:
3272:
3269:
3264:
3260:
3256:
3251:
3246:
3242:
3238:
3233:
3229:
3224:
3220:
3217:
3212:
3208:
3204:
3201:
3198:
3195:
3192:
3189:
3186:
3183:
3180:
3141:has dimension
3132:divides evenly
3125:
3104:
3095:
3071:
3064:
3061:
3043:
3040:
2984:
2979:
2973:
2970:
2969:
2966:
2963:
2962:
2960:
2955:
2950:
2947:
2944:
2939:
2933:
2928:
2922:
2919:
2916:
2915:
2912:
2909:
2908:
2906:
2901:
2896:
2893:
2890:
2885:
2860:
2856:
2852:
2847:
2843:
2840:
2837:
2834:
2830:
2790:
2785:
2781:
2777:
2774:
2771:
2768:
2765:
2762:
2757:
2751:
2748:
2745:
2742:
2740:
2737:
2736:
2733:
2730:
2728:
2725:
2722:
2719:
2718:
2716:
2711:
2708:
2705:
2702:
2699:
2696:
2693:
2690:
2652:
2647:
2641:
2638:
2636:
2633:
2632:
2629:
2626:
2624:
2621:
2620:
2618:
2613:
2610:
2589:
2582:
2575:
2566:
2558:
2557:
2548:
2546:
2535:
2532:
2529:
2526:
2521:
2517:
2513:
2510:
2507:
2504:
2501:
2496:
2492:
2488:
2485:
2482:
2479:
2474:
2470:
2466:
2463:
2460:
2457:
2454:
2451:
2448:
2445:
2442:
2428:linear terms,
2365:is always (−1)
2319:
2318:
2309:
2307:
2296:
2293:
2290:
2287:
2284:
2281:
2278:
2275:
2272:
2254:are values of
2238:of the matrix
2232:if and only if
2215:Main article:
2212:
2209:
2184:
2183:
2174:
2172:
2161:
2157:
2153:
2149:
2144:
2140:
2137:
2134:
2131:
2127:
2063:
2062:
2053:
2051:
2040:
2036:
2032:
2029:
2025:
2021:
2017:
2013:
1978:
1973:
1969:
1963:
1960:
1956:
1950:
1945:
1942:
1939:
1935:
1931:
1926:
1922:
1916:
1913:
1909:
1905:
1902:
1899:
1894:
1890:
1884:
1881:
1877:
1873:
1868:
1864:
1858:
1855:
1851:
1847:
1842:
1838:
1815:
1807:
1803:
1799:
1798:
1795:
1792:
1791:
1786:
1782:
1778:
1777:
1772:
1768:
1764:
1763:
1761:
1756:
1751:
1743:
1739:
1735:
1734:
1731:
1728:
1727:
1722:
1718:
1714:
1713:
1708:
1704:
1700:
1699:
1697:
1690:
1682:
1679:
1675:
1671:
1669:
1666:
1662:
1659:
1655:
1651:
1647:
1644:
1640:
1636:
1635:
1632:
1629:
1627:
1624:
1622:
1619:
1617:
1614:
1613:
1608:
1605:
1601:
1597:
1595:
1592:
1588:
1584:
1580:
1576:
1572:
1568:
1567:
1562:
1559:
1555:
1551:
1549:
1546:
1542:
1538:
1534:
1530:
1526:
1522:
1521:
1519:
1497:
1493:
1489:
1485:
1481:
1440:
1437:
1432:
1429:
1426:
1417:In this case,
1404:
1400:
1396:
1393:
1389:
1350:
1345:
1339:
1336:
1333:
1332:
1329:
1326:
1325:
1322:
1319:
1316:
1315:
1313:
1308:
1304:
1289:
1283:
1280:
1279:
1276:
1273:
1270:
1269:
1266:
1263:
1262:
1260:
1255:
1251:
1206:applications.
1192:
1189:
1171:published the
1094:Alfred Clebsch
1041:Joseph Fourier
1005:principal axes
997:Leonhard Euler
978:linear algebra
973:
970:
969:
968:
949:
930:
887:
883:
879:
876:
872:
868:
824:
819:
816:
812:
808:
805:
800:
797:
793:
786:
783:
779:
766:eigenfunctions
749:
746:
742:
669:. In general,
650:
646:
642:
639:
636:
632:
628:
625:
567:principal axes
532:
529:
479:
475:
472:
468:
464:
435:
401:
379:
375:
354:
329:
326:
305:
280:
276:
272:
260:
257:
202:
169:
165:
162:
158:
154:
134:
110:
89:
30:linear algebra
15:
9:
6:
4:
3:
2:
25472:
25461:
25458:
25456:
25455:Matrix theory
25453:
25451:
25448:
25446:
25443:
25441:
25438:
25437:
25435:
25420:
25419:
25410:
25408:
25407:
25398:
25396:
25395:
25386:
25384:
25383:
25378:
25372:
25371:
25368:
25362:
25359:
25357:
25354:
25352:
25349:
25347:
25344:
25342:
25339:
25335:
25332:
25331:
25330:
25327:
25326:
25324:
25322:
25318:
25312:
25309:
25307:
25304:
25302:
25299:
25297:
25294:
25292:
25289:
25287:
25284:
25283:
25281:
25279:
25278:Computational
25275:
25267:
25264:
25262:
25259:
25257:
25254:
25253:
25252:
25249:
25247:
25244:
25242:
25239:
25237:
25234:
25232:
25229:
25227:
25224:
25222:
25219:
25217:
25214:
25212:
25209:
25207:
25204:
25202:
25199:
25197:
25194:
25193:
25191:
25189:
25185:
25179:
25176:
25174:
25171:
25169:
25166:
25164:
25161:
25159:
25156:
25155:
25153:
25151:
25147:
25141:
25138:
25136:
25133:
25131:
25128:
25126:
25123:
25122:
25120:
25118:
25117:Number theory
25114:
25108:
25105:
25103:
25100:
25098:
25095:
25093:
25090:
25088:
25085:
25083:
25080:
25078:
25075:
25074:
25072:
25070:
25066:
25060:
25057:
25055:
25052:
25050:
25049:Combinatorics
25047:
25046:
25044:
25042:
25038:
25032:
25029:
25027:
25024:
25022:
25019:
25017:
25014:
25012:
25009:
25007:
25004:
25002:
25001:Real analysis
24999:
24997:
24994:
24993:
24991:
24989:
24985:
24979:
24976:
24974:
24971:
24969:
24966:
24964:
24961:
24959:
24956:
24954:
24951:
24949:
24946:
24944:
24941:
24940:
24938:
24936:
24932:
24926:
24923:
24921:
24918:
24916:
24913:
24911:
24908:
24906:
24903:
24901:
24898:
24897:
24895:
24893:
24889:
24883:
24880:
24878:
24875:
24871:
24868:
24866:
24863:
24862:
24861:
24858:
24857:
24854:
24849:
24841:
24836:
24834:
24829:
24827:
24822:
24821:
24818:
24806:
24798:
24797:
24794:
24788:
24785:
24783:
24782:Sparse matrix
24780:
24778:
24775:
24773:
24770:
24768:
24765:
24764:
24762:
24760:
24756:
24750:
24747:
24745:
24742:
24740:
24737:
24735:
24732:
24730:
24727:
24725:
24722:
24721:
24719:
24717:constructions
24716:
24712:
24706:
24705:Outermorphism
24703:
24701:
24698:
24696:
24693:
24691:
24688:
24686:
24683:
24681:
24678:
24676:
24673:
24671:
24668:
24666:
24665:Cross product
24663:
24661:
24658:
24657:
24655:
24653:
24649:
24643:
24640:
24638:
24635:
24633:
24632:Outer product
24630:
24628:
24625:
24623:
24620:
24618:
24615:
24613:
24612:Orthogonality
24610:
24609:
24607:
24605:
24601:
24595:
24592:
24590:
24589:Cramer's rule
24587:
24585:
24582:
24580:
24577:
24575:
24572:
24570:
24567:
24565:
24562:
24560:
24559:Decomposition
24557:
24555:
24552:
24551:
24549:
24547:
24543:
24538:
24528:
24525:
24523:
24520:
24518:
24515:
24513:
24510:
24508:
24505:
24503:
24500:
24498:
24495:
24493:
24490:
24488:
24485:
24483:
24480:
24478:
24475:
24473:
24470:
24468:
24465:
24463:
24460:
24458:
24455:
24453:
24450:
24448:
24445:
24443:
24440:
24438:
24435:
24434:
24432:
24428:
24422:
24419:
24417:
24414:
24413:
24410:
24406:
24399:
24394:
24392:
24387:
24385:
24380:
24379:
24376:
24370:
24366:
24362:
24359:
24357:
24354:
24353:
24347:
24346:
24345:
24339:
24319:
24316:
24313:
24305:
24302:
24300:
24296:
24293:
24290:
24287:
24284:
24281:
24278:
24275:
24274:
24267:
24264:
24256:
24253:December 2019
24246:
24242:
24241:inappropriate
24238:
24234:
24228:
24226:
24219:
24210:
24209:
24197:
24195:0-03-010567-6
24191:
24187:
24182:
24179:
24177:0-9614088-5-5
24173:
24169:
24164:
24157:
24156:
24150:
24146:
24142:
24138:
24137:Sixty Symbols
24134:
24129:
24126:
24122:
24117:
24112:
24108:
24104:
24100:
24096:
24089:
24084:
24083:
24070:
24065:
24061:
24057:
24053:
24048:
24041:
24040:
24034:
24023:
24019:
24014:
24003:
23999:
23994:
23983:
23979:
23978:"Eigenvector"
23974:
23970:
23966:
23962:
23958:
23954:
23950:
23946:
23942:
23938:
23934:
23930:
23926:
23922:
23917:
23912:
23907:
23902:
23898:
23893:
23890:
23886:
23882:
23878:
23874:
23870:
23866:
23862:
23857:
23854:
23852:0-486-63518-X
23848:
23844:
23839:
23835:
23831:
23827:
23823:
23819:
23815:
23810:
23805:
23802:(1): 015005.
23801:
23797:
23793:
23788:
23785:
23779:
23775:
23770:
23767:
23761:
23757:
23753:
23748:
23745:
23741:
23737:
23733:
23728:
23725:
23719:
23715:
23710:
23706:
23700:
23696:
23695:
23689:
23686:
23682:
23678:
23674:
23669:
23666:
23664:0-486-41147-8
23660:
23656:
23652:
23648:
23644:
23639:
23636:
23632:
23628:
23624:
23620:
23616:
23611:
23608:
23606:0-19-501496-0
23602:
23598:
23593:
23590:
23584:
23580:
23575:
23571:
23570:
23564:
23560:
23559:
23553:
23549:
23544:
23540:
23536:
23531:
23528:
23524:
23520:
23516:
23512:
23508:
23504:
23500:
23499:
23493:
23490:
23484:
23480:
23476:
23472:
23468:
23465:
23463:0-13-537102-3
23459:
23455:
23450:
23446:
23441:
23437:
23433:
23428:
23424:
23419:
23415:
23411:
23410:
23404:
23401:
23399:0-201-01984-1
23395:
23391:
23387:
23382:
23379:
23375:
23371:
23367:
23362:
23357:
23353:
23349:
23345:
23341:
23337:
23332:
23325:
23321:
23317:
23313:
23309:
23304:
23299:
23295:
23291:
23284:
23279:
23276:
23274:0-534-93219-3
23270:
23265:
23264:
23257:
23254:
23250:
23246:
23242:
23237:
23233:
23228:
23224:
23220:
23216:
23212:
23208:
23204:
23200:
23196:
23192:
23188:
23183:
23178:
23174:
23170:
23166:
23161:
23157:
23156:
23150:
23147:
23145:0-395-14017-X
23141:
23137:
23132:
23131:
23124:
23121:
23119:0-471-84819-0
23115:
23111:
23107:
23102:
23099:
23095:
23090:
23089:
23077:
23072:
23065:
23060:
23053:
23048:
23041:
23036:
23028:
23024:
23017:
23010:
23005:
22998:
22993:
22986:
22981:
22974:
22969:
22962:
22957:
22955:
22947:
22942:
22935:
22930:
22928:
22926:
22924:
22916:
22911:
22904:
22899:
22893:
22889:
22885:
22880:
22873:
22868:
22861:
22856:
22849:
22844:
22837:
22832:
22825:
22820:
22813:
22808:
22806:
22798:
22793:
22786:
22781:
22774:
22773:Herstein 1964
22769:
22762:
22757:
22750:
22749:Herstein 1964
22745:
22738:
22733:
22731:
22723:
22718:
22711:
22706:
22704:
22702:
22694:
22693:Fraleigh 1976
22689:
22683:, p. 38.
