19755:
24548:
25388:
10244:
713:
16695:
16702:
9845:
700:
24349:
1837:
1221:
25400:
24812:
8894:
19090:
16725:
1522:
25424:
16709:
10239:{\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}}}
25412:
16716:
8629:
9563:
24225:
21264:
12535:
22184:(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.)
21933:
12082:
12886:
11385:
11202:
729:. 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
10699:
9297:
1832:{\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}}}
3383:
12334:
17856:
8525:= . 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:
17660:
14239:
9088:
8889:{\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}}}
11894:
14062:
18782:
18964:
18656:
12676:
11207:
11024:
4223:
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
733:
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
7997:
21223:. 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
3556:
22002:
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
16348:
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
3012:
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
1216:
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
1370:
246:
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.
3004:
20307:
10537:
15836:
10905:
6110:
5485:
13236:
12671:
3185:
2810:
11573:
11482:
9268:
21321:, 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
16639:
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.
14117:
9686:
7851:
15385:
11017:
17066:
8935:
9558:{\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}}}
2555:
17713:
13931:
16934:
17505:
14848:
15470:
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
3048:
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.
12530:{\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}},}
6242:
3767:
3412:
16653:
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.
17499:
16545:
8405:
1998:
22178:"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'
19720:
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
12195:
11755:
10526:
9834:
7548:
4463:
1255:
18662:
18847:
2888:
18542:
15930:
16022:
262:). 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
18528:
18424:
18354:
13897:
12077:{\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}},}
7882:
22257:
15727:
10776:
12979:
13039:
18192:
18088:
18018:
5365:
15048:
13073:
11653:
7179:
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
4219:
15198:
12550:
10366:
12881:{\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}.}
10424:
10306:
19741:
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
11380:{\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}}.}
11197:{\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}},}
6529:
4296:
12290:
11850:
16853:
16312:
14699:
7043:
16106:
4513:
18841:
2695:
844:
21385:
2181:
8634:
8593:
2881:
2672:
716:
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
707:
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
22108:"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."
15688:
9160:
2060:
21870:, 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
21112:
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
14972:
5964:
22902:
16988:
16785:
9584:
22210:"Insbesondere in dieser ersten Mitteilung gelange ich zu Formeln, die die Entwickelung einer willkürlichen Funktion nach gewissen ausgezeichneten Funktionen, die ich 'Eigenfunktionen' nenne, liefern: ..."
5175:
4356:
21002:. 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
9850:
7162:
725:, 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
21058:
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,
10912:
10694:{\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}}}
6922:
6389:
20219:
22110:(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.)
20771:
19255:
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.
17337:
5097:
15103:
8174:
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
737:
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
17236:
13749:
6812:
670:
21110:
21056:
18852:
18667:
18547:
18469:
18365:
18295:
18133:
18029:
17959:
17718:
17510:
17426:
15950:
15855:
14122:
13936:
10542:
9302:
7281:
5370:
3417:
907:
11487:
11396:
7350:
1424:
500:
190:
18250:
13499:
5843:
5270:
4736:
17178:
11891:
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,
7769:
24247:
20151:
19072:
17914:
17707:
17415:
15213:
4984:
3378:{\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})}.}
1517:
21438:
16994:
5340:
1462:
17851:{\displaystyle {\begin{aligned}\lambda _{1}&=e^{\varphi }\\&=\cosh \varphi +\sinh \varphi \\\lambda _{2}&=e^{-\varphi }\\&=\cosh \varphi -\sinh \varphi \end{aligned}}}
21773:
20481:
15409:
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
13580:
6950:
4611:
3693:
19027:
5929:
5886:
2447:
20379:
4824:
4569:
4392:
4103:
21720:
19631:
16428:
15611:.) 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
1842:
20918:
19441:
12111:
11671:
10442:
9750:
8443:
7487:
2316:
20973:
20809:
20706:
17655:{\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}}}
17279:
17116:
16859:
13382:
8166:
8111:
21667:
19825:
14234:{\displaystyle {\begin{aligned}T(\mathbf {u} +\mathbf {v} )&=\lambda (\mathbf {u} +\mathbf {v} ),\\T(\alpha \mathbf {v} )&=\lambda (\alpha \mathbf {v} ).\end{aligned}}}
21525:
21496:
21467:
20007:
19869:
16577:
13434:
8204:
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.
6711:
5722:
5647:
5615:
3882:
773:
21177:
21144:
9139:
7634:
7605:
250:
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
16397:
15846:
22220:"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, ... "
19159:
18458:
18284:
8052:
5562:
5042:
400:
21022:
17368:
16603:
16454:
14718:
8427:
5297:
456:
422:
375:
326:
110:
23334:
20034:
9083:{\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}}}
300:
21179:. Furthermore, one of the striking properties of open eigenchannels, beyond the perfect transmittance, is the statistically robust spatial profile of the eigenchannels.
18122:
17948:
16221:
16172:
13833:
6979:
4690:
24341:
20644:
20601:
20539:
15511:
15465:
13673:
7418:
7216:
6760:
5797:
22330:
19560:
10760:
6145:
5117:
222:
154:
22213:(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
20054:
15627:
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
15567:
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
7482:
7372:
5224:
4922:
4893:
19495:
16256:
14057:{\displaystyle {\begin{aligned}T(\mathbf {x} +\mathbf {y} )&=T(\mathbf {x} )+T(\mathbf {y} ),\\T(\alpha \mathbf {x} )&=\alpha T(\mathbf {x} ),\end{aligned}}}
6469:
4762:
21922:
21895:
21868:
21841:
21814:
21614:
21587:
21560:
21076:
19522:
19127:
17421:
16482:
15722:
8503:
8347:
6586:
6313:
21816:) 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
16053:
7092:
20666:
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.
19369:
18777:{\displaystyle {\begin{aligned}\mathbf {u} _{1}&={\begin{bmatrix}1\\-i\end{bmatrix}}\\\mathbf {u} _{2}&={\begin{bmatrix}1\\+i\end{bmatrix}}\end{aligned}}}
11592:
6838:
18959:{\displaystyle {\begin{aligned}\mathbf {u} _{1}&={\begin{bmatrix}1\\1\end{bmatrix}}\\\mathbf {u} _{2}&={\begin{bmatrix}1\\-1\end{bmatrix}}\end{aligned}}}
15534:
13263:
10530:
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 −
4400:
4152:
349:
22294:
22223:(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, ... )
20938:
20873:
20853:
20833:
20664:
20563:
20504:
20423:
20082:
19958:
19934:
19914:
19891:
19691:
19671:
19651:
19580:
19465:
19335:
18651:{\displaystyle {\begin{aligned}\mathbf {u} _{1}&={\begin{bmatrix}1\\0\end{bmatrix}}\\\mathbf {u} _{2}&={\begin{bmatrix}0\\1\end{bmatrix}}\end{aligned}}}
16474:
16332:
16195:
16146:
16126:
15609:
15585:
15561:
15439:
10719:
8339:
7674:
7654:
7576:
7462:
7438:
7392:
7311:
7236:
7063:
6858:
6734:
6664:
6637:
6606:
6556:
6439:
6412:
6333:
6286:
6266:
6138:
5957:
5771:
5742:
5687:
5667:
5586:
5529:
5509:
5360:
5198:
5004:
4864:
4844:
4651:
4631:
4533:
130:
24008:
22133:(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.)
15401:
The calculation of eigenvalues and eigenvectors is a topic where theory, as presented in elementary linear algebra textbooks, is often very far from practice.
6643:, positive-semidefinite, negative-definite, or negative-semidefinite, then every eigenvalue is positive, non-negative, negative, or non-positive, respectively.
4231:
19074:; 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.
12206:
11766:
15941:
7992:{\displaystyle AQ={\begin{bmatrix}\lambda _{1}\mathbf {v} _{1}&\lambda _{2}\mathbf {v} _{2}&\cdots &\lambda _{n}\mathbf {v} _{n}\end{bmatrix}}.}
18464:
18360:
18290:
258:. 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 (
23624:
Knox-Robinson, C.; Gardoll, Stephen J. (1998), "GIS-stereoplot: an interactive stereonet plotting module for ArcView 3.0 geographic information system",
20999:
3551:{\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}}}
3063:
of a matrix is the list of eigenvalues, repeated according to multiplicity; in an alternative notation the set of eigenvalues with their multiplicities.
782:
22205:
22197:
20673:
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
14466:) does not exist. The converse is true for finite-dimensional vector spaces, but not for infinite-dimensional vector spaces. In general, the operator (
8536:
2615:
1076:
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
527:
of the vector space. Hence, in a finite-dimensional vector space, it is equivalent to define eigenvalues and eigenvectors using either the language of
19746:
gives sufficient conditions for a Markov chain to have a unique dominant eigenvalue, which governs the convergence of the system to a steady state.
15634:
13582:
is the eigenfunction of the derivative operator. In this case the eigenfunction is itself a function of its associated eigenvalue. In particular, for
12921:
20385:(increasing across: s, p, d, ...). The illustration shows the square of the absolute value of the wavefunctions. Brighter areas correspond to higher
12984:
3040:, 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
18128:
18024:
17954:
14329:, i.e., the maximum number of linearly independent eigenvectors associated with that eigenvalue. By the definition of eigenvalues and eigenvectors,
12981:
and is therefore 1-dimensional. Similarly, the geometric multiplicity of the eigenvalue 3 is 1 because its eigenspace is spanned by just one vector
21282:
14981:
1365:{\displaystyle \mathbf {x} ={\begin{bmatrix}1\\-3\\4\end{bmatrix}}\quad {\mbox{and}}\quad \mathbf {y} ={\begin{bmatrix}-20\\60\\-80\end{bmatrix}}.}
20234:
so that the system can be represented as linear summation of the eigenvectors. The eigenvalue problem of complex structures is often solved using
2999:{\displaystyle \mathbf {v} _{\lambda =1}={\begin{bmatrix}1\\-1\end{bmatrix}},\quad \mathbf {v} _{\lambda =3}={\begin{bmatrix}1\\1\end{bmatrix}}.}
15111:
10313:
24848:
22675:
20294:
tensor with the eigenvalues on the diagonal and eigenvectors as a basis. Because it is diagonal, in this orientation, the stress tensor has no
19693:
th smallest eigenvalue of the
Laplacian. The first principal eigenvector of the graph is also referred to merely as the principal eigenvector.
10374:
10256:
22131:"Non autem repugnat tres esse eiusmodi positiones plani HM, quia in aequatione cubica radices tres esse possunt, et tres tangentis t valores."
20160:
15483:). Even for matrices whose elements are integers the calculation becomes nontrivial, because the sums are very long; the constant term is the
15616:
13050:
is 2, which is the smallest it could be for a matrix with two distinct eigenvalues. Geometric multiplicities are defined in a later section.
6474:
16791:
21325:
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.
16261:
14581:
12894:
of each eigenvalue is 2; in other words they are both double roots. The sum of the algebraic multiplicities of all distinct eigenvalues is
6984:
630:
22165:, pp. 807–808 Augustin Cauchy (1839) "Mémoire sur l'intégration des équations linéaires" (Memoir on the integration of linear equations),
21535:. 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
16058:
15831:{\displaystyle {\begin{bmatrix}4&1\\6&3\end{bmatrix}}{\begin{bmatrix}x\\y\end{bmatrix}}=6\cdot {\begin{bmatrix}x\\y\end{bmatrix}}}
10900:{\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}}.}
4468:
23347:"On the definition and the computation of the basic reproduction ratio R0 in models for infectious diseases in heterogeneous populations"
22094:
In 1751, Leonhard Euler proved that any body has a principal axis of rotation: Leonhard Euler (presented: October 1751; published: 1760)
18788:
14522:
One can generalize the algebraic object that is acting on the vector space, replacing a single operator acting on a vector space with an
7247:
873:
21331:
7316:
6105:{\displaystyle \operatorname {tr} (A)=\sum _{i=1}^{n}a_{ii}=\sum _{i=1}^{n}\lambda _{i}=\lambda _{1}+\lambda _{2}+\cdots +\lambda _{n}.}
2131:
1393:
19194:
19161:
direction and of 1 in the orthogonal direction. The vectors shown are unit eigenvectors of the (symmetric, positive-semidefinite)
13439:
5480:{\displaystyle {\begin{aligned}\gamma _{A}&=\sum _{i=1}^{d}\gamma _{A}(\lambda _{i}),\\d&\leq \gamma _{A}\leq n,\end{aligned}}}
2834:
13231:{\displaystyle |v_{i,j}|^{2}={\frac {\prod _{k}{(\lambda _{i}-\lambda _{k}(M_{j}))}}{\prod _{k\neq i}{(\lambda _{i}-\lambda _{k})}}},}
13066:
th component of a normalized eigenvector can be calculated using only the matrix eigenvalues and the eigenvalues of the corresponding
2018:
24343:
size (for a square matrix), then fill out the entries numerically and click on the Go button. It can accept complex numbers as well.)
23506:
Graham, D.; Midgley, N. (2000), "Graphical representation of particle shape using triangular diagrams: an Excel spreadsheet method",
19165:
scaled by the square root of the corresponding eigenvalue. Just as in the one-dimensional case, the square root is taken because the
14860:
16940:
16737:
16349:(optionally normalizing the vector to keep its elements of reasonable size); this makes the vector converge towards an eigenvector.
1175:. For some time, the standard term in English was "proper value", but the more distinctive term "eigenvalue" is the standard today.