22682:
22677:
22670:
22666:
22663:
22662:
22655:
22648:
22647:
22640:
22633:
22628:
22621:
22616:
22609:
22604:
22597:
22592:
22585:
22580:
22573:
22568:
22561:
22556:
22549:
22544:
22537:
22532:
22530:
22522:
22517:
22510:
22505:
22503:
22501:
22499:
22491:
22486:
22479:
22474:
22472:
22470:
22468:
22460:
22455:
22448:
22443:
22428:
22424:
22418:
22416:
22408:
22403:
22397:, p. 38.
22396:
22391:
22389:
22381:
22380:Herstein 1964
22376:
22374:
22362:
22361:
22353:
22346:
22341:
22337:
22325:into account.
22324:
22303:
22299:
22292:
22272:
22265:truncated to
22264:
22260:
22254:
22248:
22244:
22240:
22239:Hefferon 2001
22236:
22232:
22226:
22218:
22214:
22210:
22205:
22200:
22196:
22192:
22188:
22184:
22183:
22179:
22172:
22168:
22165:
22161:
22157:
22153:
22148:
22140:
22136:
22132:
22128:
22127:Arthur Cayley
22124:
22121:
22117:
22113:
22109:
22105:
22101:
22098:
22094:
22090:
22086:
22082:
22081:
22077:
22073:
22063:
22060:
22058:
22055:
22053:
22050:
22048:
22045:
22043:
22040:
22038:
22035:
22033:
22030:
22028:
22025:
22023:
22020:
22018:
22015:
22013:
22010:
22008:
22007:Eigenoperator
22005:
22003:
22000:
21999:
21993:
21990:
21985:
21983:
21980:to faces for
21979:
21975:
21971:
21967:
21963:
21959:
21958:
21953:
21949:
21945:
21941:
21935:
21927:
21923:
21914:
21898:
21894:
21871:
21867:
21844:
21840:
21817:
21813:
21790:
21786:
21775:
21765:
21749:
21745:
21741:
21736:
21732:
21728:
21723:
21719:
21696:
21692:
21688:
21683:
21679:
21675:
21670:
21666:
21643:
21639:
21635:
21630:
21626:
21622:
21617:
21613:
21590:
21586:
21563:
21559:
21536:
21532:
21523:
21519:
21501:
21472:
21443:
21414:
21410:
21406:
21401:
21397:
21393:
21388:
21384:
21361:
21351:
21346:
21336:
21331:
21315:
21313:
21309:
21305:
21294:
21291:
21283:
21280:December 2023
21273:
21269:
21263:
21260:This section
21258:
21249:
21248:
21240:
21238:
21234:
21230:
21227:
21223:
21219:
21215:
21211:
21207:
21203:
21202:Fock operator
21199:
21195:
21191:
21188:, within the
21187:
21183:
21179:
21169:
21155:
21152:
21143:
21122:
21119:
21110:
21081:
21054:
21027:
20990:
20986:
20983:are randomly
20982:
20978:
20974:
20965:
20946:
20916:
20891:
20878:
20870:
20867:
20851:
20831:
20811:
20803:
20782:
20744:
20731:
20728:
20720:
20707:
20700:
20699:
20698:
20679:
20661:
20656:
20642:
20620:
20616:
20607:
20603:
20599:
20598:Hilbert space
20595:
20577:
20573:
20564:
20559:
20557:
20541:
20533:
20515:
20511:
20502:
20498:
20482:
20456:
20452:
20448:
20445:
20440:
20436:
20432:
20425:
20424:
20423:
20421:
20417:
20401:
20389:
20385:
20381:
20377:
20373:
20357:
20353:
20350:
20346:
20343:
20339:
20336:
20333:
20330:
20322:
20318:
20314:
20313:hydrogen atom
20310:
20306:
20302:
20301:wavefunctions
20297:
20288:
20286:
20282:
20278:
20274:
20267:Stress tensor
20264:
20262:
20258:
20255:of moment of
20254:
20250:
20246:
20242:
20238:
20228:
20226:
20222:
20217:
20215:
20210:
20197:
20194:
20191:
20187:
20183:
20180:
20177:
20174:
20171:
20168:
20163:
20159:
20154:
20145:
20129:
20126:
20123:
20120:
20117:
20111:
20108:
20102:
20099:
20093:
20090:
20084:
20076:
20060:
20052:
20048:
20032:
20010:
20006:
19985:
19982:
19977:
19973:
19969:
19966:
19963:
19956:
19952:
19936:
19928:
19912:
19892:
19883:
19869:
19860:
19847:
19844:
19841:
19838:
19832:
19829:
19823:
19803:
19800:
19797:
19794:
19791:
19785:
19782:
19776:
19768:
19764:
19760:
19754:
19745:
19736:
19734:
19729:
19725:
19718:Markov chains
19715:
19713:
19708:
19704:
19700:
19696:
19692:
19688:
19683:
19669:
19649:
19629:
19602:
19598:
19591:
19588:
19582:
19578:
19558:
19536:
19532:
19528:
19525:
19521:
19498:
19494:
19471:
19468:
19464:
19443:
19435:
19417:
19413:
19409:
19406:
19402:
19398:
19393:
19389:
19385:
19382:
19378:
19374:
19371:
19363:
19347:
19344:
19341:
19333:
19329:
19313:
19306:
19302:
19298:
19288:
19286:
19282:
19278:
19274:
19270:
19266:
19262:
19261:Q methodology
19258:
19254:
19250:
19245:
19243:
19239:
19235:
19231:
19227:
19223:
19219:
19215:
19211:
19208:
19204:
19200:
19196:
19192:
19189:
19185:
19179:
19175:
19169:
19161:
19157:
19153:
19134:
19131:
19128:
19102:
19099:
19096:
19085:
19080:
19071:
19069:
19064:
19050:
19047:
19044:
19041:
19038:
19035:
19032:
19029:
19003:
18995:
18992:
18989:
18983:
18980:
18977:
18974:
18967:
18963:
18936:
18930:
18927:
18920:
18914:
18909:
18907:
18900:
18884:
18878:
18871:
18865:
18860:
18858:
18851:
18833:
18817:
18811:
18804:
18798:
18793:
18788:
18774:
18754:
18748:
18745:
18738:
18732:
18727:
18725:
18718:
18702:
18696:
18693:
18686:
18680:
18675:
18673:
18666:
18648:
18628:
18622:
18615:
18609:
18604:
18602:
18595:
18579:
18573:
18566:
18560:
18555:
18553:
18546:
18528:
18525:
18523:Eigenvectors
18521:
18502:
18499:
18497:
18490:
18486:
18478:
18475:
18473:
18466:
18462:
18450:
18436:
18433:
18428:
18424:
18416:
18398:
18395:
18393:
18386:
18382:
18374:
18371:
18369:
18362:
18358:
18346:
18328:
18325:
18323:
18316:
18312:
18304:
18301:
18299:
18292:
18288:
18276:
18262:
18259:
18254:
18250:
18242:
18223:
18219:
18212:
18209:
18204:
18200:
18185:
18166:
18163:
18161:
18154:
18150:
18142:
18139:
18137:
18130:
18126:
18114:
18100:
18097:
18092:
18088:
18080:
18062:
18059:
18057:
18050:
18046:
18038:
18035:
18033:
18026:
18022:
18010:
17992:
17989:
17987:
17980:
17976:
17968:
17965:
17963:
17956:
17952:
17940:
17926:
17923:
17918:
17914:
17906:
17887:
17883:
17876:
17873:
17868:
17864:
17849:
17830:
17827:
17824:
17821:
17818:
17815:
17812:
17809:
17807:
17797:
17794:
17790:
17786:
17784:
17777:
17773:
17765:
17762:
17759:
17756:
17753:
17750:
17747:
17744:
17742:
17732:
17728:
17724:
17722:
17715:
17711:
17699:
17685:
17682:
17677:
17673:
17669:
17664:
17660:
17652:
17634:
17631:
17628:
17625:
17622:
17619:
17616:
17613:
17610:
17608:
17598:
17595:
17592:
17588:
17584:
17582:
17575:
17571:
17563:
17560:
17557:
17554:
17551:
17548:
17545:
17542:
17539:
17537:
17527:
17524:
17520:
17516:
17514:
17507:
17503:
17491:
17471:
17467:
17463:
17461:
17454:
17450:
17440:
17436:
17432:
17430:
17423:
17419:
17407:
17393:
17390:
17385:
17381:
17377:
17372:
17368:
17360:
17344:
17340:
17332:Eigenvalues,
17330:
17315:
17312:
17309:
17303:
17297:
17294:
17291:
17288:
17283:
17279:
17271:
17255:
17247:
17244:
17241:
17228:
17214:
17211:
17208:
17202:
17196:
17193:
17190:
17187:
17182:
17178:
17170:
17151:
17147:
17143:
17140:
17129:
17125:
17121:
17118:
17108:
17092:
17084:
17081:
17078:
17065:
17059:
17042:
17036:
17033:
17030:
17025:
17022:
17019:
17012:
17009:
17006:
17001:
16998:
16995:
16989:
16980:
16964:
16958:
16953:
16946:
16941:
16935:
16926:
16910:
16904:
16901:
16898:
16893:
16890:
16887:
16880:
16877:
16874:
16871:
16866:
16863:
16860:
16854:
16845:
16829:
16821:
16817:
16811:
16804:
16797:
16793:
16786:
16777:
16761:
16755:
16750:
16743:
16738:
16732:
16723:
16719:
16715:
16711:
16706:
16702:
16699:
16695:
16692:
16688:
16685:
16681:
16679:Illustration
16677:
16674:
16669:
16664:
16656:
16651:
16650:
16644:
16630:
16626:
16624:
16620:
16616:
16613:
16609:
16605:
16595:
16571:
16553:
16513:
16496:
16491:
16478:
16475:
16468:
16467:
16466:
16452:
16421:
16401:
16398:
16373:
16370:
16362:
16359:
16356:
16353:
16341:
16334:
16324:
16310:
16280:
16274:
16271:
16268:
16263:
16257:
16234:
16231:
16228:
16225:
16222:
16219:
16199:
16196:
16193:
16173:
16164:
16150:
16147:
16144:
16124:
16104:
16074:
16068:
16065:
16060:
16054:
16031:
16028:
16025:
16022:
16013:
15993:
15990:
15988:
15983:
15980:
15977:
15974:
15971:
15964:
15961:
15959:
15954:
15951:
15948:
15945:
15942:
15935:
15901:
15898:
15895:
15893:
15888:
15885:
15882:
15879:
15876:
15869:
15866:
15863:
15861:
15856:
15853:
15850:
15847:
15840:
15832:
15827:
15812:
15806:
15799:
15793:
15788:
15785:
15782:
15777:
15771:
15764:
15758:
15751:
15745:
15740:
15733:
15728:
15722:
15700:
15697:
15694:
15691:
15688:
15679:
15664:
15658:
15653:
15646:
15641:
15635:
15630:
15627:
15619:
15609:
15607:
15606:exact formula
15603:
15587:
15579:
15563:
15555:
15539:
15531:
15526:
15512:
15509:
15489:
15486:
15483:
15475:
15471:
15467:
15463:
15457:
15443:
15440:
15437:
15417:
15403:
15401:
15391:
15387:
15363:
15360:
15357:
15352:
15348:
15344:
15338:
15335:
15330:
15327:
15319:
15315:
15311:
15308:
15305:
15297:
15294:
15291:
15287:
15283:
15278:
15273:
15270:
15267:
15263:
15254:
15251:
15248:
15244:
15240:
15232:
15228:
15224:
15219:
15214:
15210:
15199:
15198:
15197:
15195:
15176:
15171:
15166:
15162:
15156:
15152:
15148:
15145:
15142:
15137:
15132:
15128:
15122:
15118:
15114:
15109:
15105:
15097:
15096:
15095:
15081:
15076:
15072:
15067:
15064:
15060:
15055:
15051:
15042:
15024:
15016:
15013:
15010:
15007:
15004:
15000:
14994:
14987:
14983:
14976:
14966:
14950:
14945:
14942:
14939:
14936:
14933:
14929:
14925:
14920:
14917:
14914:
14911:
14908:
14904:
14897:
14894:
14888:
14883:
14880:
14877:
14873:
14869:
14864:
14861:
14858:
14854:
14845:
14826:
14823:
14820:
14815:
14811:
14807:
14804:
14799:
14796:
14793:
14789:
14785:
14782:
14779:
14774:
14771:
14768:
14764:
14758:
14754:
14750:
14745:
14742:
14739:
14735:
14729:
14725:
14721:
14716:
14712:
14704:
14703:
14702:
14700:
14696:
14677:
14672:
14669:
14666:
14662:
14656:
14652:
14648:
14645:
14642:
14637:
14634:
14631:
14627:
14621:
14617:
14613:
14608:
14605:
14602:
14598:
14592:
14588:
14584:
14579:
14575:
14567:
14566:
14565:
14563:
14560:The simplest
14553:
14551:
14547:
14543:
14541:
14540:weight spaces
14537:
14533:
14528:
14526:
14522:
14518:
14514:
14508:
14498:
14496:
14492:
14488:
14484:
14480:
14477:
14473:
14468:
14466:
14462:
14458:
14454:
14450:
14446:
14442:
14438:
14434:
14430:
14424:
14414:
14412:
14408:
14404:
14400:
14396:
14392:
14388:
14384:
14380:
14376:
14372:
14368:
14364:
14359:
14357:
14353:
14349:
14345:
14341:
14337:
14332:
14330:
14325:
14321:
14317:
14313:
14309:
14304:
14300:
14297:
14292:
14290:
14286:
14282:
14278:
14274:
14270:
14266:
14263:
14259:
14255:
14251:
14247:
14243:
14239:
14235:
14230:
14213:
14202:
14196:
14193:
14191:
14178:
14172:
14165:
14154:
14143:
14140:
14138:
14125:
14114:
14102:
14098:
14094:
14090:
14086:
14082:
14078:
14074:
14070:
14066:
14062:
14058:
14053:
14036:
14022:
14019:
14016:
14014:
14001:
13995:
13988:
13974:
13971:
13957:
13954:
13952:
13939:
13928:
13915:
13913:
13909:
13905:
13901:
13897:
13893:
13888:
13875:
13871:
13862:
13859:
13845:
13842:
13833:
13829:
13826:
13818:
13808:
13806:
13802:
13798:
13795:
13791:
13787:
13783:
13779:
13775:
13771:
13767:
13763:
13759:
13750:
13743:
13741:
13727:
13719:
13716:
13702:
13695:
13694:
13691:
13689:
13685:
13681:
13677:
13673:
13669:
13664:
13651:
13648:
13642:
13639:
13636:
13628:
13624:
13620:
13616:
13612:
13609:
13605:
13601:
13591:
13589:
13588:eigenfunction
13584:
13582:
13578:
13574:
13558:
13553:
13550:
13546:
13539:
13533:
13530:
13524:
13518:
13511:
13507:
13503:
13499:
13495:
13490:
13477:
13471:
13465:
13462:
13459:
13453:
13447:
13441:
13438:
13434:
13408:
13405:
13401:
13384:
13382:
13378:
13373:
13357:
13351:
13348:
13345:
13339:
13333:
13330:
13323:
13319:
13315:
13311:
13307:
13303:
13299:
13295:
13291:
13287:
13283:
13277:
13276:Eigenfunction
13267:
13265:
13261:
13257:
13239:
13235:
13214:
13202:
13198:
13194:
13189:
13185:
13175:
13172:
13169:
13165:
13150:
13146:
13137:
13133:
13129:
13124:
13120:
13110:
13106:
13099:
13094:
13082:
13079:
13076:
13072:
13058:
13054:
13050:
13040:
13037:
13033:
13007:
13001:
12996:
12991:
12986:
12980:
12947:
12941:
12936:
12933:
12928:
12923:
12917:
12906:
12901:
12899:
12895:
12890:
12886:
12882:
12877:
12864:
12859:
12851:
12848:
12845:
12837:
12829:
12826:
12823:
12817:
12812:
12806:
12803:
12800:
12795:
12790:
12785:
12778:
12773:
12770:
12767:
12762:
12757:
12750:
12745:
12740:
12737:
12734:
12729:
12722:
12717:
12712:
12707:
12704:
12701:
12695:
12690:
12684:
12681:
12678:
12675:
12649:
12644:
12638:
12633:
12628:
12623:
12616:
12611:
12606:
12601:
12594:
12589:
12584:
12579:
12572:
12567:
12562:
12557:
12551:
12546:
12543:
12529:
12526:
12513:
12508:
12502:
12495:
12488:
12482:
12477:
12470:
12466:
12454:
12449:
12443:
12440:
12433:
12426:
12420:
12415:
12408:
12404:
12392:
12387:
12379:
12376:
12367:
12364:
12357:
12351:
12346:
12339:
12335:
12318:
12316:
12307:
12297:
12287:
12281:
12268:
12262:
12259:
12256:
12247:
12244:
12241:
12232:
12229:
12226:
12220:
12214:
12211:
12208:
12205:
12191:
12186:
12173:
12168:
12162:
12157:
12152:
12145:
12140:
12135:
12128:
12123:
12118:
12112:
12107:
12104:
12095:
12093:
12089:
12088:
12076:
12073:
12060:
12055:
12049:
12042:
12035:
12029:
12024:
12017:
12013:
12001:
11996:
11990:
11983:
11976:
11970:
11965:
11958:
11954:
11942:
11937:
11931:
11924:
11917:
11911:
11906:
11899:
11895:
11878:
11876:
11867:
11857:
11847:
11841:
11828:
11822:
11819:
11816:
11807:
11804:
11801:
11792:
11789:
11786:
11780:
11774:
11771:
11768:
11765:
11751:
11746:
11733:
11728:
11722:
11717:
11712:
11705:
11700:
11695:
11688:
11683:
11678:
11672:
11667:
11664:
11656:
11655:
11644:
11631:
11626:
11619:
11615:
11604:
11597:
11593:
11574:
11567:
11541:
11533:
11529:
11521:
11517:
11511:
11505:
11499:
11492:
11488:
11450:
11442:
11438:
11430:
11426:
11420:
11414:
11408:
11401:
11397:
11381:
11376:
11363:
11358:
11350:
11346:
11336:
11332:
11324:
11318:
11313:
11308:
11304:
11300:
11295:
11289:
11280:
11276:
11266:
11262:
11255:
11250:
11245:
11237:
11233:
11223:
11219:
11211:
11205:
11200:
11180:
11175:
11167:
11163:
11153:
11149:
11141:
11135:
11130:
11125:
11121:
11117:
11112:
11106:
11097:
11093:
11083:
11079:
11072:
11067:
11062:
11054:
11050:
11040:
11036:
11028:
11022:
11017:
11008:
10995:
10990:
10986:
10982:
10977:
10972:
10968:
10963:
10958:
10954:
10950:
10945:
10940:
10936:
10931:
10928:
10925:
10920:
10916:
10910:
10906:
10896:
10883:
10878:
10872:
10865:
10858:
10852:
10847:
10844:
10841:
10836:
10830:
10823:
10816:
10810:
10805:
10800:
10794:
10787:
10780:
10774:
10769:
10758:
10753:
10738:
10735:
10732:
10727:
10723:
10713:
10697:
10671:
10667:
10661:
10658:
10653:
10650:
10645:
10642:
10637:
10632:
10628:
10624:
10622:
10615:
10611:
10601:
10597:
10591:
10588:
10583:
10580:
10575:
10572:
10570:
10563:
10559:
10551:
10548:
10546:
10539:
10535:
10522:
10517:
10504:
10499:
10493:
10488:
10483:
10476:
10471:
10466:
10459:
10454:
10449:
10443:
10438:
10435:
10428:
10425:Consider the
10418:
10392:
10386:
10381:
10376:
10370:
10334:
10328:
10323:
10320:
10315:
10309:
10274:
10268:
10263:
10258:
10252:
10240:
10235:
10218:
10215:
10212:
10209:
10206:
10201:
10197:
10193:
10190:
10185:
10181:
10177:
10174:
10164:
10161:
10155:
10152:
10149:
10140:
10137:
10134:
10118:
10115:
10112:
10106:
10104:
10096:
10091:
10085:
10082:
10079:
10074:
10069:
10062:
10057:
10054:
10051:
10046:
10039:
10034:
10029:
10026:
10023:
10017:
10012:
10008:
10002:
9996:
9991:
9986:
9979:
9974:
9969:
9962:
9957:
9952:
9946:
9941:
9938:
9933:
9927:
9922:
9917:
9910:
9905:
9900:
9893:
9888:
9883:
9877:
9871:
9867:
9865:
9857:
9854:
9851:
9848:
9830:
9825:
9812:
9807:
9801:
9796:
9791:
9784:
9779:
9774:
9767:
9762:
9757:
9751:
9746:
9743:
9729:
9725:
9718:
9713:
9707:
9703:
9697:
9693:
9688:
9686:
9682:
9677:
9662:
9656:
9649:
9643:
9638:
9633:
9625:
9621:
9611:
9607:
9600:
9595:
9590:
9587:
9584:
9566:
9559:
9554:
9537:
9534:
9532:
9525:
9521:
9517:
9514:
9509:
9505:
9501:
9494:
9491:
9488:
9486:
9479:
9475:
9471:
9468:
9463:
9459:
9455:
9452:
9443:
9437:
9430:
9424:
9419:
9414:
9406:
9402:
9392:
9388:
9381:
9374:
9368:
9365:
9360:
9353:
9348:
9345:
9339:
9334:
9332:
9325:
9322:
9319:
9306:
9303:
9300:
9297:
9282:
9281:
9274:
9268:
9266:
9262:
9244:
9238:
9235:
9228:
9222:
9217:
9212:
9204:
9200:
9196:
9187:
9183:
9176:
9171:
9166:
9163:
9160:
9142:
9135:
9130:
9117:
9114:
9109:
9105:
9101:
9098:
9093:
9089:
9085:
9064:
9058:
9051:
9045:
9040:
9035:
9027:
9023:
9013:
9009:
9002:
8995:
8989:
8984:
8977:
8972:
8966:
8961:
8956:
8953:
8950:
8937:
8934:
8931:
8920:
8919:
8912:
8906:
8904:
8898:
8891:
8885:
8868:
8862:
8859:
8856:
8847:
8844:
8841:
8835:
8833:
8823:
8819:
8815:
8812:
8809:
8806:
8803:
8800:
8798:
8788:
8782:
8779:
8776:
8771:
8764:
8759:
8756:
8753:
8747:
8742:
8738:
8732:
8726:
8721:
8714:
8709:
8703:
8698:
8695:
8690:
8684:
8679:
8672:
8667:
8661:
8655:
8651:
8649:
8641:
8638:
8635:
8632:
8614:
8609:
8607:
8604: −
8603:
8599:
8595:
8594:
8589:
8584:
8571:
8566:
8560:
8555:
8548:
8543:
8537:
8532:
8529:
8517:
8511:
8507:
8501:
8497:
8480:
8472:
8467:
8460:
8455:
8447:
8438:
8433:
8419:
8396:. A value of
8363:
8354:
8333:
8317:
8309:
8303:
8293:
8291:
8287:
8283:
8279:
8275:
8271:
8266:
8264:
8260:
8256:
8252:
8248:
8244:
8240:
8236:
8232:
8227:
8223:
8218:
8214:
8210:
8207:
8203:
8199:
8194:
8192:
8188:
8185:. The matrix
8184:
8183:
8178:
8174:
8170:
8166:
8162:
8144:
8138:
8135:
8132:
8127:
8124:
8120:
8112:
8111:
8110:
8108:
8089:
8084:
8081:
8077:
8070:
8067:
8064:
8057:
8056:
8055:
8053:
8049:
8030:
8024:
8021:
8018:
8015:
8008:
8007:
8006:
8004:
8001:th column of
8000:
7995:
7975:
7970:
7962:
7950:
7946:
7940:
7933:
7921:
7917:
7909:
7897:
7893:
7886:
7881:
7878:
7875:
7868:
7867:
7866:
7864:
7860:
7856:
7852:
7848:
7829:
7824:
7816:
7804:
7797:
7783:
7770:
7765:
7762:
7755:
7754:
7753:
7751:
7747:
7743:
7740:
7739:square matrix
7735:
7731:
7724:
7717:
7712:
7708:
7701:
7694:
7690:
7686:
7682:
7676:
7666:
7652:
7632:
7604:
7575:
7554:
7546:
7545:
7539:
7526:
7509:
7506:
7481:
7460:
7440:
7432:
7416:
7396:
7393:
7390:
7370:
7350:
7341:
7328:
7320:
7317:
7314:
7289:
7281:
7278:vectors that
7277:
7272:
7259:
7251:
7248:
7240:
7232:
7231:
7214:
7194:
7191:
7188:
7180:
7176:
7172:
7165:
7132:
7128:
7121:
7118:
7115:
7112:
7104:
7100:
7093:
7067:
7061:
7041:
7018:
7015:
7010:
7006:
7002:
6999:
6996:
6993:
6990:
6985:
6981:
6957:
6954:
6951:
6948:
6923:
6920:
6897:
6894:
6889:
6885:
6881:
6878:
6875:
6872:
6869:
6864:
6860:
6836:
6816:
6813:
6810:
6785:
6781:
6777:
6774:
6771:
6766:
6762:
6738:
6735:
6732:
6712:
6704:
6689:
6686:
6676:
6672:
6658:
6642:
6634:
6631:
6615:
6607:
6604:
6600:
6584:
6562:
6558:
6550:
6534:
6526:
6523:
6503:
6499:
6495:
6490:
6487:
6484:
6477:
6473:
6469:
6445:
6442:
6438:
6417:
6409:
6406:
6390:
6382:
6365:
6360:
6356:
6352:
6349:
6346:
6341:
6336:
6332:
6311:
6289:
6285:
6264:
6244:
6236:
6220:
6215:
6211:
6207:
6202:
6198:
6192:
6188:
6184:
6179:
6175:
6169:
6164:
6161:
6158:
6154:
6150:
6144:
6131:
6130:
6116:
6108:
6104:
6088:
6083:
6079:
6075:
6072:
6069:
6064:
6060:
6056:
6051:
6047:
6043:
6038:
6034:
6028:
6023:
6020:
6017:
6013:
6009:
6004:
6001:
5997:
5991:
5986:
5983:
5980:
5976:
5972:
5966:
5960:
5957:
5950:
5949:
5935:
5927:
5923:
5922:
5921:
5902:
5898:
5889:
5885:
5859:
5855:
5846:
5842:
5819:
5815:
5811:
5808:
5805:
5800:
5796:
5775:
5772:
5769:
5749:
5720:
5698:
5683:
5681:
5665:
5645:
5623:
5608:
5591:
5564:
5556:
5555:
5554:
5540:
5537:
5532:
5528:
5507:
5487:
5479:
5459:
5456:
5453:
5448:
5444:
5440:
5438:
5433:
5426:
5418:
5414:
5405:
5401:
5395:
5390:
5387:
5384:
5380:
5376:
5374:
5367:
5363:
5338:
5313:
5309:
5300:
5296:
5273:
5269:
5246:
5242:
5238:
5235:
5232:
5227:
5223:
5202:
5199:
5196:
5176:
5167:
5150:
5142:
5138:
5134:
5128:
5120:
5116:
5108:must satisfy
5095:
5070:
5062:
5058:
5049:
5046:
5043:
5017:
5014:
5011:
5008:
4982:
4959:
4956:
4953:
4950:
4941:
4935:
4932:
4929:
4926:
4900:
4897:
4894:
4891:
4871:
4868:
4865:
4862:
4842:
4822:
4799:
4796:
4793:
4790:
4784:
4781:
4778:
4772:
4769:
4766:
4763:
4740:
4737:
4734:
4709:
4701:
4697:
4692:
4688:
4668:
4665:
4660:
4656:
4652:
4649:
4629:
4609:
4586:
4578:
4574:
4570:
4567:
4544:
4536:
4532:
4511:
4489:
4479:
4476:
4471:
4461:
4436:
4428:
4424:
4412:
4409:
4405:
4400:
4386:eigenvectors
4385:
4367:
4359:
4355:
4331:
4323:
4319:
4315:
4309:
4301:
4297:
4287:
4274:
4271:
4265:
4257:
4253:
4249:
4243:
4235:
4231:
4227:
4224:
4216:
4210:
4197:
4191:
4188:
4185:
4182:
4176:
4173:
4170:
4167:
4164:
4158:
4150:
4146:
4137:
4133:
4129:
4125:
4121:
4117:
4113:
4109:
4105:
4101:
4097:
4078:
4070:
4066:
4058:
4054:
4050:
4046:
4041:
4039:
4035:
4031:
4028:
4024:
4020:
4017:. As long as
4016:
4010:
4007:
4003:
3999:
3996:
3992:
3986:
3982:
3979:
3973:
3968:
3964:
3959:
3955:
3949:
3945:
3941:
3937:
3933:
3929:
3923:
3919:
3915:
3908:
3904:
3900:
3895:
3891:
3887:
3883:
3879:
3874:
3858:
3843:
3839:
3835:
3831:
3828:. In general
3827:
3823:
3819:
3815:
3811:
3807:
3803:
3799:
3795:
3791:
3787:
3783:
3779:
3775:
3771:
3767:
3763:
3758:
3745:
3741:
3732:
3723:
3719:
3716:
3713:
3710:
3706:
3702:
3693:
3689:
3686:
3678:
3677:
3672:
3668:
3665:
3662:, define the
3661:
3657:
3653:
3649:
3639:
3637:
3632:
3628:
3623:
3619:
3614:
3610:
3605:
3601:
3596:
3592:
3587:
3583:
3579:
3574:
3570:
3565:
3561:
3556:
3552:
3547:
3530:
3527:
3524:
3520:
3515:
3511:
3507:
3501:
3497:
3491:
3486:
3483:
3480:
3476:
3472:
3470:
3463:
3459:
3451:
3448:
3445:
3437:
3433:
3424:
3420:
3416:
3414:
3409:
3397:
3393:
3392:
3387:
3383:
3379:
3374:
3361:
3351:
3347:
3338:
3334:
3325:
3322:
3317:
3313:
3306:
3296:
3292:
3283:
3279:
3270:
3267:
3262:
3258:
3244:
3240:
3231:
3227:
3218:
3215:
3210:
3206:
3199:
3193:
3190:
3187:
3184:
3170:
3166:
3162:
3158:
3157:
3152:
3148:
3144:
3140:
3135:
3133:
3128:
3124:
3120:
3116:
3112:
3107:
3103:
3098:
3094:
3091:
3087:
3083:
3079:
3074:
3070:
3060:
3058:
3053:
3051:
3050:
3039:
3036:
3032:
3028:
3023:
3021:
3017:
3013:
3009:
3005:
3000:
2995:
2982:
2977:
2971:
2964:
2958:
2953:
2948:
2945:
2942:
2931:
2926:
2920:
2917:
2910:
2904:
2899:
2894:
2891:
2888:
2854:
2845:
2841:
2838:
2835:
2832:
2828:
2818:
2814:
2801:
2788:
2783:
2779:
2775:
2772:
2769:
2766:
2763:
2760:
2755:
2749:
2746:
2743:
2738:
2731:
2726:
2723:
2720:
2714:
2709:
2703:
2700:
2697:
2694:
2680:
2674:
2670:
2663:
2650:
2645:
2639:
2634:
2627:
2622:
2616:
2611:
2608:
2599:
2597:
2592:
2588:
2581:
2574:
2569:
2565:
2556:
2549:
2547:
2533:
2527:
2524:
2519:
2515:
2508:
2502:
2499:
2494:
2490:
2480:
2477:
2472:
2468:
2461:
2455:
2452:
2449:
2446:
2433:
2432:
2429:
2427:
2423:
2419:
2415:
2411:
2407:
2403:
2398:
2396:
2392:
2388:
2384:
2383:
2378:
2374:
2373:
2368:
2364:
2360:
2356:
2352:
2348:
2344:
2340:
2336:
2332:
2331:
2326:
2317:
2310:
2308:
2294:
2291:
2285:
2282:
2279:
2276:
2263:
2262:
2259:
2257:
2253:
2247:
2243:
2237:
2233:
2230:
2226:
2225:
2218:
2208:
2206:
2202:
2182:
2175:
2173:
2159:
2151:
2142:
2138:
2135:
2132:
2129:
2125:
2117:
2116:
2113:
2111:
2110:
2104:
2098:
2094:
2093:
2088:
2076:
2071:
2061:
2054:
2052:
2038:
2030:
2027:
2019:
2011:
2004:
2003:
2000:
1989:
1976:
1971:
1967:
1961:
1958:
1954:
1948:
1943:
1940:
1937:
1933:
1929:
1924:
1920:
1914:
1911:
1907:
1903:
1900:
1897:
1892:
1888:
1882:
1879:
1875:
1871:
1866:
1862:
1856:
1853:
1849:
1845:
1840:
1836:
1813:
1805:
1801:
1793:
1784:
1780:
1770:
1766:
1759:
1754:
1749:
1741:
1737:
1729:
1720:
1716:
1706:
1702:
1695:
1688:
1680:
1677:
1673:
1667:
1660:
1657:
1653:
1645:
1642:
1638:
1630:
1625:
1620:
1615:
1606:
1603:
1599:
1593:
1586:
1582:
1574:
1570:
1560:
1557:
1553:
1547:
1540:
1536:
1528:
1524:
1517:
1495:
1487:
1479:
1454:
1438:
1435:
1430:
1427:
1424:
1415:
1402:
1394:
1391:
1374:
1370:
1366:
1361:
1348:
1343:
1337:
1334:
1327:
1320:
1317:
1311:
1306:
1287:
1281:
1274:
1271:
1264:
1258:
1253:
1228:
1224:
1220:
1216:
1211:
1207:
1202:
1198:
1188:
1186:
1182:
1178:
1174:
1170:
1165:
1163:
1159:
1155:
1151:
1147:
1146:David Hilbert
1142:
1140:
1136:
1132:
1128:
1124:
1123:
1118:
1113:
1111:
1107:
1103:
1099:
1095:
1091:
1087:
1083:
1078:
1076:
1072:
1068:
1064:
1060:
1059:
1054:
1050:
1049:heat equation
1047:to solve the
1046:
1042:
1037:
1035:
1034:
1029:
1025:
1021:
1017:
1012:
1010:
1006:
1002:
998:
993:
991:
987:
983:
982:matrix theory
979:
966:
962:
958:
954:
950:
947:
943:
939:
935:
931:
928:
924:
923:
922:
920:
915:
913:
912:diagonalizing
909:
905:
901:
885:
877:
874:
866:
858:
854:
850:
846:
842:
838:
822:
817:
814:
810:
806:
803:
798:
795:
791:
784:
781:
777:
767:
747:
744:
740:
729:
724:
721:
717:
716:shear mapping
713:
703:
695:
694:shear mapping
690:
686:
684:
680:
676:
672:
668:
667:eigenequation
664:
648:
640:
637:
623:
615:
611:
607:
603:
599:
594:
592:
588:
584:
580:
576:
572:
568:
564:
563:
558:
554:
550:
549:
544:
540:
539:
528:
526:
521:
519:
515:
511:
509:
504:
501:
497:
492:
473:
470:
462:
424:
373:
327:
324:
278:
274:
270:
256:
254:
250:
246:
242:
237:
234:
230:
226:
222:
218:
214:
200:
192:
188:
184:
163:
160:
152:
132:
124:
108:
78:
74:
70:
66:
65:
56:
35:
31:
26:
22:
25416:
25404:
25392:
25373:
25306:Optimization
25168:Differential
25092:Differential
25059:Order theory
25054:Graph theory
24958:Group theory
24715:Vector space
24516:
24447:Vector space
24365:James Demmel
24343:
24335:
24259:
24250:
24235:by removing
24222:
24185:
24167:
24154:
24136:
24098:
24094:
24059:
24055:
24038:
24025:. Retrieved
24021:
24005:. Retrieved
24001:
23998:"Eigenvalue"
23985:. Retrieved
23981:
23928:
23924:
23896:
23864:
23860:
23842:
23799:
23795:
23773:
23755:
23731:
23713:
23693:
23676:
23672:
23646:
23618:
23614:
23596:
23578:
23568:
23557:
23538:
23534:
23502:
23496:
23478:
23453:
23435:
23431:
23413:
23407:
23385:
23343:
23339:
23296:(1): 31–58.
23293:
23289:
23262:
23243:, New York:
23240:
23231:
23172:
23168:
23154:
23129:
23105:
23097:
23071:
23059:
23047:
23035:
23026:
23016:
23004:
22992:
22980:
22968:
22941:
22917:, p. 189 §8.
22910:
22898:
22879:
22867:
22855:
22843:
22831:
22819:
22792:
22780:
22768:
22756:
22744:
22717:
22688:
22676:
22660:
22654:
22645:
22639:
22627:
22615:
22603:
22596:Francis 1961
22591:
22584:Aldrich 2006
22579:
22567:
22555:
22543:
22516:
22509:Hawkins 1975
22490:Hawkins 1975
22485:
22454:
22442:
22430:. Retrieved
22426:
22402:
22359:
22352:
22340:
22253:
22225:
22203:
22190:
22178:
22170:
22166:
22164:From p. 827:
22159:
22155:
22147:
22139:pp. 225–226.
22134:
22130:
22119:
22115:
22107:
22096:
22088:
22076:
22017:Eigenmoments
21988:
21986:
21955:
21944:brightnesses
21937:
21777:
21518:compass rose
21316:
21308:glacial till
21301:
21286:
21277:
21261:
21192:theory, the
21190:Hartree–Fock
21175:
20971:
20801:
20764:
20657:
20560:
20532:wavefunction
20474:
20393:
20305:bound states
20270:
20234:
20218:
20211:
19905:dimensions,
19884:
19861:
19766:
19756:
19724:Markov chain
19721:
19707:Markov chain
19684:
19433:
19361:
19294:
19264:
19246:
19205:, where the
19181:
19086:centered at
19065:
18966:discriminant
18959:
16641:
16633:Applications
16627:
16604:QR algorithm
16601:
16568:denotes the
16538:
16422:
16336:
16165:
16014:
15828:
15680:
15615:
15612:Eigenvectors
15527:
15458:
15409:
15397:
15389:
15196:of the form
15191:
15040:
14964:
14843:
14841:
14698:
14697:in terms of
14694:
14692:
14559:
14544:
14539:
14535:
14529:
14519:acting on a
14510:
14490:
14486:
14482:
14478:
14469:
14464:
14460:
14456:
14452:
14448:
14440:
14436:
14432:
14428:
14426:
14410:
14406:
14402:
14398:
14394:
14390:
14386:
14382:
14378:
14374:
14370:
14362:
14360:
14355:
14351:
14347:
14343:
14335:
14333:
14328:
14323:
14319:
14315:
14311:
14307:
14302:
14298:
14295:
14293:
14288:
14284:
14280:
14276:
14272:
14268:
14264:
14261:
14257:
14253:
14249:
14245:
14241:
14237:
14233:
14231:
14100:
14096:
14092:
14088:
14084:
14080:
14076:
14072:
14068:
14064:
14060:
14056:
14054:
13916:
13911:
13907:
13903:
13899:
13895:
13891:
13889:
13816:
13814:
13804:
13800:
13796:
13793:
13789:
13785:
13781:
13777:
13773:
13769:
13765:
13761:
13757:
13755:
13744:
13687:
13683:
13679:
13675:
13671:
13667:
13665:
13626:
13622:
13618:
13610:
13603:
13597:
13585:
13580:
13576:
13572:
13501:
13497:
13493:
13491:
13390:
13380:
13376:
13374:
13317:
13313:
13305:
13301:
13290:Banach space
13281:
13279:
13259:
13057:minor matrix
13052:
13046:
13035:
13031:
12904:
12902:
12897:
12893:
12888:
12884:
12880:
12878:
12535:
12527:
12319:
12314:
12305:
12295:
12285:
12282:
12189:
12187:
12096:
12091:
12084:
12082:
12074:
11879:
11874:
11865:
11855:
11845:
11842:
11749:
11747:
11652:
11650:
11572:
11565:
11379:
11377:
11009:
10897:
10756:
10754:
10520:
10518:
10424:
10238:
10236:
9828:
9826:
9735:
9723:
9716:
9711:
9705:
9701:
9695:
9691:
9689:
9684:
9680:
9678:
9564:
9557:
9555:
9278:
9277:, equation (
9272:
9269:
9264:
9260:
9140:
9133:
9131:
8916:
8915:, equation (
8910:
8907:
8902:
8896:
8889:
8886:
8612:
8610:
8605:
8601:
8597:
8591:
8587:
8585:
8521:
8509:
8505:
8499:
8495:
8436:
8305:
8281:
8267:
8262:
8258:
8254:
8250:
8246:
8242:
8238:
8234:
8230:
8225:
8221:
8216:
8212:
8208:
8205:
8201:
8197:
8195:
8190:
8186:
8180:
8176:
8172:
8167:and it is a
8160:
8159:
8106:
8104:
8051:
8047:
8045:
8002:
7998:
7993:
7990:
7862:
7858:
7854:
7850:
7846:
7844:
7749:
7745:
7741:
7733:
7729:
7722:
7715:
7710:
7706:
7699:
7692:
7688:
7684:
7680:
7678:
7542:
7540:
7430:
7342:
7279:
7275:
7273:
7228:
7178:
7177:vector that
7174:
7170:
7167:
6605:real matrix.
6257:th power of
5741:
5679:
5168:
4524:whose first
4454:, such that
4288:
4214:
4211:
4135:
4131:
4127:
4123:
4119:
4115:
4111:
4107:
4103:
4099:
4095:
4056:
4052:
4048:
4044:
4042:
4037:
4033:
4029:
4026:
4022:
4018:
4008:
4005:
4001:
3997:
3994:
3990:
3984:
3980:
3977:
3971:
3966:
3962:
3957:
3947:
3943:
3939:
3935:
3931:
3927:
3921:
3917:
3913:
3906:
3902:
3898:
3893:
3889:
3885:
3877:
3875:
3841:
3833:
3829:
3825:
3821:
3817:
3813:
3809:
3805:
3801:
3797:
3793:
3789:
3781:
3777:
3773:
3769:
3765:
3759:
3674:
3670:
3666:
3659:
3655:
3651:
3647:
3645:
3635:
3630:
3626:
3621:
3617:
3612:
3608:
3603:
3599:
3594:
3590:
3585:
3581:
3577:
3572:
3568:
3567:) = 1, then
3563:
3559:
3554:
3550:
3548:
3395:
3389:
3385:
3381:
3377:
3375:
3168:
3164:
3160:
3154:
3150:
3146:
3142:
3138:
3136:
3126:
3122:
3118:
3114:
3105:
3101:
3096:
3092:
3089:
3085:
3081:
3077:
3072:
3068:
3066:
3054:
3047:
3045:
3024:
3015:
3007:
2998:
2996:
2816:
2812:
2802:
2678:
2672:
2668:
2664:
2600:
2595:
2590:
2586:
2579:
2572:
2567:
2563:
2561:
2550:
2425:
2417:
2413:
2409:
2405:
2399:
2394:
2390:
2386:
2380:
2379:. Equation (
2376:
2370:
2366:
2362:
2358:
2355:coefficients
2350:
2346:
2338:
2328:
2322:
2311:
2255:
2251:
2245:
2241:
2228:
2222:
2220:
2204:
2187:
2176:
2107:
2105:
2096:
2090:
2086:
2074:
2069:
2066:
2055:
1990:
1455:
1416:
1362:
1232:
1226:
1222:
1218:
1214:
1204:
1177:QR algorithm
1173:power method
1166:
1157:
1143:
1120:
1114:
1079:
1056:
1038:
1031:
1027:
1023:
1013:
994:
975:
964:
960:
952:
945:
941:
937:
933:
926:
918:
916:
903:
899:
856:
852:
848:
844:
840:
836:
725:
719:
709:
678:
670:
666:
662:
613:
609:
605:
601:
597:
595:
571:rigid bodies
560:
546:
536:
534:
522:
510:-dimensional
507:
499:
495:
493:
366:(denoted by
263:Consider an
262:
253:steady state
238:
215:
190:
186:
182:
68:
33:
27:
25:
25418:WikiProject
25261:Game theory
25241:Probability
24978:Homological
24968:Multilinear
24948:Commutative
24925:Type theory
24892:Foundations
24848:mathematics
24695:Multivector
24660:Determinant
24617:Dot product
24462:Linear span
24299:3Blue1Brown
24141:Brady Haran
24027:27 November
22888:Shilov 1977
22884:Nering 1970
22874:, p. 186 §8
22785:Nering 1970
22761:Nering 1970
22478:Nering 1970
22395:Nering 1970
22243:Beezer 2006
22235:Shilov 1977
22195:From p. 51:
21989:eigenvoices
21220:method. In
20563:bound state
20497:Hamiltonian
20380:measurement
20243:define the
19927:mass matrix
19330:due to its
19082:PCA of the
17063:polynomial
16340:A variation
16166:The matrix
15604:. Even the
15474:determinant
15406:Eigenvalues
15380:Calculation
13676:eigenvector
13506:integrating
8921:) becomes,
7173:, namely a
6751:matrix and
6383:The matrix
6107:determinant
5609:A basis of
4384:orthonormal
4015:commutative
3117:such that (
3035:real matrix
2562:where each
2236:determinant
2075:eigenvector
1100:. Finally,
1090:unit circle
1088:lie on the
927:eigensystem
673:may be any
221:dimensional
34:eigenvector
25434:Categories
25246:Statistics
25125:Arithmetic
25087:Arithmetic
24953:Elementary
24920:Set theory
24729:Direct sum
24564:Invertible
24467:Linear map
24062:: 91–134.