24406:
24305:
24299:
21328:
The output for the orientation tensor is in the three orthogonal (perpendicular) axes of space. The three eigenvectors are ordered
20091:
12666:{\displaystyle A={\begin{bmatrix}2&0&0&0\\1&2&0&0\\0&1&3&0\\0&0&1&3\end{bmatrix}},}
5122:
4303:
21843:
is the average number of people that one typical infectious person will infect. The generation time of an infection is the time,
19780:) of vibration, and the eigenvectors are the shapes of these vibrational modes. In particular, undamped vibration is governed by
14357:
eigenvalues are always linearly independent. Therefore, the sum of the dimensions of the eigenspaces cannot exceed the dimension
7097:
1486:
24739:
24353:
22142:
Report of the Thirty-second meeting of the
British Association for the Advancement of Science; held at Cambridge in October 1862
24797:
23529:
23293:
20224:
6863:
6338:
25470:
23915:
Van
Mieghem, Piet (18 January 2014). "Graph eigenvectors, fundamental weights and centrality metrics for nodes in networks".
23792:
23732:
23508:
23497:
20714:
17285:
13525:
5047:
24028:
23656:(2000), "Mathematical Handbook for Scientists and Engineers: Definitions, Theorems, and Formulas for Reference and Review",
15057:
6531:
and each eigenvalue's geometric multiplicity coincides. Moreover, since the characteristic polynomial of the inverse is the
4738:. This can be seen by evaluating what the left-hand side does to the first column basis vectors. By reorganizing and adding
2805:{\displaystyle \det(A-\lambda I)={\begin{vmatrix}2-\lambda &1\\1&2-\lambda \end{vmatrix}}=3-4\lambda +\lambda ^{2}.}
25211:
17184:
13709:
11568:{\displaystyle \mathbf {v} _{\lambda _{3}}={\begin{bmatrix}1&\lambda _{3}&\lambda _{2}\end{bmatrix}}^{\textsf {T}}}
11477:{\displaystyle \mathbf {v} _{\lambda _{2}}={\begin{bmatrix}1&\lambda _{2}&\lambda _{3}\end{bmatrix}}^{\textsf {T}}}
6765:
3066:
An important quantity associated with the spectrum is the maximum absolute value of any eigenvalue. This is known as the
1171:, which means "own", to denote eigenvalues and eigenvectors in 1904, though he may have been following a related usage by
23713:
22047:
21081:
21027:
9263:{\displaystyle \mathbf {v} _{\lambda =1}={\begin{bmatrix}v_{1}\\-v_{1}\end{bmatrix}}={\begin{bmatrix}1\\-1\end{bmatrix}}}
19033:
is not an integer multiple of 180°. Therefore, except for these special cases, the two eigenvalues are complex numbers,
13337:
3178:
linear terms with some terms potentially repeating, the characteristic polynomial can also be written as the product of
1068:
469:
159:
25356:
24841:
23774:
23597:
19783:
19094:
18206:
15050:
in terms of its once-lagged value, and taking the characteristic equation of this system's matrix. This equation gives
14853:
which can be found by stacking into matrix form a set of equations consisting of the above difference equation and the
9681:{\displaystyle \mathbf {v} _{\lambda =3}={\begin{bmatrix}v_{1}\\v_{1}\end{bmatrix}}={\begin{bmatrix}1\\1\end{bmatrix}}}
7846:{\displaystyle Q={\begin{bmatrix}\mathbf {v} _{1}&\mathbf {v} _{2}&\cdots &\mathbf {v} _{n}\end{bmatrix}}.}
5802:
5229:
4695:
31:
24293:
19970:
19830:
17122:
25428:
24273:
24204:
24186:
23861:
23673:
23615:
23472:
23408:
23283:
23154:
23128:
21300:
19212:
of eigenvectors, each of which has a nonnegative eigenvalue. The orthogonal decomposition of a PSD matrix is used in
9092:
936:
The set of all eigenvectors of a linear transformation, each paired with its corresponding eigenvalue, is called the
535:
70:
15380:{\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.}
8597:
The figure on the right shows the effect of this transformation on point coordinates in the plane. The eigenvectors
24892:
24787:
23870:
Sneed, E. D.; Folk, R. L. (1958), "Pebbles in the lower
Colorado River, Texas, a study of particle morphogenesis",
21992:
20390:
19036:
16613:
Efficient, accurate methods to compute eigenvalues and eigenvectors of arbitrary matrices were not known until the
11012:{\displaystyle \lambda _{2}\lambda _{3}=1,\quad \lambda _{2}^{2}=\lambda _{3},\quad \lambda _{3}^{2}=\lambda _{2}.}
2335:
83:
21527:
is the tertiary, in terms of strength. The clast orientation is defined as the direction of the eigenvector, on a
17061:{\displaystyle {\begin{bmatrix}\cosh \varphi &\sinh \varphi \\\sinh \varphi &\cosh \varphi \end{bmatrix}}}
25306:
24749:
24685:
20507:
17870:
17666:
17374:
15475:. However, this approach is not viable in practice because the coefficients would be contaminated by unavoidable
4927:
1130:
studied eigenvalue problems similar to those of Sturm; the discipline that grew out of their work is now called
25404:
21390:
21243:
19965:
19769:
14542:
14517:
14486:
13643:
8526:
8175:
7685:
5302:
3121:
2550:{\displaystyle \det(A-\lambda I)=(\lambda _{1}-\lambda )(\lambda _{2}-\lambda )\cdots (\lambda _{n}-\lambda ),}
2412:
1431:
870:, then the eigenvalue equation for a linear transformation above can be rewritten as the matrix multiplication
692:
may be negative, in which case the eigenvector reverses direction as part of the scaling, or it may be zero or
601:
15479:, and the roots of a polynomial can be an extremely sensitive function of the coefficients (as exemplified by
25460:
24834:
22182:, le degré de cette équation étant précisément l'order de l'équation différentielle qu'il s'agit d'intégrer."
21725:
20439:
6927:
5931:
is the eigenvalue's algebraic multiplicity. The following are properties of this matrix and its eigenvalues:
4574:
18981:
12105:. As with diagonal matrices, the eigenvalues of triangular matrices are the elements of the main diagonal.
5891:
5848:
25416:
24399:
22140:. See: A. Cayley (1862) "Report on the progress of the solution of certain special problems of dynamics,"
22062:
21972:
20386:
20337:
20154:
19743:
19244:
19224:
19205:
19201:
19184:
19178:
16929:{\displaystyle {\begin{bmatrix}\cos \theta &-\sin \theta \\\sin \theta &\cos \theta \end{bmatrix}}}
16621:
with the LU decomposition results in an algorithm with better convergence than the QR algorithm. For large
4767:
4538:
4361:
4072:
3783:). On the other hand, by definition, any nonzero vector that satisfies this condition is an eigenvector of
22103:
22095:
21672:
19754:
16405:
589:
25331:
24887:
24632:
24482:
23682:
Kublanovskaya, Vera N. (1962), "On some algorithms for the solution of the complete eigenvalue problem",
20885:
19585:
19378:
3182:
terms each corresponding to a distinct eigenvalue and raised to the power of the algebraic multiplicity,
2612:
As a brief example, which is described in more detail in the examples section later, consider the matrix
2277:
17:
20943:
20779:
20676:
17242:
17079:
8126:
8071:
24902:
24537:
24431:
21619:
20604:
20255:
16618:
3041:
1132:
1015:
577:
546:
Eigenvalues and eigenvectors feature prominently in the analysis of linear transformations. The prefix
24243:
21501:
21472:
21443:
21278:
20882:, the infinite-dimensional analog of Hermitian matrices. As in the matrix case, in the equation above
16553:
13405:
6673:
5698:
5623:
5591:
3858:
744:
25450:
25316:
25288:
24925:
24777:
24426:
23572:, Wiley series in mathematical and computational biology, West Sussex, England: John Wiley & Sons
22369:
22129:, which proves that a body has three principal axes of rotation. He then states (on the same page):
21980:
21784:
21149:
21116:
20331:
19342:
19295:
15480:
14843:{\displaystyle \lambda ^{k}-a_{1}\lambda ^{k-1}-a_{2}\lambda ^{k-2}-\cdots -a_{k-1}\lambda -a_{k}=0,}
14508:
inverse. The spectrum of an operator always contains all its eigenvalues but is not limited to them.
14455:
10248:
The roots of the characteristic polynomial are 2, 1, and 11, which are the only three eigenvalues of
7610:
7581:
2382:
2227:
928:
Eigenvalues and eigenvectors give rise to many closely related mathematical concepts, and the prefix
24317:
from
Symbolab (Click on the bottom right button of the 2×12 grid to select a matrix size. Select an
24098:
22670:
16357:
1073:
25361:
24769:
24652:
24255:
24251:
24235:
24155:
22202:
Nachrichten von der
Gesellschaft der Wissenschaften zu Göttingen, Mathematisch-Physikalische Klasse
19738:
19712:
of the World Wide Web graph gives the page ranks as its components. This vector corresponds to the
19279:
19259:
19132:
18430:
18256:
15564:
14560:
8022:
7174:
6640:
5534:
5009:
380:
23175:
21005:
17346:
16586:
16437:
11668:. The eigenvalues of a diagonal matrix are the diagonal elements themselves. Consider the matrix
8410:
5275:
2826:. The eigenvectors corresponding to each eigenvalue can be found by solving for the components of
439:
405:
358:
309:
93:
25455:
25246:
25236:
25206:
25140:
24875:
24815:
24744:
24522:
24392:
22333:
22174:
21995:
purposes. Research related to eigen vision systems determining hand gestures has also been made.
20235:
20012:
19768:
Eigenvalue problems occur naturally in the vibration analysis of mechanical structures with many
19713:
19213:
8284:
6237:{\displaystyle \det(A)=\prod _{i=1}^{n}\lambda _{i}=\lambda _{1}\lambda _{2}\cdots \lambda _{n}.}
3772:
2605:, which may not all have distinct values, are roots of the polynomial and are the eigenvalues of
1063:
277:
20426:
19251:
by the principal components. Principal component analysis of the correlation matrix provides an
18094:
17920:
16200:
16151:
6955:
4656:
3762:{\displaystyle E=\left\{\mathbf {v} :\left(A-\lambda I\right)\mathbf {v} =\mathbf {0} \right\}.}
1178:
The first numerical algorithm for computing eigenvalues and eigenvectors appeared in 1929, when
25465:
25344:
25241:
25221:
25216:
25145:
24870:
24579:
24512:
24502:
24320:
22052:
20622:
20616:
20579:
20517:
19697:
16223:. A similar calculation shows that the corresponding eigenvectors are the nonzero solutions of
15490:
15444:
14523:
13516:
13067:
8300:
7397:
7195:
6739:
5936:
5776:
2353:
967:
524:
23931:
23703:
22299:
22149:
19653:
th principal eigenvector of a graph is defined as either the eigenvector corresponding to the
19527:
17494:{\displaystyle {\begin{aligned}\lambda _{1}&=k_{1}\\\lambda _{2}&=k_{2}\end{aligned}}}
16540:{\displaystyle \lambda ={\frac {\mathbf {v} ^{*}A\mathbf {v} }{\mathbf {v} ^{*}\mathbf {v} }}}
14396:, or equivalently if the direct sum of the eigenspaces associated with all the eigenvalues of
13303:. A widely used class of linear transformations acting on infinite-dimensional spaces are the
10729:
8400:{\displaystyle \mathbf {x} ^{\textsf {T}}H\mathbf {x} /\mathbf {x} ^{\textsf {T}}\mathbf {x} }
5102:
4024:. This can be checked by noting that multiplication of complex matrices by complex numbers is
207:
139:
25371:
25301:
25178:
25102:
25041:
25026:
25021:
24998:
24880:
24594:
24589:
24584:
24517:
24462:
24314:
22204:(News of the Philosophical Society at Göttingen, mathematical-physical section), pp. 49–91.