23809:1702.05395
23621:(3): 243,
23303:1908.03795
23182:2004.12167
23134:, Boston:
22915:Roman 2008
22872:Roman 2008
22722:Anton 1987
22632:Meyer 2000
22572:Kline 1972
22560:Kline 1972
22548:Kline 1972
22536:Kline 1972
22521:Kline 1972
22241:, p. 364;
22237:, p. 109;
22231:Roman 2008
22152:Kline 1972
22012:Eigenplane
21974:biometrics
21972:branch of
21957:eigenfaces
21926:Eigenfaces
21917:Eigenfaces
21226:orthogonal
20981:microwaves
20866:observable
20802:eigenstate
20249:rigid body
19925:becomes a
19687:centrality
19197:yields an
19172:See also:
18187:Geometric
17851:Algebraic
15713:, that is
14445:one-to-one
14399:eigenbasis
14397:called an
14340:direct sum
13900:eigenspace
13766:eigenvalue
13690:such that
13617:, and let
9283:) becomes
7162:See also:
6405:invertible
5680:eigenbasis
4094:. Because
3896:, written
3814:eigenspace
2335:polynomial
2323:Using the
2221:Equation (
2106:Equation (
2087:eigenvalue
1379:such that
1195:See also:
1028:eigenvalue
1001:rigid body
965:eigenbasis
938:eigenspace
317:of length
259:Definition
219:are multi-
183:eigenvalue
25173:Geometric
25163:Algebraic
25102:Euclidean
25077:Algebraic
24973:Universal
24759:Numerical
24522:Transpose
24317:×
24237:excessive
24125:1874/2663
24007:19 August
23953:1539-4794
23911:1401.4580
23889:129658242
23834:119330480
23527:128825838
23361:1874/8051
23320:213918682
23223:216553547
23207:0031-9007
22862:, p. 217.
22523:, p. 673.
22432:19 August
22333:Citations
22257:By doing
22102:In 1755,
22093:On p. 212
21934:Eigenface
21407:≥
21394:≥
21229:basis set
21214:iteration
21144:τ
21111:τ
21082:†
21055:τ
21028:†
20985:scattered
20952:⟩
20943:Ψ
20897:⟩
20888:Ψ
20788:⟩
20779:Ψ
20750:⟩
20741:Ψ
20726:⟩
20717:Ψ
20685:⟩
20676:Ψ
20617:ψ
20608:in which
20606:basis set
20574:ψ
20512:ψ
20453:ψ
20437:ψ
20358:…
20237:mechanics
20175:ω
20160:ω
20112:˙
20094:¨
20033:ω
20007:ω
19974:ω
19842:−
19833:¨
19786:¨
19753:Vibration
19592:
19526:−
19513:, and in
19436:), where
19407:−
19383:−
19375:−
19345:−
19265:practical
19253:data sets
19188:symmetric
19051:θ
19048:
19039:±
19036:θ
19033:
18996:θ
18993:
18981:−
18928:−
18694:−
18487:γ
18463:γ
18425:γ
18383:γ
18359:γ
18313:γ
18289:γ
18251:γ
18220:λ
18213:γ
18201:γ
18151:μ
18127:μ
18089:μ
18047:μ
18023:μ
17977:μ
17953:μ
17915:μ
17884:λ
17877:μ
17865:μ
17831:φ
17828:
17822:−
17819:φ
17816:
17798:φ
17795:−
17774:λ
17766:φ
17763:
17754:φ
17751:
17733:φ
17712:λ
17674:λ
17661:λ
17635:θ
17632:
17623:−
17620:θ
17617:
17599:θ
17593:−
17572:λ
17564:θ
17561:
17549:θ
17546:
17528:θ
17504:λ
17451:λ
17420:λ
17382:λ
17369:λ
17341:λ
17310:λ
17304:φ
17298:
17289:−
17280:λ
17245:−
17242:λ
17209:λ
17203:θ
17197:
17188:−
17179:λ
17144:−
17141:λ
17122:−
17119:λ
17082:−
17079:λ
17037:φ
17034:
17026:φ
17023:
17013:φ
17010:
17002:φ
16999:
16905:θ
16902:
16894:θ
16891:
16881:θ
16878:
16872:−
16867:θ
16864:
16612:Hermitian
16554:∗
16514:∗
16492:∗
16476:λ
16402:∈
16399:μ
16371:−
16360:μ
16357:−
16269:−
16194:λ
16145:λ
15978:−
15943:−
15789:⋅
15580:of order
15528:Explicit
15487:×
15441:×
15309:⋯
15295:−
15271:−
15252:−
15163:λ
15146:⋯
15129:λ
15073:λ
15065:…
15052:λ
15008:−
14995:⋯
14963:giving a
14937:−
14912:−
14895:…
14881:−
14862:−
14808:−
14805:λ
14797:−
14786:−
14783:⋯
14780:−
14772:−
14765:λ
14751:−
14743:−
14736:λ
14722:−
14713:λ
14670:−
14646:⋯
14635:−
14606:−
14493:) has no
14443:) is not
14344:different
14289:eigenline
14252:, namely
14232:So, both
14203:α
14197:λ
14179:α
14144:λ
14091:, namely
14020:α
14002:α
13863:λ
13720:λ
13646:→
13586:The main
13551:λ
13463:λ
13349:λ
13256:submatrix
13199:λ
13195:−
13186:λ
13173:≠
13166:∏
13134:λ
13130:−
13121:λ
13107:∏
12934:−
12852:λ
12849:−
12830:λ
12827:−
12807:λ
12804:−
12774:λ
12771:−
12741:λ
12738:−
12708:λ
12705:−
12682:λ
12679:−
12467:λ
12441:−
12405:λ
12365:−
12336:λ
12263:λ
12260:−
12248:λ
12245:−
12233:λ
12230:−
12212:λ
12209:−
12014:λ
11955:λ
11896:λ
11823:λ
11820:−
11808:λ
11805:−
11793:λ
11790:−
11772:λ
11769:−
11627:∗
11616:λ
11594:λ
11530:λ
11518:λ
11489:λ
11439:λ
11427:λ
11398:λ
11347:λ
11333:λ
11314:⋅
11305:λ
11277:λ
11263:λ
11234:λ
11220:λ
11164:λ
11150:λ
11131:⋅
11122:λ
11094:λ
11080:λ
11051:λ
11037:λ
10987:λ
10969:λ
10955:λ
10937:λ
10917:λ
10907:λ
10848:⋅
10736:−
10659:−
10646:−
10638:∗
10629:λ
10612:λ
10576:−
10560:λ
10536:λ
10321:−
10213:λ
10207:−
10198:λ
10182:λ
10178:−
10162:−
10156:λ
10153:−
10141:λ
10138:−
10119:λ
10116:−
10086:λ
10083:−
10058:λ
10055:−
10030:λ
10027:−
9942:λ
9939:−
9855:λ
9852:−
9585:λ
9515:−
9453:−
9366:−
9346:−
9320:λ
9301:−
9236:−
9197:−
9161:λ
8951:λ
8935:−
8860:−
8857:λ
8845:−
8842:λ
8820:λ
8813:λ
8807:−
8783:λ
8780:−
8760:λ
8757:−
8699:λ
8696:−
8639:λ
8636:−
8308:Hermitian
8270:defective
8142:Λ
8125:−
8082:−
8074:Λ
8028:Λ
7947:λ
7941:⋯
7918:λ
7894:λ
7805:⋯
7510:κ
7461:κ
7394:×
7351:κ
7321:κ
7252:λ
7192:×
7129:λ
7116:…
7101:λ
7019:α
7007:λ
7000:…
6994:α
6982:λ
6949:α
6924:∈
6921:α
6886:λ
6879:…
6861:λ
6782:λ
6775:…
6763:λ
6736:×
6673:λ
6603:symmetric
6599:Hermitian
6563:∗
6500:λ
6488:…
6474:λ
6443:−
6357:λ
6350:…
6333:λ
6212:λ
6208:⋯
6199:λ
6189:λ
6176:λ
6155:∏
6080:λ
6073:⋯
6061:λ
6048:λ
6035:λ
6014:∑
5977:∑
5961:
5899:λ
5886:μ
5856:λ
5843:μ
5816:λ
5809:…
5797:λ
5773:×
5529:γ
5454:≤
5445:γ
5441:≤
5415:λ
5402:γ
5381:∑
5364:γ
5310:λ
5297:γ
5270:λ
5243:λ
5236:…
5224:λ
5200:≤
5151:λ
5139:γ
5135:≥
5129:λ
5117:μ
5096:λ
5071:λ
5059:γ
5050:λ
5047:−
5044:ξ
5015:ξ
5012:−
4957:ξ
4954:−
4933:ξ
4930:−
4898:ξ
4895:−
4869:ξ
4866:−
4797:ξ
4794:−
4770:ξ
4767:−
4738:ξ
4735:−
4710:λ
4698:γ
4689:λ
4587:λ
4575:γ
4571:−
4545:λ
4533:γ
4480:λ
4437:λ
4425:γ
4410:…
4368:λ
4356:γ
4332:λ
4320:μ
4316:≤
4310:λ
4298:γ
4272:≤
4266:λ
4254:μ
4250:≤
4244:λ
4232:γ
4228:≤
4189:λ
4186:−
4177:
4171:−
4159:λ
4147:γ
4079:λ
4067:γ
3717:λ
3714:−
3512:λ
3498:μ
3477:∑
3460:μ
3446:≤
3434:λ
3421:μ
3417:≤
3348:λ
3335:μ
3326:λ
3323:−
3314:λ
3307:⋯
3293:λ
3280:μ
3271:λ
3268:−
3259:λ
3241:λ
3228:μ
3219:λ
3216:−
3207:λ
3191:λ
3188:−
2943:λ
2918:−
2889:λ
2839:λ
2836:−
2780:λ
2773:λ
2767:−
2750:λ
2747:−
2727:λ
2724:−
2701:λ
2698:−
2528:λ
2525:−
2516:λ
2509:⋯
2503:λ
2500:−
2491:λ
2481:λ
2478:−
2469:λ
2453:λ
2450:−
2420:, can be
2283:λ
2280:−
2136:λ
2133:−
2095:) is the
2031:λ
1934:∑
1901:⋯
1794:⋮
1730:⋮
1668:⋯
1631:⋮
1626:⋱
1621:⋮
1616:⋮
1594:⋯
1548:⋯
1431:−
1425:λ
1395:λ
1373:collinear
1335:−
1318:−
1272:−
1233:Consider
1187:in 1961.
940:, or the
878:λ
815:λ
807:λ
796:λ
712:Mona Lisa
641:λ
555:with the
474:λ
275:×
229:stretches
201:λ
164:λ
133:λ
25394:Category
25150:Topology
25097:Discrete
25082:Analytic
25069:Geometry
25041:Discrete
24996:Calculus
24988:Analysis
24943:Abstract
24882:Glossary
24865:Timeline
24805:Category
24744:Subspace
24739:Quotient
24690:Bivector
24604:Bilinear
24546:Matrices
24421:Glossary
24143:for the
23987:4 August
23969:45359403
23961:17700768
23744:76091646
23541:: 1–29,
23477:(1996),
23378:22275430
23324:Archived
23215:33124845
22665:Archived
21996:See also
21946:of each
20309:electron
20281:diagonal
19695:PageRank
19160:variance
16663:Rotation
15924:that is
15462:accuracy
14099: ∈
14071: ∈
14063: ∈
13792:, while
5169:Suppose
3049:spectrum
3018:are all
2422:factored
2341:and the
1369:parallel
1137:studied
1135:Poincaré
955:forms a
692:In this
531:Overview
518:matrices
417:, where
249:feedback
25406:Commons
25188:Applied
25158:General
24935:Algebra
24860:History
24416:Outline
24231:Please
24223:use of
24103:Bibcode
23933:Bibcode
23869:Bibcode
23814:Bibcode
23651:Bibcode
23623:Bibcode
23507:Bibcode
23370:2117040
23253:58-7924
23187:Bibcode
23085:Sources
22634:, §7.3.
22622:, §7.3.
21304:geology
21266:Please
21235:called
20257:inertia
19705:of the
19224:or the
16721:Matrix
16655:Scaling
14495:bounded
14405:. When
14103:, then
14095:,
14059:,
13764:is the
13615:scalars
13320:is the
13286:Hilbert
13254:is the
8306:In the
8177:similar
8005:. Then
7728:, ...,
7705:, ...,
7207:matrix
6829:(where
6657:unitary
5553:, then
4622:. Then
4120:nullity
3910:, then
3784:is the
3658:matrix
3650:of the
3084:matrix
2585:, ...,
2412:matrix
2389:or the
2333:) is a
2192:is the
2085:is the
1468:matrix
1213:Matrix
1127:Schwarz
1039:Later,
972:History
855:matrix
683:complex
557:English
553:cognate
425:, then
291:matrix
241:geology
225:rotates
25107:Finite
24963:Linear
24870:Future
24846:Major
24700:Tensor
24512:Kernel
24442:Vector
24437:Scalar
24350:Theory
24192:
24174:
23967:
23959:
23951:
23899:, SIAM
23887:
23849:
23832:
23780:
23762:
23742:
23720:
23701:
23661:
23603:
23585:
23525:
23485:
23460:
23396:
23376:
23368:
23318:
23271:
23251:
23221:
23213:
23205:
23142:
23116:
22080:Note:
21578:, and
21312:clasts
21194:atomic
21182:atomic
20979:, and
20864:is an
20800:is an
20765:where
20556:energy
20530:, the
20495:, the
20475:where
20388:proton
20370:) and
20307:of an
20277:stress
20275:, the
20253:tensor
20251:. The
19998:where
19691:Google
19622:. The
19551:, the
19291:Graphs
19275:; cf.
19207:sample
19195:matrix
19193:(PSD)
17236:
17073:
16617:, the
16539:where
14901:
14892:
14521:module
14365:is an
14271:, and
13674:is an
13504:) and
13300:. Let
13227:where
13047:For a
12892:= 4 =
12303:, and
12085:lower
11863:, and
10710:is an
10690:where
8284:has a
7343:where
7175:column
6324:, are
4913:, and
4815:since
3882:closed
3796:, and
3762:kernel
3088:. The
2353:. Its
2343:degree
2188:where
2073:is an
1154:German
1092:, and
919:eigen-
902:is an
675:scalar
589:, and
543:German
538:eigen-
451:, and
423:scalar
233:shears
73:vector
25334:lists
24877:Lists
24850:areas
24569:Minor
24554:Block
24492:Basis
24159:(PDF)
24091:(PDF)
24043:(PDF)
23965:S2CID
23906:arXiv
23885:S2CID
23830:S2CID
23804:arXiv
23736:Wiley
23523:S2CID
23374:S2CID
23327:(PDF)
23316:S2CID
23298:arXiv
23286:(PDF)
23219:S2CID
23177:arXiv
23110:Wiley
22511:, §3.
22492:, §2.
22364:(PDF)
22261:over
22182:See:
22068:Notes
21948:pixel
21067:, of
20973:Light
20311:in a
20285:shear
20247:of a
19364:) or
19301:graph
19259:. In
19186:of a
19135:0.478
19129:0.878
18964:with
18189:mult.
17853:mult.
14515:– an
14240:and α
13803:with
13625:into
13608:field
11010:Then
10714:with
7383:is a
7179:right
6725:is a
5926:trace
5520:. If
4138:) as
3840:, so
3786:union
3580:. If
2067:then
1158:eigen
1156:word
957:basis
730:like
720:along
706:them.
559:word
548:eigen
545:word
514:basis
421:is a
344:with
231:, or
189:, or
121:, is
71:is a
67:) or
64:-gən-
32:, an
24724:Dual
24579:Rank
24190:ISBN
24172:ISBN
24029:2019
24009:2020
23989:2019
23957:PMID
23949:ISSN
23847:ISBN
23778:ISBN
23760:ISBN
23740:LCCN
23718:ISBN
23699:ISBN
23659:ISBN
23601:ISBN
23583:ISBN
23483:ISBN
23458:ISBN
23394:ISBN
23366:PMID
23269:ISBN
23249:LCCN
23211:PMID
23203:ISSN
23140:ISBN
23114:ISBN
22434:2020
21742:>
21729:>
21689:>
21522:360°
21208:via
21196:and
21184:and
21135:and
20824:and
20658:The
20635:and
20503:and
20386:, a
20299:The
19929:and
19765:(or
19182:The
19176:and
17825:sinh
17813:cosh
17760:sinh
17748:cosh
17295:cosh
17031:cosh
17020:sinh
17007:sinh
16996:cosh
14538:and
14530:The
14294:The
14079:and
14067:and
14055:for
11571:and
11473:and
11193:and
10359:and
9721:and
9700:and
9270:For
8908:For
8894:and
7687:has
7453:and
7280:left
7083:are
6970:are
6460:are
6105:The
5924:The
5742:Let
5189:has
4174:rank
4122:of (
4025:and
4000:) =
3983:) ∈
3970:and
3938:) =
3920:) ∈
3888:and
3145:and
3067:Let
3046:The
3010:are
2807:and
2408:-by-
2400:The
2234:the
2203:and
1995:and
1199:and
1183:and
988:and
914:it.
498:-by-
24239:or
24121:hdl
24111:doi
24099:123
24064:doi
24060:692
23941:doi
23877:doi
23822:doi
23681:doi
23631:doi
23543:doi
23515:doi
23440:doi
23418:doi
23356:hdl
23348:doi
23308:doi
23195:doi
23173:125
21938:In
21520:of
21302:In
21270:to
21176:In
21148:min
21115:max
20929:to
20804:of
20418:in
20271:In
20235:In
19885:In
19816:or
19693:'s
19589:deg
19295:In
19283:in
19271:of
19045:sin
19030:cos
18990:sin
17629:sin
17614:cos
17558:sin
17543:cos
17194:cos
16899:cos
16888:sin
16875:sin
16861:cos
16572:of
16423:If
14427:If
14369:of
13906:of
13902:or
13768:of
13678:of
13613:of
13296:on
13288:or
12670:det
12311:= 3
12301:= 2
12291:= 1
12200:det
12192:is
11871:= 3
11861:= 2
11851:= 1
11760:det
11752:is
10219:22.
9843:det
9831:is
9139:= −
8627:det
7857:by
7433:of
7276:row
7233:),
6705:If
6655:is
6635:If
6608:If
6597:is
6527:If
6410:If
6403:is
6139:det
6109:of
5928:of
5478:sum
5288:is
5003:det
4945:det
4921:det
3820:of
3816:or
3808:).
3679:),
3664:set
3654:by
3549:If
3398:as
3376:If
3179:det
3080:by
2809:λ=3
2805:λ=1
2689:det
2681:is
2441:det
2393:of
2375:of
2271:det
2196:by
1508:or
1464:by
1371:or
1296:and
1051:by
980:or
944:of
851:by
839:by
665:or
562:own
243:to
62:EYE
28:In
25436::
24139:.
24135:.
24119:,
24109:,
24097:,
24093:,
24058:.
24054:.
24020:.
24000:.
23980:.
23963:.
23955:.
23947:.
23939:.
23929:32
23927:.
23923:.
23883:,
23875:,
23865:66
23863:,
23828:.
23820:.
23812:.
23800:89
23798:.
23794:.
23738:,
23675:,
23657:,
23629:,
23619:24
23617:,
23537:,
23521:,
23513:,
23503:25
23501:,
23473:;
23434:,
23412:,
23392:,
23372:,
23364:,
23354:,
23344:28
23342:,
23338:,
23322:.
23314:.
23306:.
23294:59
23292:.
23288:.
23247:,
23217:.
23209:.
23201:.
23193:.
23185:.
23171:.
23167:.
23138:,
23112:,
23025:.
22953:^
22922:^
22804:^
22729:^
22700:^
22528:^
22497:^
22466:^
22425:.
22414:^
22387:^
22372:^
22158:,
22135:32
22133:,
21551:,
21429:;
21239:.
20975:,
20964:.
20558:.
20422::
20263:.
20198:0.
20146:,
19949:a
19722:A
19287:.
16594:.
16323:.
16163:.
15402:.
15364:0.
14552:.
14527:.
14491:λI
14489:−
14461:λI
14459:−
14453:λI
14451:−
14441:λI
14439:−
14291:.
14267:∈
14260:,
14256:+
14236:+
13914:.
13894:.
13807:.
13776:.
13686:∈
13670:∈
13629:,
13494:dt
13383:.
13059:,
12900:.
12317:.
12293:,
11877:.
11853:,
10210:35
10194:14
10165:16
9726:=3
9719:=1
9708:=3
9698:=1
9563:=
9275:=3
8913:=1
8905:.
8899:=3
8892:=1
8615:,
8606:λI
8512:=3
8502:=1
8439:=
8292:.
8226:PD
8224:=
8222:AP
8219:,
8209:AP
8109:,
8054:,
7994:ii
7752:,
7721:,
7698:,
5958:tr
5351:,
5166:.
4653::=
4217:.
4136:λI
4134:−
4128:λI
4126:−
4116:λI
4114:−
4104:λI
4102:−
4040:.
4021:+
3965:∈
3946:+
3934:+
3916:+
3905:∈
3901:,
3873:.
3806:λI
3804:−
3770:λI
3768:−
3638:.
3607:,
3380:=
3149:≤
3130:)
3121:−
2673:λI
2671:−
2598:.
2578:,
2397:.
2246:λI
2244:−
2103:.
1587:22
1575:21
1541:12
1529:11
1472:,
1453:.
1439:20
1338:80
1328:60
1321:20
1125:.
1077:.
1061:.
1036:.
1007:.
992:.
697:1.
685:.
593:.
585:,
581:,
577:,
527:.
491:.
227:,
213:.
185:,
125:,
55:-/
52:ən
46:aɪ
24839:e
24832:t
24825:v
24397:e
24390:t
24383:v
24320:n
24314:n
24266:)
24260:(
24255:)
24251:(
24247:.
24229:.
24147:.
24123::
24113::
24105::
24072:.
24066::
24031:.
24011:.
23991:.
23971:.
23943::
23935::
23914:.
23908::
23879::
23871::
23836:.
23824::
23816::
23806::
23707:.
23683::
23677:1
23653::
23633::
23625::
23545::
23539:2
23517::
23509::
23442::
23436:4
23420::
23414:4
23358::
23350::
23310::
23300::
23225:.