22114:
22012:
21239:
21228:
20879:
20511:
20231:
20039:
19307:
15631:
with known coefficients. For example, once it is known that 6 is an eigenvalue of the matrix
15204:
14535:
13332:
13304:
13274:
8192:
7467:
7357:
6532:
5203:
4898:
4869:
3964:
3892:
1993:{\displaystyle w_{i}=A_{i1}v_{1}+A_{i2}v_{2}+\cdots +A_{in}v_{n}=\sum _{j=1}^{n}A_{ij}v_{j}.}
1375:
1172:
1149:
1141:
1108:
1026:
1019:
1000:
922:
738:
133:
19470:
16226:
13901:
which is the union of the zero vector with the set of all eigenvectors associated with
12190:{\displaystyle A={\begin{bmatrix}1&0&0\\1&2&0\\2&3&3\end{bmatrix}}.}
11750:{\displaystyle A={\begin{bmatrix}1&0&0\\0&2&0\\0&0&3\end{bmatrix}}.}
10521:{\displaystyle A={\begin{bmatrix}0&1&0\\0&0&1\\1&0&0\end{bmatrix}}.}
9829:{\displaystyle A={\begin{bmatrix}2&0&0\\0&3&4\\0&4&9\end{bmatrix}}.}
7543:{\displaystyle A^{\textsf {T}}\mathbf {u} ^{\textsf {T}}=\kappa \mathbf {u} ^{\textsf {T}}.}
6444:
4741:
3036:
The non-real roots of a real polynomial with real coefficients can be grouped into pairs of
25351:
25231:
25226:
25150:
25051:
24604:
24569:
24556:
24447:
24113:
23943:
23879:
23824:
23746:
23661:
23633:
23517:
23419:
23197:
23146:
23120:
22269:
22198:"Grundzüge einer allgemeinen Theorie der linearen Integralgleichungen. (Erste Mitteilung)"
22072:
22032:
21900:
21873:
21846:
21819:
21792:
21592:
21565:
21538:
21220:
21216:
21061:
20670:
20334:. They are associated with eigenvalues interpreted as their energies (increasing downward:
19500:
19100:
15695:
15396:
14531:
13625:
13520:
8453:
8061:
are linearly independent, Q is invertible. Right multiplying both sides of the equation by
6564:
6291:
5511:'s eigenvalues, or equivalently the maximum number of linearly independent eigenvectors of
4458:{\displaystyle {\boldsymbol {v}}_{1},\,\ldots ,\,{\boldsymbol {v}}_{\gamma _{A}(\lambda )}}
4025:
3059:
1211:
1055:
992:
918:
685:
528:
433:
234:
quantities with magnitude and direction, often pictured as arrows. A linear transformation
16029:
15441:
can be determined by finding the roots of the characteristic polynomial. This is easy for
8200:
is the change of basis matrix of the similarity transformation. Essentially, the matrices
7068:
4571:
columns are these eigenvectors, and whose remaining columns can be any orthonormal set of
4062:, or equivalently the maximum number of linearly independent eigenvectors associated with
3067:
8:
25366:
25276:
25198:
25097:
25031:
24988:
24978:
24958:
24782:
24662:
24637:
24487:
23485:
23141:
A First Course In Linear
Algebra: with Optional Introduction to Groups, Rings, and Fields
22273:
20295:
19722:
19348:
16683:
16673:
16580:
14572:
14527:
14482:
8321:
case, eigenvalues can be given a variational characterization. The largest eigenvalue of
6817:
6559:
3142:
1379:
235:
24117:
23947:
23883:
23828:
23802:
23665:
23637:
23521:
23201:
22655:
19274:, the eigenvalues of the correlation matrix determine the Q-methodologist's judgment of
19243:
equal to one). For the covariance or correlation matrix, the eigenvectors correspond to
15516:
331:
25392:
25311:
25251:
25183:
25173:
25112:
25087:
24963:
24920:
24915:
24492:
24379:
23975:
23916:
23895:
23840:
23814:
23533:
23384:
23326:
23308:
23229:
23187:
23139:
22279:
22226:
For the origin and evolution of the terms eigenvalue, characteristic value, etc., see:
22042:
21976:
21247:
20923:
20858:
20838:
20818:
20649:
20548:
20489:
20408:
20287:
20067:
19943:
19919:
19899:
19876:
19676:
19656:
19636:
19565:
19450:
19320:
19283:
19248:
19236:
19166:
18972:
16665:
16459:
16317:
16180:
16131:
16111:
15925:{\displaystyle \left\{{\begin{aligned}4x+y&=6x\\6x+3y&=6y\end{aligned}}\right.}
15594:
15570:
15546:
15540:
15424:
14377:
13241:
10704:
10437:
8324:
8296:
8288:
7659:
7639:
7561:
7447:
7423:
7377:
7296:
7221:
7048:
6843:
6719:
6649:
6622:
6591:
6541:
6424:
6397:
6318:
6271:
6251:
6123:
5942:
5756:
5727:
5672:
5652:
5571:
5514:
5494:
5345:
5183:
4989:
4849:
4829:
4636:
4616:
4518:
3796:
3652:
1195:
1191:
1092:
239:
115:
24287:
24126:
23645:
21961:. The dimension of this vector space is the number of pixels. The eigenvectors of the
16017:{\displaystyle \left\{{\begin{aligned}-2x+y&=0\\6x-3y&=0\end{aligned}}\right.}
11589:, respectively. The two complex eigenvectors also appear in a complex conjugate pair,
25387:
25107:
25092:
25036:
24983:
24690:
24647:
24574:
24467:
24371:
24200:
24182:
23967:
23959:
23899:
23857:
23844:
23788:
23770:
23750:
23728:
23709:
23695:
23669:
23611:
23593:
23558:
23537:
23493:
23468:
23404:
23376:
23330:
23279:
23272:
23259:
23255:
23233:
23221:
23213:
23150:
23124:
23033:
22057:
21962:
21232:
21208:
21196:
21188:
20995:
20430:
20057:
19773:
19232:
19220:
19162:
16629:
16350:
12097:
11664:
8179:
6415:
5488:
3674:
3037:
3014:
1179:
1160:
1096:
255:
30:"Characteristic root" redirects here. For the root of a characteristic equation, see
23979:
23388:
18523:{\displaystyle {\begin{aligned}\gamma _{1}&=1\\\gamma _{2}&=1\end{aligned}}}
18419:{\displaystyle {\begin{aligned}\gamma _{1}&=1\\\gamma _{2}&=1\end{aligned}}}
18349:{\displaystyle {\begin{aligned}\gamma _{1}&=1\\\gamma _{2}&=1\end{aligned}}}
12918:
of the eigenvalue 2 is only 1, because its eigenspace is spanned by just one vector
3405:). The size of each eigenvalue's algebraic multiplicity is related to the dimension
1145:
712:
25321:
25296:
25168:
25016:
24953:
24695:
24599:
24452:
24164:
24131:
24121:
24074:
23951:
23887:
23832:
23691:
23641:
23553:
23525:
23450:
23428:
23366:
23358:
23318:
23209:
23205:
22096:"Du mouvement d'un corps solide quelconque lorsqu'il tourne autour d'un axe mobile"
21988:
21950:
21200:
20382:
20085:
19961:
19709:
19338:
19315:
19252:
19209:
19198:
16633:
16622:
15612:
15588:
14556:
14505:
13892:{\displaystyle E=\left\{\mathbf {v} :T(\mathbf {v} )=\lambda \mathbf {v} \right\},}
13059:
8318:
8280:
6613:
6609:
5959:, defined as the sum of its diagonal elements, is also the sum of all eigenvalues,
3030:
1207:
1127:
1120:
1116:
1112:
1085:
1077:
1043:
585:
567:
227:
192:. It is often important to know these vectors in linear algebra. The corresponding
50:
24143:
22117:
proved that any body has three principal axes of rotation: Johann Andreas Segner,
12101:, while a matrix whose elements below the main diagonal are all zero is called an
8299:
and therefore admits a basis of generalized eigenvectors and a decomposition into
947:
corresponding to the same eigenvalue, together with the zero vector, is called an
734:
eigenvalue equal to one, because the mapping does not change their length either.
25261:
25188:
25117:
24910:
24754:
24547:
24507:
24497:
23988:
23836:
23653:
23292:
Denton, Peter B.; Parke, Stephen J.; Tao, Terence; Zhang, Xining (January 2022).
22679:
22433:
22200:(Fundamentals of a general theory of linear integral equations. (First report)),
21532:
20283:
20064:
are different from the principal compliance modes, which are the eigenvectors of
19291:
19287:
19240:
19228:
19217:
19188:
19078:
16343:
15841:
15476:
14433:
13618:
10773:= 1, any vector with three equal nonzero entries is an eigenvector. For example,
8312:
4358:, consider how the definition of geometric multiplicity implies the existence of
3848:
3022:
2211:
1164:
1137:
1081:
1030:
584:, eigenvalues and eigenvectors have a wide range of applications, for example in
553:
24366:
23176:"Fluctuations and Correlations of Transmission Eigenchannels in Diffusive Media"
20270:
is a key quantity required to determine the rotation of a rigid body around its
12974:{\displaystyle {\begin{bmatrix}0&1&-1&1\end{bmatrix}}^{\textsf {T}}}
8463:
7607:, with the same eigenvalue. Furthermore, since the characteristic polynomial of
25339:
25266:
24973:
24759:
24680:
24415:
23856:, Translated and edited by Richard A. Silverman, New York: Dover Publications,
23762:
23400:
23294:"Eigenvectors from Eigenvalues: A Survey of a Basic Identity in Linear Algebra"
22067:
22037:
21204:
21192:
20612:
20394:
20271:
20251:
20061:
19267:
19077:
A linear transformation that takes a square to a rectangle of the same area (a
15410:
13308:
13034:{\displaystyle {\begin{bmatrix}0&0&0&1\end{bmatrix}}^{\textsf {T}}}
10722:
8342:
6667:
4394:
3033:, which include the rationals, the eigenvalues must also be algebraic numbers.
2242:
1104:
1051:
1007:
996:
988:
693:
593:
40:
24079:
24062:
23567:
23174:
Bender, Nicholas; Yamilov, Alexey; Yilmaz, Hasan; Cao, Hui (14 October 2020).
22685:
22451:
18187:{\displaystyle {\begin{aligned}\mu _{1}&=1\\\mu _{2}&=1\end{aligned}}}
18083:{\displaystyle {\begin{aligned}\mu _{1}&=1\\\mu _{2}&=1\end{aligned}}}
18013:{\displaystyle {\begin{aligned}\mu _{1}&=1\\\mu _{2}&=1\end{aligned}}}
12673:
has a characteristic polynomial that is the product of its diagonal elements,
25444:
25127:
25059:
25011:
24792:
24715:
24675:
24642:
24622:
23963:
23481:
23455:
23433:
23217:
22137:
22017:
21212:
20987:
20920:
is understood to be the vector obtained by application of the transformation
20608:
20327:
20323:
19271:
16678:
16625:
15628:
15043:{\displaystyle {\begin{bmatrix}x_{t}&\cdots &x_{t-k+1}\end{bmatrix}}}
13598:
13296:
13295:
remains valid even if the underlying vector space is an infinite-dimensional
13286:
7749:
2432:
1156:
1059:
1022:
realized that the principal axes are the eigenvectors of the inertia matrix.
776:
726:
704:
699:
558:
513:
243:
21953:, processed images of faces can be seen as vectors whose components are the
16636:
to compute eigenvalues and eigenvectors, among several other possibilities.
15467:
matrices, but the difficulty increases rapidly with the size of the matrix.
11648:{\displaystyle \mathbf {v} _{\lambda _{2}}=\mathbf {v} _{\lambda _{3}}^{*}.}
10431:
3971:
is a linear subspace, it is closed under scalar multiplication. That is, if
25069:
25064:
24968:
24725:
24614:
24564:
24457:
24375:
23971:
23225:
22098:(On the movement of any solid body while it rotates around a moving axis),
22027:
21528:
21318:
20542:
20425:
is represented in terms of a differential operator is the time-independent
20311:
19734:
19717:
19311:
19290:). More generally, principal component analysis can be used as a method of
18976:
16614:
14388:
to such a subspace is diagonalizable. Moreover, if the entire vector space
13609:
The concept of eigenvalues and eigenvectors extends naturally to arbitrary
13300:
13280:
12324:. These roots are the diagonal elements as well as the eigenvalues of
11884:. These roots are the diagonal elements as well as the eigenvalues of
8002:
With this in mind, define a diagonal matrix Λ where each diagonal element Λ
4214:{\displaystyle \gamma _{A}(\lambda )=n-\operatorname {rank} (A-\lambda I).}
1383:
1187:
1183:
1163:
by viewing the operators as infinite matrices. He was the first to use the
858:
by 1 matrices. If the linear transformation is expressed in the form of an
263:
24826:
24048:
23578:
23380:
23056:
22100:
Histoire de l'Académie royale des sciences et des belles lettres de Berlin
20238:, but neatly generalize the solution to scalar-valued vibration problems.
20230:
The orthogonality properties of the eigenvectors allows decoupling of the
16401:
this causes it to converge to an eigenvector of the eigenvalue closest to
15193:{\displaystyle x_{t}=c_{1}\lambda _{1}^{t}+\cdots +c_{k}\lambda _{k}^{t}.}
13821:
13291:
The definitions of eigenvalue and eigenvectors of a linear transformation
13273:
th row and column from the original matrix. This identity also extends to
12890:
The roots of this polynomial, and hence the eigenvalues, are 2 and 3. The
10361:{\displaystyle {\begin{bmatrix}0&-2&1\end{bmatrix}}^{\textsf {T}}}
1033:, and generalized it to arbitrary dimensions. Cauchy also coined the term
572:
25271:
24935:
24858:
24705:
24670:
24627:
24472:
24309:
24151:
23955:
23803:"Light fields in complex media: Mesoscopic scattering meets wave control"
20573:
20315:
19964:. Admissible solutions are then a linear combination of solutions to the
19937:
15484:
12094:
A matrix whose elements above the main diagonal are all zero is called a
10419:{\displaystyle {\begin{bmatrix}0&1&2\end{bmatrix}}^{\textsf {T}}}
10301:{\displaystyle {\begin{bmatrix}1&0&0\end{bmatrix}}^{\textsf {T}}}
6117:
3045:
2365:
2246:
1100:
24348:
23322:
22977:
6524:{\textstyle {\frac {1}{\lambda _{1}}},\ldots ,{\frac {1}{\lambda _{n}}}}
4291:{\displaystyle 1\leq \gamma _{A}(\lambda )\leq \mu _{A}(\lambda )\leq n}
548:
466:
is the corresponding eigenvalue. This relationship can be expressed as:
25256:
25135:
24930:
24734:
24477:
23727:, Philadelphia: Society for Industrial and Applied Mathematics (SIAM),
23362:
22122:
22022:
21984:
21965:
associated with a large set of normalized pictures of faces are called
21954:
21236:
20876:
20259:
16694:
14350:
13610:
13320:
12285:{\displaystyle \det(A-\lambda I)=(1-\lambda )(2-\lambda )(3-\lambda ),}
11845:{\displaystyle \det(A-\lambda I)=(1-\lambda )(2-\lambda )(3-\lambda ),}
6535:
of the original, the eigenvalues share the same algebraic multiplicity.
2345:
1011:
917:
by 1 matrix. For a matrix, eigenvalues and eigenvectors can be used to
581:
87:
23530:
10.1002/1096-9837(200012)25:13<1473::AID-ESP158>3.0.CO;2-C
23080:
22864:
20298:
components; the components it does have are the principal components.
16848:{\displaystyle {\begin{bmatrix}k_{1}&0\\0&k_{2}\end{bmatrix}}}
14978:-dimensional system of the first order in the stacked variable vector
13503:
This differential equation can be solved by multiplying both sides by
2885:
In this example, the eigenvectors are any nonzero scalar multiples of
1220:
627:, called an eigenvalue. This condition can be written as the equation
534:
The following section gives a more general viewpoint that also covers
24532:
24135:
23463:
Friedberg, Stephen H.; Insel, Arnold J.; Spence, Lawrence E. (1989),
23371:
21967:
21944:
21936:
21224:
20991:
20247:
19763:
19089:
16307:{\displaystyle {\begin{bmatrix}b&-3b\end{bmatrix}}^{\textsf {T}}}
14694:{\displaystyle x_{t}=a_{1}x_{t-1}+a_{2}x_{t-2}+\cdots +a_{k}x_{t-k}.}
13266:
8898:
Setting the characteristic polynomial equal to zero, it has roots at
7484:
is its associated eigenvalue. Taking the transpose of this equation,
7038:{\displaystyle \{\lambda _{1}+\alpha ,\ldots ,\lambda _{k}+\alpha \}}
2814:
Setting the characteristic polynomial equal to zero, it has roots at
722:
597:
517:
231:
24306:
Eigenvectors and eigenvalues | Essence of linear algebra, chapter 10
23104:
22743:
22741:
22227:
22106:, Euler proves that any body contains a principal axis of rotation:
16101:{\displaystyle {\begin{bmatrix}a&2a\end{bmatrix}}^{\textsf {T}}}
8283:. For defective matrices, the notion of eigenvectors generalizes to
4508:{\displaystyle A{\boldsymbol {v}}_{k}=\lambda {\boldsymbol {v}}_{k}}
3653:
Eigenspaces, geometric multiplicity, and the eigenbasis for matrices
25160:
25079:
25006:
24700:
24254:
external links, and converting useful links where appropriate into
23891:
23819:
23313:
23192:
23013:
22852:
20319:
20291:
19873:
That is, acceleration is proportional to position (i.e., we expect
19705:
19263:
19170:
18836:{\displaystyle \mathbf {u} _{1}={\begin{bmatrix}1\\0\end{bmatrix}}}
15619:
for the roots of a degree 3 polynomial is numerically impractical.
15472:
14286:
is closed under addition and scalar multiplication. The eigenspace
846:
Alternatively, the linear transformation could take the form of an
839:{\displaystyle {\frac {d}{dx}}e^{\lambda x}=\lambda e^{\lambda x}.}
623:
to the eigenvector only scales the eigenvector by the scalar value
259:
24290:– non-technical introduction from PhysLink.com's "Ask the Experts"
23932:"Focusing coherent light through opaque strongly scattering media"
23921:
23544:
Hawkins, T. (1975), "Cauchy and the spectral theory of matrices",
22989:
22950:
22818:
22816:
21380:{\displaystyle \mathbf {v} _{1},\mathbf {v} _{2},\mathbf {v} _{3}}
16724:
16701:
12907:, the order of the characteristic polynomial and the dimension of
2176:{\displaystyle \left(A-\lambda I\right)\mathbf {v} =\mathbf {0} ,}
576:) for 'proper', 'characteristic', 'own'. Originally used to study
24945:
24384:
22738:
21314:
20267:
14511:
8588:{\displaystyle A={\begin{bmatrix}2&1\\1&2\end{bmatrix}}.}
2876:{\displaystyle \left(A-\lambda I\right)\mathbf {v} =\mathbf {0} }
2667:{\displaystyle A={\begin{bmatrix}2&1\\1&2\end{bmatrix}}.}
563:
251:
23760:
23754:
23441:
Francis, J. G. F. (1962), "The QR Transformation, II (part 2)",
22691:
22457:
20306:
15683:{\displaystyle A={\begin{bmatrix}4&1\\6&3\end{bmatrix}}}
2221:
2055:{\displaystyle A\mathbf {v} =\mathbf {w} =\lambda \mathbf {v} ,}
24710:
23492:(3rd ed.), Baltimore, MD: Johns Hopkins University Press,
23417:
Francis, J. G. F. (1961), "The QR Transformation, I (part 1)",
23263:
23169:, Free online book under GNU licence, University of Puget Sound
23068:
22967:
22965:
22813:
22716:
22714:
22712:
22542:
22540:
22463:
22136:
The relevant passage of Segner's work was discussed briefly by
21322:
20566:
20545:, is one of its eigenfunctions corresponding to the eigenvalue
20398:
20263:
19701:
14967:{\displaystyle x_{t-1}=x_{t-1},\ \dots ,\ x_{t-k+1}=x_{t-k+1},}
14553:
are the analogs of eigenvectors and eigenspaces, respectively.
8442:
7285:
The eigenvalue and eigenvector problem can also be defined for
3124:
of the characteristic polynomial, that is, the largest integer
22907:
22386:
22384:
21924:
is then the largest eigenvalue of the next generation matrix.
20405:
An example of an eigenvalue equation where the transformation
16983:{\displaystyle {\begin{bmatrix}1&k\\0&1\end{bmatrix}}}
16780:{\displaystyle {\begin{bmatrix}k&0\\0&k\end{bmatrix}}}
14298:. If that subspace has dimension 1, it is sometimes called an
11662:
Matrices with entries only along the main diagonal are called
4692:, we get a matrix whose top left block is the diagonal matrix
1144:
on general domains towards the end of the 19th century, while
62:
24063:"Eigenvector components of symmetric, graph-related matrices"
23787:(3rd ed.), New York: Springer Science + Business Media,
23346:
22624:
21958:
21932:
20983:
14342:) ≥ 1 because every eigenvalue has at least one eigenvector.
13386:
The functions that satisfy this equation are eigenvectors of
7679:
3025:
or even if they are all integers. However, if the entries of
23044:
23001:
22962:
22709:
22564:
22552:
22537:
19758:
Mode shape of a tuning fork at eigenfrequency 440.09 Hz
12539:
respectively, as well as scalar multiples of these vectors.
12086:
respectively, as well as scalar multiples of these vectors.
8622:
Taking the determinant to find characteristic polynomial of
5170:{\displaystyle \mu _{A}(\lambda )\geq \gamma _{A}(\lambda )}
4351:{\displaystyle \gamma _{A}(\lambda )\leq \mu _{A}(\lambda )}
2582:
may be real but in general is a complex number. The numbers
23105:"Eigenvalue, eigenfunction, eigenvector, and related terms"
22940:
22938:
22936:
22934:
22381:
22228:
Earliest Known Uses of Some of the Words of Mathematics (E)
20223:
This can be reduced to a generalized eigenvalue problem by
16708:
16011:
15919:
12542:
7157:{\displaystyle \{P(\lambda _{1}),\ldots ,P(\lambda _{k})\}}
3847:
by 1 matrices. A property of the nullspace is that it is a
1201:
56:
24046:
23164:
23086:
22919:
22600:
22349:
21975:. They are very useful for expressing any face image as a
21235:, one often represents the Hartree–Fock equation in a non-
13277:, and has been rediscovered many times in the literature.
8287:
and the diagonal matrix of eigenvalues generalizes to the
8248:
whose eigenvalue is the corresponding diagonal element of
5724:
can be written as a linear combination of eigenvectors of
4224:
that an eigenvalue's algebraic multiplicity cannot exceed
22765:
22726:
22612:
20572:
However, in the case where one is interested only in the
10909:
For the complex conjugate pair of imaginary eigenvalues,
10432:
Three-dimensional matrix example with complex eigenvalues
6917:{\displaystyle \{\lambda _{1}+1,\ldots ,\lambda _{k}+1\}}
6612:, then every eigenvalue is real. The same is true of any
6384:{\displaystyle \lambda _{1}^{k},\ldots ,\lambda _{n}^{k}}
59:
22931:
22515:
22513:
22511:
22509:
22125:
p. xxviiii ), Segner derives a third-degree equation in
21616:
are dictated by the nature of the sediment's fabric. If
20214:{\displaystyle \left(\omega ^{2}m+\omega c+k\right)x=0.}
16715:
14412:
can be formed from linearly independent eigenvectors of
13281:
Eigenvalues and eigenfunctions of differential operators
12547:
As in the previous example, the lower triangular matrix
23684:
USSR Computational Mathematics and Mathematical Physics
23344:
23109:
Earliest Known Uses of Some of the Words of Mathematics
23062:
22840:
22828:
22777:
22753:
22697:
22576:
22525:
22484:
22482:
22480:
22478:
20766:{\displaystyle H|\Psi _{E}\rangle =E|\Psi _{E}\rangle }
17332:{\displaystyle \lambda ^{2}-2\cosh(\varphi )\lambda +1}
16476:, then the corresponding eigenvalue can be computed as
15543:
for the roots of a polynomial exist only if the degree
13822:
Eigenspaces, geometric multiplicity, and the eigenbasis
7578:
is the same as the transpose of a right eigenvector of
5748:
5092:{\displaystyle (\xi -\lambda )^{\gamma _{A}(\lambda )}}
3799:
of the zero vector with the set of all eigenvectors of
779:
that are scaled by that differential operator, such as
24029:"Neutrinos Lead to Unexpected Discovery in Basic Math"
23173:
22983:
22401:
22399:
19588:
18928:
18879:
18812:
18746:
18694:
18623:
18574:
17003:
16949:
16868:
16800:
16746:
16271:
16068:
15807:
15772:
15736:
15649:
15098:{\displaystyle \lambda _{1},\,\ldots ,\,\lambda _{k},}
14990:
13410:
13244:
12994:
12931:
12709:
12565:
12496:
12434:
12365:
12126:
12043:
11984:
11925:
11686:
11519:
11428:
11332:
11269:
11219:
11149:
11086:
11036:
10866:
10824:
10788:
10457:
10384:
10323:
10266:
10031:
9960:
9891:
9765:
9657:
9614:
9438:
9395:
9353:
9236:
9190:
9059:
9016:
8980:
8761:
8717:
8675:
8551:
8505:
preserves the direction of purple vectors parallel to
7900:
7784:
6477:
4653:
has full rank and is therefore invertible. Evaluating
2972:
2918:
2728:
2630:
1773:
1709:
1531:
1325:
1305:
1272:
775:, in which case the eigenvectors are functions called
749:
619:
is applied to it, does not change direction. Applying
24323:
23765:; Vetterling, William T.; Flannery, Brian P. (2007),
23467:(2nd ed.), Englewood Cliffs, NJ: Prentice Hall,
22801:
22789:
22636:
22506:
22494:
22411:
22302:
22282:
21903:
21876:
21849:
21822:
21795:
21728:
21675:
21622:
21595:
21568:
21541:
21504:
21475:
21446:
21393:
21334:
21152:
21119:
21084:
21064:
21030:
21008:
20946:
20926:
20888:
20861:
20841:
20821:
20782:
20717:
20679:
20652:
20625:
20582:
20576:
solutions of the Schrödinger equation, one looks for
20551:
20520:
20492:
20442:
20411:
20340:
20163:
20094:
20070:
20042:
20015:
19973:
19946:
19922:
19902:
19879:
19833:
19786:
19679:
19659:
19639:
19568:
19530:
19503:
19473:
19453:
19381:
19351:
19323:
19135:
19103:
19039:
18984:
18850:
18791:
18665:
18545:
18467:
18433:
18363:
18293:
18259:
18209:
18131:
18097:
18027:
17957:
17923:
17873:
17716:
17669:
17508:
17424:
17377:
17349:
17288:
17245:
17231:{\displaystyle \lambda ^{2}-2\cos(\theta )\lambda +1}
17187:
17125:
17082:
16997:
16943:
16862:
16794:
16740:
16589:
16556:
16485:
16462:
16440:
16408:
16360:
16320:
16264:
16229:
16203:
16183:
16154:
16134:
16114:
16061:
16032:
15944:
15849:
15730:
15698:
15692:
we can find its eigenvectors by solving the equation
15637:
15597:
15573:
15549:
15519:
15493:
15447:
15427:
15216:
15114:
15060:
14984:
14863:
14721:
14584:
14120:
13934:
13836:
13744:{\displaystyle T(\mathbf {v} )=\lambda \mathbf {v} .}
13712:
13646:
13528:
13442:
13408:
13340:
13076:
12987:
12924:
12679:
12553:
12337:
12209:
12114:
11897:
11769:
11674:
11595:
11490:
11399:
11210:
11027:
10915:
10779:
10732:
10707:
10540:
10445:
10377:
10316:
10259:
9848:
9753:
9587:
9300:
9163:
9095:
8938:
8632:
8539:
8456:
8413:
8350:
8327:
8129:
8074:
8025:
7885:
7772:
7662:
7642:
7613:
7584:
7564:
7558:), it follows immediately that a left eigenvector of
7490:
7470:
7450:
7426:
7400:
7380:
7360:
7319:
7299:
7250:
7224:
7198:
7100:
7071:
7051:
6987:
6958:
6930:
6866:
6846:
6820:
6807:{\displaystyle \{\lambda _{1},\ldots ,\lambda _{k}\}}
6768:
6742:
6722:
6676:
6652:
6625:
6594:
6567:
6544:
6447:
6427:
6400:
6341:
6321:
6294:
6274:
6254:
6148:
6126:
5967:
5945:
5894:
5851:
5805:
5779:
5759:
5730:
5701:
5675:
5655:
5626:
5594:
5574:
5537:
5517:
5497:
5368:
5348:
5305:
5278:
5232:
5206:
5186:
5125:
5105:
5050:
5012:
4992:
4930:
4901:
4872:
4852:
4832:
4770:
4744:
4698:
4659:
4639:
4619:
4577:
4541:
4521:
4471:
4403:
4364:
4306:
4234:
4155:
4075:
3861:
3843:
is a complex number and the eigenvectors are complex
3696:
3415:
3188:
2891:
2837:
2698:
2618:
2450:
2280:
2134:
2021:
1845:
1525:
1489:
1434:
1396:
1258:
876:
785:
747:
665:{\displaystyle T(\mathbf {v} )=\lambda \mathbf {v} ,}
633:
472:
442:
408:
383:
361:
334:
312:
280:
210:
162:
142:
118:
96:
71:
24050:
Estimation of 3D motion and structure of human faces
23986:
23623:
23462:
23278:(5th ed.), Boston: Prindle, Weber and Schmidt,
23019:
22870:
22654:
Cornell University Department of Mathematics (2016)
22475:
22469:
20290:
tensor is symmetric and so can be decomposed into a
16026:
Both equations reduce to the single linear equation
14559:
is a tensor-multiple of itself and is considered in
13767:
This equation is called the eigenvalue equation for
6814:
are its eigenvalues, then the eigenvalues of matrix
995:. Historically, however, they arose in the study of
24352:Wikiversity uses introductory physics to introduce
24047:Xirouhakis, A.; Votsis, G.; Delopoulus, A. (2004),
23702:Lipschutz, Seymour; Lipson, Marc (12 August 2002).