23197::
23189::
23179::
23078:.
23066:.
23029:.
22936:.
22838:.
22826:.
22814:.
22799:.
22610:.
22586:.
22461:.
22436:.
22409:.
22309:)
22304:4
22300:n
22296:(
22293:O
22273:n
22160:8
22116:t
21899:0
21895:R
21872:G
21868:t
21845:G
21841:t
21818:0
21814:R
21791:0
21787:R
21750:3
21746:E
21737:2
21733:E
21724:1
21720:E
21697:3
21693:E
21684:2
21680:E
21676:=
21671:1
21667:E
21644:3
21640:E
21636:=
21631:2
21627:E
21623:=
21618:1
21614:E
21591:3
21587:E
21564:2
21560:E
21537:1
21533:E
21502:3
21497:v
21473:2
21468:v
21444:1
21439:v
21415:3
21411:E
21402:2
21398:E
21389:1
21385:E
21362:3
21357:v
21352:,
21347:2
21342:v
21337:,
21332:1
21327:v
21293:)
21287:(
21282:)
21278:(
21264:.
21156:0
21153:=
21123:1
21120:=
21088:t
21077:t
21034:t
21023:t
21000:t
20947:E
20938:|
20917:H
20892:E
20883:|
20879:H
20852:H
20832:E
20812:H
20783:E
20774:|
20745:E
20736:|
20732:E
20729:=
20721:E
20712:|
20708:H
20680:E
20671:|
20643:H
20621:E
20578:E
20542:E
20516:E
20483:H
20457:E
20449:E
20446:=
20441:E
20433:H
20402:T
20390:.
20354:,
20351:3
20347:,
20344:2
20340:,
20337:1
20334:=
20331:n
20195:=
20192:x
20188:)
20184:k
20181:+
20178:c
20172:+
20169:m
20164:2
20155:(
20130:0
20127:=
20124:x
20121:k
20118:+
20109:x
20103:c
20100:+
20091:x
20085:m
20061:k
20011:2
19986:x
19983:m
19978:2
19970:=
19967:x
19964:k
19937:k
19913:m
19893:n
19870:x
19848:x
19845:k
19839:=
19830:x
19824:m
19804:0
19801:=
19798:x
19795:k
19792:+
19783:x
19777:m
19670:k
19650:k
19630:k
19608:)
19603:i
19599:v
19595:(
19583:/
19579:1
19559:i
19537:2
19533:/
19529:1
19522:D
19499:i
19495:v
19472:i
19469:i
19465:D
19444:D
19418:2
19414:/
19410:1
19403:D
19399:A
19394:2
19390:/
19386:1
19379:D
19372:I
19348:A
19342:D
19314:A
19162:.
19138:)
19132:,
19126:(
19106:)
19103:3
19100:,
19097:1
19094:(
19042:i
19020:θ
19004:2
19000:)
18987:(
18984:4
18978:=
18975:D
18937:]
18931:1
18921:1
18915:[
18910:=
18901:2
18896:u
18885:]
18879:1
18872:1
18866:[
18861:=
18852:1
18847:u
18818:]
18812:0
18805:1
18799:[
18794:=
18789:1
18784:u
18755:]
18749:i
18746:+
18739:1
18733:[
18728:=
18719:2
18714:u
18703:]
18697:i
18687:1
18681:[
18676:=
18667:1
18662:u
18629:]
18623:1
18616:0
18610:[
18605:=
18596:2
18591:u
18580:]
18574:0
18567:1
18561:[
18556:=
18547:1
18542:u
18503:1
18500:=
18491:2
18479:1
18476:=
18467:1
18437:1
18434:=
18429:1
18399:1
18396:=
18387:2
18375:1
18372:=
18363:1
18329:1
18326:=
18317:2
18305:1
18302:=
18293:1
18263:2
18260:=
18255:1
18229:)
18224:i
18216:(
18210:=
18205:i
18191:,
18167:1
18164:=
18155:2
18143:1
18140:=
18131:1
18101:2
18098:=
18093:1
18063:1
18060:=
18051:2
18039:1
18036:=
18027:1
17993:1
17990:=
17981:2
17969:1
17966:=
17957:1
17927:2
17924:=
17919:1
17893:)
17888:i
17880:(
17874:=
17869:i
17855:,
17810:=
17791:e
17787:=
17778:2
17757:+
17745:=
17729:e
17725:=
17716:1
17686:1
17683:=
17678:2
17670:=
17665:1
17626:i
17611:=
17596:i
17589:e
17585:=
17576:2
17555:i
17552:+
17540:=
17525:i
17521:e
17517:=
17508:1
17472:2
17468:k
17464:=
17455:2
17441:1
17437:k
17433:=
17424:1
17394:k
17391:=
17386:2
17378:=
17373:1
17345:i
17316:1
17313:+
17307:)
17301:(
17292:2
17284:2
17256:2
17252:)
17248:1
17239:(
17215:1
17212:+
17206:)
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17191:2
17183:2
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17135:)
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17093:2
17089:)
17085:k
17076:(
17043:]
16990:[
16965:]
16959:1
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16947:k
16942:1
16936:[
16911:]
16855:[
16830:]
16822:2
16818:k
16812:0
16805:0
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16794:k
16787:[
16762:]
16756:k
16751:0
16744:0
16739:k
16733:[
16581:v
16549:v
16520:v
16509:v
16501:v
16497:A
16487:v
16479:=
16453:A
16432:v
16419:.
16406:C
16388:;
16374:1
16367:)
16363:I
16354:A
16351:(
16311:b
16288:T
16281:]
16275:b
16272:3
16264:b
16258:[
16235:0
16232:=
16229:y
16226:+
16223:x
16220:3
16200:1
16197:=
16174:A
16151:6
16148:=
16125:A
16105:a
16082:T
16075:]
16069:a
16066:2
16061:a
16055:[
16032:x
16029:2
16026:=
16023:y
15994:0
15991:=
15984:y
15981:3
15975:x
15972:6
15965:0
15962:=
15955:y
15952:+
15949:x
15946:2
15936:{
15902:y
15899:6
15896:=
15889:y
15886:3
15883:+
15880:x
15877:6
15870:x
15867:6
15864:=
15857:y
15854:+
15851:x
15848:4
15841:{
15813:]
15807:y
15800:x
15794:[
15786:6
15783:=
15778:]
15772:y
15765:x
15759:[
15752:]
15746:3
15741:6
15734:1
15729:4
15723:[
15701:v
15698:6
15695:=
15692:v
15689:A
15665:]
15659:3
15654:6
15647:1
15642:4
15636:[
15631:=
15628:A
15588:n
15564:n
15540:n
15513:!
15510:n
15490:n
15484:n
15444:2
15438:2
15418:A
15361:=
15358:x
15353:0
15349:a
15345:+
15339:t
15336:d
15331:x
15328:d
15320:1
15316:a
15312:+
15306:+
15298:1
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15288:t
15284:d
15279:x
15274:1
15268:k
15264:d
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15215:k
15211:d
15177:.
15172:t
15167:k
15157:k
15153:c
15149:+
15143:+
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15115:=
15110:t
15106:x
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15077:k
15068:,
15061:,
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15025:]
15017:1
15014:+
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14988:t
14984:x
14977:[
14965:k
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14943:+
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14930:x
14926:=
14921:1
14918:+
14915:k
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14878:t
14874:x
14870:=
14865:1
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14844:k
14827:,
14824:0
14821:=
14816:k
14812:a
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14790:a
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14585:=
14580:t
14576:x
14487:T
14483:λ
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14324:T
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14115:T
14101:E
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3093:μ
3086:A
3082:n
3078:n
3073:i
3069:λ
3016:A
3008:A
2999:A
2983:.
2978:]
2972:1
2965:1
2959:[
2954:=
2949:3
2946:=
2938:v
2932:,
2927:]
2921:1
2911:1
2905:[
2900:=
2895:1
2892:=
2884:v
2872:.
2859:0
2855:=
2851:v
2846:)
2842:I
2833:A
2829:(
2817:v
2813:A
2789:.
2784:2
2776:+
2770:4
2764:3
2761:=
2756:|
2744:2
2739:1
2732:1
2721:2
2715:|
2710:=
2707:)
2704:I
2695:A
2692:(
2679:A
2675:)
2669:A
2667:(
2651:.
2646:]
2640:2
2635:1
2628:1
2623:2
2617:[
2612:=
2609:A
2596:A
2591:n
2587:λ
2583:2
2580:λ
2576:1
2573:λ
2568:i
2564:λ
2555:)
2553:4
2551:(
2534:,
2531:)
2520:n
2512:(
2506:)
2495:2
2487:(
2484:)
2473:1
2465:(
2462:=
2459:)
2456:I
2447:A
2444:(
2426:n
2418:n
2414:A
2410:n
2406:n
2395:A
2382:3
2377:A
2367:λ
2363:n
2359:A
2351:A
2347:n
2339:λ
2330:3
2316:)
2314:3
2312:(
2295:0
2292:=
2289:)
2286:I
2277:A
2274:(
2256:λ
2252:A
2248:)
2242:A
2240:(
2229:v
2224:2
2205:0
2198:n
2194:n
2190:I
2181:)
2179:2
2177:(
2160:,
2156:0
2152:=
2148:v
2143:)
2139:I
2130:A
2126:(
2109:1
2101:A
2092:1
2083:λ
2079:A
2070:v
2060:)
2058:1
2056:(
2039:,
2035:v
2028:=
2024:w
2020:=
2016:v
2012:A
1997:w
1993:v
1977:.
1972:j
1968:v
1962:j
1959:i
1955:A
1949:n
1944:1
1941:=
1938:j
1930:=
1925:n
1921:v
1915:n
1912:i
1908:A
1904:+
1898:+
1893:2
1889:v
1883:2
1880:i
1876:A
1872:+
1867:1
1863:v
1857:1
1854:i
1850:A
1846:=
1841:i
1837:w
1814:]
1806:n
1802:w
1785:2
1781:w
1771:1
1767:w
1760:[
1755:=
1750:]
1742:n
1738:v
1721:2
1717:v
1707:1
1703:v
1696:[
1689:]
1681:n
1678:n
1674:A
1661:2
1658:n
1654:A
1646:1
1643:n
1639:A
1607:n
1604:2
1600:A
1583:A
1571:A
1561:n
1558:1
1554:A
1537:A
1525:A
1518:[
1496:,
1492:w
1488:=
1484:v
1480:A
1470:A
1466:n
1462:n
1458:n
1436:1
1428:=
1403:.
1399:y
1392:=
1388:x
1377:λ
1349:.
1344:]
1312:[
1307:=
1303:y
1288:]
1282:4
1275:3
1265:1
1259:[
1254:=
1250:x
1239:n
1235:n
1229:.
1227:A
1223:x
1219:x
1215:A
967:.
961:T
953:T
946:T
934:T
904:n
900:v
886:,
882:v
875:=
871:v
867:A
857:A
853:n
849:n
845:n
841:n
837:n
823:.
818:x
811:e
804:=
799:x
792:e
785:x
782:d
778:d
748:x
745:d
741:d
679:λ
671:λ
649:,
645:v
638:=
635:)
631:v
627:(
624:T
614:λ
610:T
606:T
602:T
598:v
551:(
508:n
500:n
496:n
478:v
471:=
467:v
463:A
453:λ
449:A
434:v
419:λ
415:λ
400:v
378:v
374:A
353:v
342:A
328:.
325:n
304:v
293:A
279:n
271:n
168:v
161:=
157:v
153:T
109:T
88:v
49:ɡ
43:ˈ
40:/
36:(
23:.
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