23583:, Colchester, VT: Online book, St Michael's College
23565:
23291:
23243:
A Practical Guide to the study of Glacial Sediments
23074:
22822:
22588:
22396:
22368:Gilbert Strang. "6: Eigenvalues and Eigenvectors".
21273:
may be too technical for most readers to understand
21215:. The corresponding eigenvalues are interpreted as
21105:{\displaystyle \mathbf {t} ^{\dagger }\mathbf {t} }
21051:{\displaystyle \mathbf {t} ^{\dagger }\mathbf {t} }
19725:. Other methods are also available for clustering.
19708:algorithm. The principal eigenvector of a modified
19258:Principal component analysis is used as a means of
14325:is the dimension of the eigenspace associated with
10252:. These eigenvalues correspond to the eigenvectors
9742:
7276:{\displaystyle A\mathbf {v} =\lambda \mathbf {v} .}
987:Eigenvalues are often introduced in the context of
902:{\displaystyle A\mathbf {v} =\lambda \mathbf {v} ,}
156:, when the linear transformation is applied to it:
53:
24335:
24096:
23767:Numerical Recipes: The Art of Scientific Computing
23271:
23138:
23137:Beauregard, Raymond A.; Fraleigh, John B. (1973),
23136:
22747:
22648:
22324:
22288:
21916:
21889:
21862:
21835:
21808:
21767:
21714:
21661:
21608:
21581:
21554:
21519:
21490:
21461:
21432:
21379:
21171:
21138:
21104:
21070:
21050:
21016:
20967:
20932:
20912:
20867:
20847:
20827:
20803:
20765:
20700:
20658:
20638:
20595:
20557:
20533:
20498:
20475:
20417:
20373:
20213:
20145:
20076:
20048:
20028:
20001:
19952:
19928:
19908:
19885:
19863:
19819:
19685:
19665:
19645:
19625:
19574:
19554:
19516:
19489:
19459:
19435:
19363:
19329:
19153:
19121:
19066:
19021:
18958:
18835:
18776:
18650:
18522:
18452:
18418:
18348:
18278:
18244:
18186:
18116:
18082:
18012:
17942:
17908:
17850:
17701:
17654:
17493:
17409:
17362:
17331:
17273:
17230:
17172:
17110:
17060:
16982:
16928:
16847:
16779:
16597:
16571:
16539:
16468:
16448:
16422:
16391:
16326:
16306:
16250:
16215:
16189:
16166:
16140:
16120:
16100:
16047:
16016:
15924:
15830:
15716:
15682:
15603:
15579:
15555:
15528:
15505:
15459:
15433:
15379:
15192:
15097:
15042:
14966:
14842:
14693:
14233:
14056:
13891:
13743:
13667:
13574:
13493:
13428:
13376:
13257:
13230:
13053:
13033:
12973:
12880:
12665:
12529:
12331:These eigenvalues correspond to the eigenvectors,
12284:
12189:
12076:
11844:
11749:
11647:
11567:
11476:
11379:
11196:
11011:
10899:
10754:
10713:
10693:
10520:
10418:
10360:
10300:
10238:
9828:
9680:
9557:
9262:
9133:
9082:
8888:
8587:
8497:
8421:
8399:
8333:
8279:A matrix that is not diagonalizable is said to be
8160:
8105:
8046:
7991:
7845:
7668:
7648:
7628:
7599:
7570:
7542:
7476:
7456:
7432:
7412:
7386:
7366:
7345:{\displaystyle \mathbf {u} A=\kappa \mathbf {u} ,}
7344:
7305:
7275:
7230:
7210:
7156:
7086:
7057:
7037:
6973:
6944:
6916:
6852:
6832:
6806:
6754:
6728:
6705:
6658:
6631:
6600:
6580:
6550:
6523:
6463:
6433:
6406:
6383:
6327:
6307:
6280:
6260:
6236:
6132:
6104:
5951:
5923:
5880:
5837:
5791:
5765:
5736:
5716:
5681:
5661:
5641:
5609:
5580:
5556:
5523:
5503:
5479:
5354:
5334:
5291:
5264:
5218:
5192:
5169:
5111:
5091:
5036:
4998:
4978:
4916:
4887:
4858:
4838:
4818:
4756:
4730:
4684:
4645:
4625:
4605:
4563:
4527:
4507:
4457:
4386:
4350:
4290:
4213:
4097:
3876:
3761:
3550:
3377:
3044:at least one of the roots is real. Therefore, any
2998:
2875:
2804:
2666:
2549:
2310:
2175:
2054:
1992:
1831:
1511:
1456:
1419:{\displaystyle \mathbf {x} =\lambda \mathbf {y} .}
1418:
1364:
1248:-dimensional vectors that are formed as a list of
901:
838:
767:
664:
495:{\displaystyle A\mathbf {v} =\lambda \mathbf {v} }
494:
450:
416:
394:
369:
343:
320:
294:
216:
185:{\displaystyle T\mathbf {v} =\lambda \mathbf {v} }
184:
148:
124:
104:
24296:– Tutorial and Interactive Program from Revoledu.
24294:Eigen Values and Eigen Vectors Numerical Examples
24238:may not follow Knowledge's policies or guidelines
23608:Mathematical thought from ancient to modern times
20708:. In this notation, the Schrödinger equation is:
19737:is represented by a matrix whose entries are the
19696:The principal eigenvector is used to measure the
19223:are PSD. This orthogonal decomposition is called
18245:{\displaystyle \gamma _{i}=\gamma (\lambda _{i})}
13494:{\displaystyle {\frac {d}{dt}}f(t)=\lambda f(t).}
8437:
8291:. Over an algebraically closed field, any matrix
7748:. The eigenvalues need not be distinct. Define a
5838:{\displaystyle \lambda _{1},\ldots ,\lambda _{n}}
5265:{\displaystyle \lambda _{1},\ldots ,\lambda _{d}}
5099:, which means that the algebraic multiplicity of
4731:{\displaystyle \lambda I_{\gamma _{A}(\lambda )}}
2415:implies that the characteristic polynomial of an
1029:saw how their work could be used to classify the
25442:
24106:Journal of Computational and Applied Mathematics
23930:Vellekoop, I. M.; Mosk, A. P. (15 August 2007).
23569:Mathematical epidemiology of infectious diseases
22663:
22121:( Halle ("Halae"), (Germany): Gebauer, 1755). (
21158:
21125:
21024:. The eigenvectors of the transmission operator
18971:The characteristic equation for a rotation is a
17173:{\displaystyle (\lambda -k_{1})(\lambda -k_{2})}
12680:
12210:
11770:
9853:
8637:
7636:is the same as the characteristic polynomial of
7313:. In this formulation, the defining equation is
6149:
5013:
4955:
4931:
3189:
2699:
2451:
2281:
24097:Golub, Gene F.; van der Vorst, Henk A. (2000),
23801:Rotter, Stefan; Gigan, Sylvain (2 March 2017).
23701:
23345:Diekmann, O; Heesterbeek, JA; Metz, JA (1990),
23270:Burden, Richard L.; Faires, J. Douglas (1993),
22913:
22657:Lower-Level Courses for Freshmen and Sophomores
19084:
16456:is (a good approximation of) an eigenvector of
13795:) is the result of applying the transformation
13315:be a linear differential operator on the space
9698:= 3, as is any scalar multiple of this vector.
9278:= 1, as is any scalar multiple of this vector.
8527:An extended version, showing all four quadrants
8306:
4141:), which relates to the dimension and rank of (
3399:linear terms and this is the same as equation (
1252:scalars, such as the three-dimensional vectors
24300:Introduction to Eigen Vectors and Eigen Values
21469:then is the primary orientation/dip of clast,
20241:
19288:criteria for determining the number of factors
19239:(in which each variable is scaled to have its
19129:with a standard deviation of 3 in roughly the
14712:is found by using its characteristic equation
14512:Associative algebras and representation theory
13397:
8429:that realizes that maximum is an eigenvector.
7168:
1037:(characteristic root), for what is now called
24842:
24400:
23929:
23681:
23566:Heesterbeek, J. A. P.; Diekmann, Odo (2000),
23505:
23480:
23301:Bulletin of the American Mathematical Society
23034:"Endogene Geologie - Ruhr-Universität Bochum"
22995:
22956:
22720:
22630:
22618:
22258:Lemma for linear independence of eigenvectors
20146:{\displaystyle m{\ddot {x}}+c{\dot {x}}+kx=0}
19067:{\displaystyle \cos \theta \pm i\sin \theta }
10180:
10137:
8116:or by instead left multiplying both sides by
3967:of matrix multiplication. Similarly, because
2222:Eigenvalues and the characteristic polynomial
2100:corresponding to that eigenvector. Equation (
1186:. One of the most popular methods today, the
531:, or the language of linear transformations.
24181:, Wellesley, MA: Wellesley-Cambridge Press,
24099:"Eigenvalue Computation in the 20th Century"
23905:
23769:(3rd ed.), Cambridge University Press,
23269:
22944:
22355:
22218:
22208:
22003:recognition systems for speaker adaptation.
21778:
20962:
20907:
20798:
20760:
20736:
20695:
16648:
14543:representation-theoretical concept of weight
14534:. The study of such actions is the field of
12089:
7151:
7101:
7032:
6988:
6911:
6867:
6801:
6769:
5568:The direct sum of the eigenspaces of all of
4613:vectors orthogonal to these eigenvectors of
4043:are not zero, they are also eigenvectors of
1080:have real eigenvalues. This was extended by
86:that has its direction unchanged by a given
24856:
24060:
23914:
23800:
22971:
22846:
22834:
19314:is defined as an eigenvalue of the graph's
17909:{\displaystyle \mu _{i}=\mu (\lambda _{i})}
17702:{\displaystyle \lambda _{1}=\lambda _{2}=1}
17410:{\displaystyle \lambda _{1}=\lambda _{2}=k}
16337:
8268:. It then follows that the eigenvectors of
6418:if and only if every eigenvalue is nonzero.
5799:matrix of complex numbers with eigenvalues
4979:{\displaystyle \det(A-\xi I)=\det(D-\xi I)}
4515:. We can therefore find a (unitary) matrix
3395:then the right-hand side is the product of
3170:) factors the characteristic polynomial of
1512:{\displaystyle A\mathbf {v} =\mathbf {w} ,}
854:matrix, in which case the eigenvectors are
24849:
24835:
24407:
24393:
23725:Matrix analysis and applied linear algebra
23249:
23031:
22669:University of Michigan Mathematics (2016)
22417:
22367:
21211:can be defined by the eigenvectors of the
15840:This matrix equation is equivalent to two
15203:A similar procedure is used for solving a
13928:By definition of a linear transformation,
11657:
8215:be a non-singular square matrix such that
7680:Diagonalization and the eigendecomposition
7676:are associated with the same eigenvalues.
5588:'s eigenvalues is the entire vector space
1467:Now consider the linear transformation of
24372:Numerical solution of eigenvalue problems
24274:Learn how and when to remove this message
24125:
24078:
24056:, National Technical University of Athens
24026:
24006:
23920:
23869:
23818:
23708:. McGraw Hill Professional. p. 111.
23592:, Waltham: Blaisdell Publishing Company,
23557:
23454:
23432:
23370:
23312:
23240:
23191:
23050:
23007:
22807:
21987:, eigenfaces provide a means of applying
21669:, the fabric is said to be isotropic. If
21433:{\displaystyle E_{1}\geq E_{2}\geq E_{3}}
21301:Learn how and when to remove this message
21285:, without removing the technical details.
21253:
20472:
20367:
20360:
20353:
19231:among variables. PCA is performed on the
16658:Eigenvalues of geometric transformations
16416:
16298:
16092:
15587:is the characteristic polynomial of some
15081:
15074:
13025:
12965:
11559:
11468:
11389:Therefore, the other two eigenvectors of
10410:
10352:
10292:
8611:for which the determinant of the matrix (
8601:of this transformation satisfy equation (
8386:
8359:
7620:
7591:
7531:
7511:
7497:
6938:
5704:
5629:
5597:
5335:{\displaystyle \gamma _{A}(\lambda _{i})}
4426:
4419:
3864:
3073:
1457:{\displaystyle \lambda =-{\frac {1}{20}}}
505:There is a direct correspondence between
23906:Trefethen, Lloyd N.; Bau, David (1997),
23651:
23587:
23576:
23394:
23087:Xirouhakis, Votsis & Delopoulus 2004
22858:
22783:
22759:
22703:
22390:
22377:(5 ed.). Wellesley-Cambridge Press.
22249:
21931:
20998:numerous times when traversing a static
20305:
20227:at the cost of solving a larger system.
19753:
19088:
14372:Any subspace spanned by eigenvectors of
14365:operates, and there cannot be more than
12543:Matrix with repeated eigenvalues example
8441:
3895:under addition. That is, if two vectors
3164:distinct eigenvalues. Whereas equation (
1219:
1202:Eigenvalues and eigenvectors of matrices
711:
698:
24162:
24027:Wolchover, Natalie (13 November 2019).
23705:Schaum's Easy Outline of Linear Algebra
23543:
23416:
23102:
22606:
22594:
22519:
22500:
22173:: 827–830, 845–865, 889–907, 931–937.
21768:{\displaystyle E_{1}>E_{2}>E_{3}}
20476:{\displaystyle H\psi _{E}=E\psi _{E}\,}
20326:can be seen as the eigenvectors of the
20301:
13575:{\displaystyle f(t)=f(0)e^{\lambda t},}
6945:{\displaystyle \alpha \in \mathbb {C} }
6441:is invertible, then the eigenvalues of
6140:is the product of all its eigenvalues,
4606:{\displaystyle n-\gamma _{A}(\lambda )}
4495:
4477:
4429:
4406:
3771:On one hand, this set is precisely the
3609:) equals the geometric multiplicity of
3052:
2261:is zero. Therefore, the eigenvalues of
1014:, and discovered the importance of the
932:is applied liberally when naming them:
14:
25443:
24798:Comparison of linear algebra libraries
24194:
24176:
23851:
23740:
23162:
22898:
22894:
22795:
22771:
22488:
22405:
22296:terms it is possible to get away with
22253:
22245:
21722:, the fabric is said to be planar. If
21242:. This particular representation is a
19029:, which is a negative number whenever
19022:{\displaystyle D=-4(\sin \theta )^{2}}
14485:eigenvalues can be generalized to the
14392:can be spanned by the eigenvectors of
12108:Consider the lower triangular matrix,
8008:is the eigenvalue associated with the
6670:, every eigenvalue has absolute value
5924:{\displaystyle \mu _{A}(\lambda _{i})}
5881:{\displaystyle \mu _{A}(\lambda _{i})}
5342:. The total geometric multiplicity of
5272:, where the geometric multiplicity of
4121:is the dimension of the nullspace of (
1119:started by Laplace, by realizing that
24830:
24388:
24199:, Belmont, CA: Thomson, Brooks/Cole,
23782:
23722:
23605:
23509:Earth Surface Processes and Landforms
23340:from the original on 19 January 2022.
23114:
23063:Diekmann, Heesterbeek & Metz 1990
22925:
22882:
22732:
22642:
22582:
22570:
22558:
22546:
22531:
22241:
22162:
21283:make it understandable to non-experts
21182:
20374:{\displaystyle n=1,\,2,\,3,\,\ldots }
19749:
19278:significance (which differs from the
16353:is to instead multiply the vector by
13693:if and only if there exists a scalar
13604:
8264:linearly independent eigenvectors of
7759:linearly independent eigenvectors of
7656:, the left and right eigenvectors of
7552:Comparing this equation to equation (
7440:satisfying this equation is called a
5669:linearly independent eigenvectors of
4819:{\displaystyle (A-\xi I)V=V(D-\xi I)}
4564:{\displaystyle \gamma _{A}(\lambda )}
4387:{\displaystyle \gamma _{A}(\lambda )}
4098:{\displaystyle \gamma _{A}(\lambda )}
4066:, is referred to as the eigenvalue's
3636:), defined in the next section, then
1115:clarified an important aspect in the
25411:
24218:
23032:Busche, Christian; Schiller, Beate.
22428:
22426:
22123:https://books.google.com/books?id=29
21715:{\displaystyle E_{1}=E_{2}>E_{3}}
21257:
19626:{\textstyle 1/{\sqrt {\deg(v_{i})}}}
19169:is more readily visualized than the
18199:
17863:
16617:was designed in 1961. Combining the
16423:{\displaystyle \mu \in \mathbb {C} }
16055:. Therefore, any vector of the form
14566:
14353:. As a consequence, eigenvectors of
13703:
8244:must therefore be an eigenvector of
7238:in the defining equation, equation (
5749:Additional properties of eigenvalues
2441:
2271:
2125:
2012:
523:vector space into itself, given any
25423:
24197:Linear algebra and its applications
24067:Linear Algebra and Its Applications
23252:The New Cassell's German Dictionary
22332:operations, but that does not take
22048:List of numerical-analysis software
21775:, the fabric is said to be linear.
20913:{\displaystyle H|\Psi _{E}\rangle }
20393:. The center of each figure is the
19436:{\displaystyle I-D^{-1/2}AD^{-1/2}}
19337:, or (increasingly) of the graph's
15404:
14545:is an analog of eigenvalues, while
14255:are either zero or eigenvectors of
13632:be a linear transformation mapping
13041:. The total geometric multiplicity
8912:, which are the two eigenvalues of
8619:) equals zero are the eigenvalues.
7045:. More generally, for a polynomial
2822:, which are the two eigenvalues of
2688:, the characteristic polynomial of
2311:{\displaystyle \det(A-\lambda I)=0}
1471:-dimensional vectors defined by an
1107:found the corresponding result for
1010:studied the rotational motion of a
516:and linear transformations from an
24:
24414:
24089:
23397:A First Course In Abstract Algebra
23245:, London: Arnold, pp. 103–107
22871:Friedberg, Insel & Spence 1989
20968:{\displaystyle |\Psi _{E}\rangle }
20953:
20898:
20804:{\displaystyle |\Psi _{E}\rangle }
20789:
20751:
20727:
20701:{\displaystyle |\Psi _{E}\rangle }
20686:
19095:multivariate Gaussian distribution
17274:{\displaystyle \ (\lambda -1)^{2}}
17111:{\displaystyle \ (\lambda -k)^{2}}
16714:
16258:, that is, any vector of the form
14704:The solution of this equation for
14474:) may not have an inverse even if
14427:
14294:is therefore a linear subspace of
13377:{\displaystyle Df(t)=\lambda f(t)}
13323:real functions of a real argument
8432:
8161:{\displaystyle Q^{-1}AQ=\Lambda .}
8152:
8106:{\displaystyle A=Q\Lambda Q^{-1},}
8084:
8038:
7702:linearly independent eigenvectors
2338:, the left-hand side of equation (
1155:At the start of the 20th century,
536:infinite-dimensional vector spaces
90:. More precisely, an eigenvector,
32:Characteristic equation (calculus)
25:
25482:
24214:
24166:An introduction to linear algebra
22423:
22217:: ... ) Later on the same page:
21662:{\displaystyle E_{1}=E_{2}=E_{3}}
20978:
20607:functions. Since this space is a
19820:{\displaystyle m{\ddot {x}}+kx=0}
19227:(PCA) in statistics. PCA studies
16608:
15105:for use in the solution equation
13783:corresponding to the eigenvector
13402:Consider the derivative operator
12199:The characteristic polynomial of
11759:The characteristic polynomial of
10428:or any nonzero multiple thereof.
9838:The characteristic polynomial of
9581:solves this equation. Therefore,
9157:solves this equation. Therefore,
8462:
8256:must be linearly independent for
4117:), the geometric multiplicity of
2372:, except that its term of degree
2010:are scalar multiples, that is if
1232:, not changing its direction, so
25422:
25410:
25399:
25398:
25386:
24811:
24810:
24788:Basic Linear Algebra Subprograms
24546:
24378:, Jack Dongarra, Axel Ruhe, and
24347:
24223:
23743:Linear Algebra and Matrix Theory
23166:A first course in linear algebra
23020:Knox-Robinson & Gardoll 1998
21520:{\displaystyle \mathbf {v} _{3}}
21507:
21491:{\displaystyle \mathbf {v} _{2}}
21478:
21462:{\displaystyle \mathbf {v} _{1}}
21449:
21367:
21352:
21337:
21262:
21098:
21087:
21044:
21033:
21010:
20277:
20002:{\displaystyle kx=\omega ^{2}mx}
19864:{\displaystyle m{\ddot {x}}=-kx}
19728:
18906:
18857:
18794:
18724:
18672:
18601:
18552:
16723:
16707:
16700:
16693:
16591:
16572:{\displaystyle \mathbf {v} ^{*}}
16559:
16530:
16519:
16511:
16497:
16442:
14217:
14193:
14169:
14161:
14140:
14132:
14040:
14016:
13992:
13975:
13954:
13946:
13877:
13863:
13849:
13734:
13720:
13613:on arbitrary vector spaces. Let
13429:{\displaystyle {\tfrac {d}{dt}}}
12471:
12409:
12340:
12018:
11959:
11900:
11620:
11598:
11493:
11402:
9743:Three-dimensional matrix example
9725:associated with the eigenvalues
9590:
9325:
9166:
8956:
8515:= and blue vectors parallel to
8415:
8393:
8380:
8369:
8353:
7968:
7939:
7915:
7822:
7803:
7789:
7525:
7505:
7335:
7321:
7266:
7255:
6706:{\displaystyle |\lambda _{i}|=1}
5717:{\displaystyle \mathbb {C} ^{n}}
5642:{\displaystyle \mathbb {C} ^{n}}
5610:{\displaystyle \mathbb {C} ^{n}}
4054:The dimension of the eigenspace
3963:. This can be checked using the
3877:{\displaystyle \mathbb {C} ^{n}}
3747:
3739:
3709:
2948:
2894:
2869:
2861:
2380:. This polynomial is called the
2336:Leibniz formula for determinants
2166:
2158:
2123:) can be stated equivalently as
2045:
2034:
2026:
1502:
1494:
1409:
1398:
1313:
1260:
1190:, was proposed independently by
1140:studied the first eigenvalue of
1069:Théorie analytique de la chaleur
959:associated with that eigenvalue.
892:
881:
768:{\displaystyle {\tfrac {d}{dx}}}
655:
641:
488:
477:
444:
410:
388:
363:
314:
178:
167:
98:
49:
25307:Computational complexity theory
24686:Seven-dimensional cross product
23351:Journal of Mathematical Biology
23075:Heesterbeek & Diekmann 2000
23025:
22888:
22876:
22262:
22256:, Theorem EDELI on p. 469; and
22240:For a proof of this lemma, see
22234:
22187:
22156:
22085:
21789:The basic reproduction number (
21227:procedure, called in this case
21172:{\displaystyle \tau _{\min }=0}
21139:{\displaystyle \tau _{\max }=1}
19700:of its vertices. An example is
19266:, such as those encountered in
16643:
16632:is one example of an efficient
15622:
15563:is 4 or less. According to the
13054:Eigenvector-eigenvalue identity
12468:
12406:
12015:
11956:
10977:
10945:
9134:{\displaystyle 1v_{1}+1v_{2}=0}
7629:{\displaystyle A^{\textsf {T}}}
7600:{\displaystyle A^{\textsf {T}}}
6639:is not only Hermitian but also
2945:
2427:, being a polynomial of degree
1311:
1303:
1095:proved that the eigenvalues of
1084:in 1855 to what are now called
974:, then this basis is called an
943:The set of all eigenvectors of
721:The example here, based on the
615:is a nonzero vector that, when
24315:Matrix Eigenvectors Calculator
24179:Introduction to linear algebra
23250:Betteridge, Harold T. (1965),
23210:10.1103/physrevlett.125.165901
22748:Beauregard & Fraleigh 1973
22371:Introduction to Linear Algebra
22361:
22319:
22306:
21244:generalized eigenvalue problem
20948:
20893:
20784:
20746:
20722:
20681:
19966:generalized eigenvalue problem
19618:
19605:
19497:equal to the degree of vertex
19148:
19136:
19116:
19104:
19081:) has reciprocal eigenvalues.
19010:
18997:
18239:
18226:
17903:
17890:
17317:
17311:
17262:
17249:
17216:
17210:
17167:
17148:
17145:
17126:
17099:
17086:
16392:{\displaystyle (A-\mu I)^{-1}}
16377:
16361:
16314:, for any nonzero real number
16108:, for any nonzero real number
15416:
15390:
14518:Weight (representation theory)
14221:
14210:
14197:
14186:
14173:
14157:
14144:
14128:
14044:
14036:
14020:
14009:
13996:
13988:
13979:
13971:
13958:
13942:
13867:
13859:
13724:
13716:
13656:
13617:be any vector space over some
13601:article gives other examples.
13553:
13547:
13538:
13532:
13485:
13479:
13467:
13461:
13371:
13365:
13353:
13347:
13327:. The eigenvalue equation for
13218:
13192:
13169:
13166:
13153:
13127:
13100:
13078:
12866:
12853:
12844:
12831:
12698:
12683:
12276:
12264:
12261:
12249:
12246:
12234:
12228:
12213:
11836:
11824:
11821:
11809:
11806:
11794:
11788:
11773:
10169:
10157:
10154:
10142:
10132:
10120:
9871:
9856:
9320:
9305:
8951:
8939:
8876:
8864:
8861:
8849:
8655:
8640:
8438:Two-dimensional matrix example
8260:to be invertible, there exist
7876:by its associated eigenvalue,
7694:form a basis, or equivalently
7686:Eigendecomposition of a matrix
7148:
7135:
7120:
7107:
7081:
7075:
6693:
6678:
6158:
6152:
5980:
5974:
5918:
5905:
5875:
5862:
5434:
5421:
5329:
5316:
5164:
5158:
5142:
5136:
5084:
5078:
5064:
5051:
5031:
5016:
4973:
4958:
4949:
4934:
4813:
4798:
4786:
4771:
4723:
4717:
4600:
4594:
4558:
4552:
4450:
4444:
4381:
4375:
4345:
4339:
4323:
4317:
4279:
4273:
4257:
4251:
4205:
4190:
4172:
4166:
4092:
4086:
3657:Given a particular eigenvalue
3453:
3440:
3367:
3354:
3340:
3320:
3312:
3299:
3285:
3265:
3260:
3247:
3233:
3213:
3207:
3192:
2717:
2702:
2541:
2522:
2516:
2497:
2494:
2475:
2469:
2454:
2413:fundamental theorem of algebra
2299:
2284:
1228:acts by stretching the vector
1054:used the work of Lagrange and
645:
637:
112:, of a linear transformation,
13:
1:
24127:10.1016/S0377-0427(00)00413-1
23646:10.1016/S0098-3004(97)00122-2
21927:
21317:, especially in the study of
19154:{\displaystyle (0.878,0.478)}
18453:{\displaystyle \gamma _{1}=1}
18279:{\displaystyle \gamma _{1}=2}
16197:above has another eigenvalue
14487:spectrum of a linear operator
14458:, and therefore its inverse (
14361:of the vector space on which
13810:is the product of the scalar
13677:We say that a nonzero vector
8207:Conversely, suppose a matrix
8047:{\displaystyle AQ=Q\Lambda .}
5557:{\displaystyle \gamma _{A}=n}
5037:{\displaystyle \det(D-\xi I)}
4986:. But from the definition of
3008:If the entries of the matrix
2088:of the linear transformation
1374:These vectors are said to be
395:{\displaystyle A\mathbf {v} }
269:
25471:Singular value decomposition
24528:Eigenvalues and eigenvectors
24355:Eigenvalues and eigenvectors
24308:– A visual explanation with
23837:10.1103/RevModPhys.89.015005
23696:10.1016/0041-5553(63)90168-X
23559:10.1016/0315-0860(75)90032-4
23241:Benn, D.; Evans, D. (2004),
22434:"Eigenvector and Eigenvalue"
22343:
22063:Quadratic eigenvalue problem
21973:principal component analysis
21017:{\displaystyle \mathbf {t} }
20155:quadratic eigenvalue problem
19296:structural equation modeling
19225:principal component analysis
19185:Positive semidefinite matrix
19179:Principal component analysis
19085:Principal component analysis
17363:{\displaystyle \lambda _{i}}
16598:{\displaystyle \mathbf {v} }
16449:{\displaystyle \mathbf {v} }
15421:The eigenvalues of a matrix
8422:{\displaystyle \mathbf {x} }
8341:is the maximum value of the
8307:Variational characterization
8272:form a basis if and only if
8190:to the diagonal matrix Λ or
7725:with associated eigenvalues
7690:Suppose the eigenvectors of
6860:is the identity matrix) are
5689:; such a basis is called an
5292:{\displaystyle \lambda _{i}}
3891:is a linear subspace, it is
3775:or nullspace of the matrix (
962:If a set of eigenvectors of
580:of the rotational motion of
458:is called an eigenvector of
451:{\displaystyle \mathbf {v} }
417:{\displaystyle \mathbf {v} }
370:{\displaystyle \mathbf {v} }
321:{\displaystyle \mathbf {v} }
105:{\displaystyle \mathbf {v} }
27:Concepts from linear algebra
7:
24302:– lecture from Khan Academy
24007:Weisstein, Eric W. (n.d.).
23626:Computers & Geosciences
23610:, Oxford University Press,
22914:Lipschutz & Lipson 2002
22006:
21939:as examples of eigenvectors
20855:represents the eigenvalue.
20242:Tensor of moment of inertia
20029:{\displaystyle \omega ^{2}}
19893:to be sinusoidal in time).
19247:and the eigenvalues to the
14400:is the entire vector space
14098:associated with eigenvalue
13757:
13398:Derivative operator example
9290:
8928:
8603:
8226:. Left multiplying both by
7554:
7240:
7169:Left and right eigenvectors
6315:, for any positive integer
6288:; i.e., the eigenvalues of
3686:
3401:
3166:
3120:) of the eigenvalue is its
3017:even if all the entries of
2563:
2392:
2340:
2324:
2234:
2189:
2119:
2102:
2068:
1159:studied the eigenvalues of
1025:In the early 19th century,
611:of a linear transformation
607:In essence, an eigenvector
541:
295:{\displaystyle n{\times }n}
134:scaled by a constant factor
10:
25487:
25357:Films about mathematicians
24367:Computation of Eigenvalues
24172:, Brigham Young University
23852:Shilov, Georgi E. (1977),
23745:(2nd ed.), New York:
23395:Fraleigh, John B. (1976),
23163:Beezer, Robert A. (2006),
23119:(5th ed.), New York:
23095:
22148:: 184–252; see especially
22119:Specimen theoriae turbinum
21942:
21782:
20250:, the eigenvectors of the
19772:. The eigenvalues are the
19761:
19467:is a diagonal matrix with
19182:
19176:
18117:{\displaystyle \mu _{1}=2}
17943:{\displaystyle \mu _{1}=2}
16619:Householder transformation
16341:
16216:{\displaystyle \lambda =1}
16167:{\displaystyle \lambda =6}
15629:system of linear equations
15394:
14515:
14492:as the set of all scalars
14431:
13284:
13062:, the norm squared of the
8446:The transformation matrix
8310:
7683:
7172:
7065:the eigenvalues of matrix
6974:{\displaystyle \alpha I+A}
5888:times in this list, where
5845:. Each eigenvalue appears
5491:of all the eigenspaces of
4685:{\displaystyle D:=V^{T}AV}
4109:is also the nullspace of (
3042:intermediate value theorem
2676:Taking the determinant of
2360:, the order of the matrix
2269:that satisfy the equation
2225:
1205:
982:
204:is the multiplying factor
29:
25380:
25330:
25287:
25197:
25159:
25126:
25078:
25050:
24997:
24944:
24926:Philosophy of mathematics
24901:
24866:
24806:
24768:
24724:
24661:
24613:
24555:
24544:
24440:
24422:
24360:
24336:{\displaystyle n\times n}
24163:Kuttler, Kenneth (2017),
24080:10.1016/j.laa.2024.03.035
23807:Reviews of Modern Physics
23399:(2nd ed.), Reading:
23117:Elementary Linear Algebra
23107:, in Miller, Jeff (ed.),
22996:Graham & Midgley 2000
22957:Vellekoop & Mosk 2007
22721:Golub & Van Loan 1996
22682:. Accessed on 2016-03-27.
22660:. Accessed on 2016-03-27.
22631:Golub & Van Loan 1996
22244:, Theorem 8.2 on p. 186;
21998:Similar to this concept,
21785:Basic reproduction number
21779:Basic reproduction number
20639:{\displaystyle \psi _{E}}
20596:{\displaystyle \psi _{E}}
20534:{\displaystyle \psi _{E}}
20332:angular momentum operator
20328:hydrogen atom Hamiltonian
19343:discrete Laplace operator
19301:
18533:
18197:
17861:
17342:
17071:
16731:
16689:
16682:
16677:
16672:
16669:
16664:
16649:Geometric transformations
15506:{\displaystyle n\times n}
15460:{\displaystyle 2\times 2}
14857: – 1 equations
14384:, and the restriction of
13668:{\displaystyle T:V\to V.}
13436:with eigenvalue equation
12090:Triangular matrix example
10438:cyclic permutation matrix
8180:similarity transformation
7413:{\displaystyle 1\times n}
7211:{\displaystyle n\times n}
6755:{\displaystyle n\times n}
5792:{\displaystyle n\times n}
3811:equals the nullspace of (
2383:characteristic polynomial
2368:depend on the entries of
2348:function of the variable
2238:) has a nonzero solution
2228:Characteristic polynomial
25362:Recreational mathematics
24195:Strang, Gilbert (2006),
24177:Strang, Gilbert (1993),
24156:University of Nottingham
24061:Van Mieghem, P. (2024).
23908:Numerical Linear Algebra
23741:Nering, Evar D. (1970),
23660:(2nd Revised ed.),
23588:Herstein, I. N. (1964),
22945:Trefethen & Bau 1997
22903:Lemma for the eigenspace
22470:Wolfram.com: Eigenvector
22356:Burden & Faires 1993
22325:{\displaystyle O(n^{4})}
22180:équation caractéristique
22078:
21979:of some of them. In the
21971:; this is an example of
20252:moment of inertia tensor
19744:Perron–Frobenius theorem
19739:transition probabilities
19555:{\displaystyle D^{-1/2}}
19280:statistical significance
19260:dimensionality reduction
16338:Simple iterative methods
15937:
15933:
14561:Langlands correspondence
14496:for which the operator (
13390:and are commonly called
10766:For the real eigenvalue
10755:{\displaystyle i^{2}=-1}
9567:Any nonzero vector with
9143:Any nonzero vector with
8285:generalized eigenvectors
8222:is some diagonal matrix
7175:left and right (algebra)
5487:is the dimension of the
5112:{\displaystyle \lambda }
4300:To prove the inequality
3855:is a linear subspace of
1066:in his famous 1822 book
217:{\displaystyle \lambda }
149:{\displaystyle \lambda }
25247:Mathematical statistics
25237:Mathematical psychology
25207:Engineering mathematics
25141:Algebraic number theory
24374:Edited by Zhaojun Bai,
23785:Advanced linear algebra
23723:Meyer, Carl D. (2000),
23180:Physical Review Letters
22972:Rotter & Gigan 2017
22334:combinatorial explosion
21191:, and in particular in
20236:finite element analysis
20049:{\displaystyle \omega }
19714:stationary distribution
19373:combinatorial Laplacian
16128:, is an eigenvector of
13275:diagonalizable matrices
13269:formed by removing the
12914:On the other hand, the
12103:upper triangular matrix
11658:Diagonal matrix example
8301:generalized eigenspaces
8252:. Since the columns of
8211:is diagonalizable. Let
8057:Because the columns of
7477:{\displaystyle \kappa }
7420:matrix. Any row vector
7367:{\displaystyle \kappa }
6248:The eigenvalues of the
5219:{\displaystyle d\leq n}
4917:{\displaystyle D-\xi I}
4888:{\displaystyle A-\xi I}
3887:Because the eigenspace
3684:that satisfy equation (
3087:be an eigenvalue of an
2398:characteristic equation
1386:, if there is a scalar
1123:can cause instability.
1109:skew-symmetric matrices
1064:separation of variables
1044:characteristic equation
1041:; his term survives in
940:of that transformation.
25393:Mathematics portal
25242:Mathematical sociology
25222:Mathematical economics
25217:Mathematical chemistry
25146:Analytic number theory
25027:Differential equations
24513:Row and column vectors
24337:
24288:What are Eigen Values?
23783:Roman, Steven (2008),
23606:Kline, Morris (1972),
23577:Hefferon, Jim (2001),
23456:10.1093/comjnl/4.4.332
23434:10.1093/comjnl/4.3.265
23115:Anton, Howard (1987),
23103:Aldrich, John (2006),
23038:www.ruhr-uni-bochum.de
22326:
22290:
22219:
22209:
22053:Nonlinear eigenproblem
21940:
21918:
21897:has passed. The value
21891:
21864:
21837:
21810:
21769:
21716:
21663:
21610:
21583:
21556:
21521:
21492:
21463:
21434:
21381:
21254:Geology and glaciology
21173:
21140:
21106:
21072:
21052:
21018:
20969:
20934:
20914:
20869:
20849:
20829:
20805:
20767:
20702:
20660:
20640:
20615:, one can introduce a
20597:
20559:
20535:
20500:
20477:
20419:
20402:
20375:
20232:differential equations
20225:algebraic manipulation
20215:
20147:
20078:
20050:
20036:is the eigenvalue and
20030:
20003:
19954:
19930:
19910:
19887:
19865:
19821:
19759:
19687:
19667:
19647:
19627:
19576:
19556:
19518:
19491:
19490:{\displaystyle D_{ii}}
19461:
19443:(sometimes called the
19437:
19371:(sometimes called the
19365:
19331:
19262:in the study of large
19174:
19155:
19123:
19068:
19023:
18960:
18837:
18778:
18652:
18524:
18454:
18420:
18350:
18280:
18246:
18188:
18118:
18084:
18014:
17944:
17910:
17852:
17703:
17656:
17495:
17411:
17364:
17333:
17275:
17232:
17174:
17112:
17062:
16984:
16930:
16849:
16781:
16719:
16599:
16573:
16541:
16470:
16450:
16424:
16393:
16328:
16308:
16252:
16251:{\displaystyle 3x+y=0}
16217:
16191:
16168:
16142:
16122:
16102:
16049:
16018:
15926:
15832:
15718:
15684:
15605:
15581:
15557:
15530:
15507:
15481:Wilkinson's polynomial
15461:
15435:
15381:
15194:
15099:
15044:
14968:
14844:
14695:
14524:algebra representation
14478:is not an eigenvalue.
14420:admits an eigenbasis,
14369:distinct eigenvalues.
14307:geometric multiplicity
14235:
14058:
13893:
13745:
13669:
13611:linear transformations
13586:= 0 the eigenfunction
13576:
13495:
13430:
13378:
13305:differential operators
13259:
13232:
13035:
12975:
12916:geometric multiplicity
12892:algebraic multiplicity
12882:
12667:
12531:
12286:
12191:
12078:
11846:
11751:
11649:
11569:
11478:
11381:
11198:
11013:
10901:
10756:
10715:
10695:
10522:
10420:
10362:
10302:
10240:
9830:
9682:
9559:
9264:
9135:
9084:
8890:
8589:
8530:
8499:
8423:
8401:
8335:
8162:
8107:
8048:
7993:
7872:scales each column of
7847:
7755:whose columns are the
7670:
7650:
7630:
7601:
7572:
7544:
7478:
7458:
7434:
7414:
7388:
7368:
7346:
7307:
7277:
7232:
7212:
7158:
7088:
7059:
7039:
6975:
6946:
6918:
6854:
6834:
6808:
6756:
6730:
6707:
6660:
6633:
6602:
6582:
6552:
6525:
6465:
6464:{\displaystyle A^{-1}}
6435:
6408:
6385:
6329:
6309:
6282:
6262:
6238:
6184:
6134:
6106:
6043:
6006:
5953:
5925:
5882:
5839:
5793:
5767:
5738:
5718:
5683:
5663:
5643:
5611:
5582:
5558:
5525:
5505:
5481:
5410:
5356:
5336:
5293:
5266:
5220:
5194:
5171:
5113:
5093:
5038:
5000:
4980:
4918:
4889:
4860:
4840:
4820:
4764:on both sides, we get
4758:
4757:{\displaystyle -\xi V}
4732:
4686:
4647:
4627:
4607:
4565:
4529:
4509:
4459:
4388:
4352:
4292:
4215:
4099:
4068:geometric multiplicity
3878:
3763:
3552:
3506:
3379:
3122:multiplicity as a root
3101:algebraic multiplicity
3074:Algebraic multiplicity
3000:
2877:
2806:
2668:
2551:
2356:of this polynomial is
2312:
2177:
2056:
1994:
1963:
1833:
1513:
1458:
1420:
1366:
1241:
1133:Sturm–Liouville theory
1091:Around the same time,
1074:Charles-François Sturm
1035:racine caractéristique
1001:differential equations
909:where the eigenvector
903:
840:
769:
718:
709:
666:
602:matrix diagonalization
496:
452:
418:
396:
371:
345:
322:
296:
228:Geometrically, vectors
218:
186:
150:
126:
106:
25372:Mathematics education
25302:Theory of computation
25022:Hypercomplex analysis
24518:Row and column spaces
24463:Scalar multiplication
24338:
24013:mathworld.wolfram.com
23993:mathworld.wolfram.com
23658:New York: McGraw-Hill
23051:Benn & Evans 2004
23008:Sneed & Folk 1958
22998:, pp. 1473–1477.
22959:, pp. 2309–2311.
22672:Math Course Catalogue
22327:
22291:
22196:David Hilbert (1904)
22115:Johann Andreas Segner
22013:Antieigenvalue theory
21935:
21919:
21917:{\displaystyle R_{0}}
21892:
21890:{\displaystyle t_{G}}
21865:
21863:{\displaystyle t_{G}}
21838:
21836:{\displaystyle R_{0}}
21811:
21809:{\displaystyle R_{0}}
21770:
21717:
21664:
21611:
21609:{\displaystyle E_{3}}
21584:
21582:{\displaystyle E_{2}}
21557:
21555:{\displaystyle E_{1}}
21522:
21498:is the secondary and
21493:
21464:
21435:
21387:by their eigenvalues
21382:
21229:self-consistent field
21217:ionization potentials
21174:
21141:
21107:
21073:
21071:{\displaystyle \tau }
21053:
21019:
20970:
20935:
20915:
20880:self-adjoint operator
20870:
20850:
20830:
20806:
20768:
20703:
20661:
20641:
20598:
20565:, interpreted as its
20560:
20536:
20512:differential operator
20501:
20478:
20420:
20376:
20309:
20216:
20153:leads to a so-called
20148:
20079:
20051:
20031:
20004:
19955:
19931:
19911:
19888:
19866:
19822:
19757:
19688:
19668:
19648:
19628:
19582:th diagonal entry is
19577:
19557:
19519:
19517:{\displaystyle v_{i}}
19492:
19462:
19438:
19366:
19332:
19310:, an eigenvalue of a
19308:spectral graph theory
19214:multivariate analysis
19202:positive semidefinite
19156:
19124:
19122:{\displaystyle (1,3)}
19092:
19069:
19024:
18961:
18838:
18779:
18653:
18525:
18455:
18421:
18351:
18281:
18247:
18189:
18119:
18085:
18015:
17945:
17911:
17853:
17704:
17657:
17496:
17412:
17365:
17334:
17276:
17233:
17175:
17113:
17063:
16985:
16931:
16850:
16782:
16718:
16600:
16574:
16542:
16471:
16451:
16425:
16394:
16329:
16309:
16253:
16218:
16192:
16169:
16143:
16123:
16103:
16050:
16019:
15927:
15833:
15719:
15717:{\displaystyle Av=6v}
15685:
15606:
15582:
15558:
15531:
15508:
15462:
15436:
15382:
15205:differential equation
15195:
15100:
15054:characteristic roots
15045:
14969:
14845:
14696:
14536:representation theory
14446:, then the operator (
14236:
14059:
13921:associated with
13894:
13746:
13670:
13577:
13496:
13431:
13379:
13333:differential equation
13260:
13233:
13036:
12976:
12883:
12668:
12532:
12287:
12192:
12079:
11847:
11752:
11650:
11570:
11479:
11382:
11199:
11014:
10902:
10757:
10716:
10696:
10523:
10421:
10363:
10303:
10241:
9831:
9690:is an eigenvector of
9683:
9560:
9270:is an eigenvector of
9265:
9136:
9085:
8891:
8607:), and the values of
8590:
8500:
8498:{\displaystyle \left}
8445:
8424:
8402:
8336:
8163:
8108:
8049:
7994:
7860:is an eigenvector of
7856:Since each column of
7848:
7671:
7651:
7631:
7602:
7573:
7545:
7479:
7459:
7435:
7415:
7389:
7369:
7347:
7308:
7278:
7233:
7213:
7159:
7089:
7060:
7040:
6976:
6952:, the eigenvalues of
6947:
6919:
6855:
6835:
6809:
6757:
6731:
6708:
6661:
6634:
6603:
6588:, or equivalently if
6583:
6581:{\displaystyle A^{*}}
6553:
6533:reciprocal polynomial
6526:
6466:
6436:
6409:
6386:
6330:
6310:
6308:{\displaystyle A^{k}}
6283:
6263:
6239:
6164:
6135:
6107:
6023:
5986:
5954:
5926:
5883:
5840:
5794:
5768:
5739:
5719:
5684:
5664:
5644:
5612:
5583:
5559:
5526:
5506:
5482:
5390:
5357:
5337:
5294:
5267:
5226:distinct eigenvalues
5221:
5195:
5172:
5114:
5094:
5039:
5001:
4981:
4919:
4890:
4861:
4841:
4821:
4759:
4733:
4687:
4648:
4628:
4608:
4566:
4530:
4510:
4460:
4389:
4353:
4293:
4216:
4100:
3985:is a complex number,
3965:distributive property
3879:
3764:
3647:semisimple eigenvalue
3553:
3486:
3380:
3001:
2878:
2807:
2669:
2552:
2313:
2178:
2092:and the scale factor
2057:
1995:
1943:
1839:where, for each row,
1834:
1514:
1459:
1421:
1367:
1236:is an eigenvector of
1223:
1173:Hermann von Helmholtz
1027:Augustin-Louis Cauchy
1020:Joseph-Louis Lagrange
1006:In the 18th century,
904:
841:
770:
739:differential operator
715:
702:
667:
497:
453:
419:
397:
372:
346:
323:
306:and a nonzero vector
297:
219:
187:
151:
127:
107:
88:linear transformation
80:characteristic vector
25461:Mathematical physics
25352:Informal mathematics
25232:Mathematical physics
25227:Mathematical finance
25212:Mathematical biology
25151:Diophantine geometry
24653:Gram–Schmidt process
24605:Gaussian elimination
24321:
24244:improve this article
24142:Hill, Roger (2009).
23956:10.1364/OL.32.002309
23546:Historia Mathematica
23486:Van Loan, Charles F.
23443:The Computer Journal
23420:The Computer Journal
23147:Houghton Mifflin Co.
22859:Korn & Korn 2000
22735:, pp. 305, 307.
22393:, pp. 228, 229.
22300:
22280:
22270:Gaussian elimination
22073:Spectrum of a matrix
22033:Eigenvalue algorithm
21901:
21874:
21847:
21820:
21793:
21726:
21673:
21620:
21593:
21566:
21539:
21502:
21473:
21444:
21391:
21332:
21150:
21117:
21082:
21062:
21028:
21006:
20944:
20924:
20886:
20859:
20839:
20819:
20780:
20715:
20677:
20650:
20623:
20611:with a well-defined
20603:within the space of
20580:
20549:
20518:
20510:, is a second-order
20490:
20440:
20427:Schrödinger equation
20409:
20338:
20314:associated with the
20302:Schrödinger equation
20161:
20092:
20084:alone. Furthermore,
20068:
20040:
20013:
19971:
19944:
19920:
19900:
19877:
19831:
19784:
19677:
19657:
19637:
19586:
19566:
19528:
19501:
19471:
19451:
19445:normalized Laplacian
19379:
19349:
19321:
19245:principal components
19133:
19101:
19037:
18982:
18848:
18789:
18663:
18543:
18537:All nonzero vectors
18465:
18431:
18361:
18291:
18257:
18207:
18129:
18095:
18025:
17955:
17921:
17871:
17714:
17667:
17506:
17422:
17375:
17347:
17286:
17243:
17185:
17123:
17080:
16995:
16941:
16860:
16792:
16738:
16587:
16554:
16483:
16460:
16438:
16406:
16358:
16318:
16262:
16227:
16201:
16181:
16152:
16132:
16112:
16059:
16048:{\displaystyle y=2x}
16030:
15942:
15847:
15728:
15696:
15635:
15595:
15571:
15565:Abel–Ruffini theorem
15547:
15536:different products.
15517:
15491:
15445:
15425:
15397:Eigenvalue algorithm
15214:
15112:
15058:
14982:
14861:
14719:
14582:
14573:difference equations
14481:For this reason, in
14442:is an eigenvalue of
14118:
14094:are eigenvectors of
13932:
13915:characteristic space
13834:
13826:Given an eigenvalue
13710:
13644:
13526:
13521:exponential function
13519:. Its solution, the
13440:
13406:
13338:
13242:
13074:
12985:
12922:
12677:
12551:
12335:
12294:which has the roots
12207:
12112:
11895:
11854:which has the roots
11767:
11672:
11593:
11488:
11397:
11393:are complex and are
11208:
11025:
10913:
10777:
10730:
10705:
10538:
10443:
10375:
10314:
10257:
9846:
9751:
9747:Consider the matrix
9721:are eigenvectors of
9585:
9298:
9161:
9093:
8936:
8630:
8537:
8533:Consider the matrix
8454:
8411:
8348:
8325:
8127:
8072:
8023:
7883:
7864:, right multiplying
7770:
7660:
7640:
7611:
7582:
7562:
7488:
7468:
7448:
7424:
7398:
7378:
7358:
7317:
7297:
7248:
7222:
7196:
7098:
7087:{\displaystyle P(A)}
7069:
7049:
6985:
6956:
6928:
6864:
6844:
6818:
6766:
6740:
6720:
6674:
6650:
6623:
6592:
6565:
6542:
6475:
6445:
6425:
6398:
6339:
6319:
6292:
6272:
6252:
6146:
6124:
5965:
5943:
5892:
5849:
5803:
5777:
5757:
5728:
5699:
5673:
5653:
5624:
5592:
5572:
5535:
5515:
5495:
5366:
5346:
5303:
5276:
5230:
5204:
5184:
5123:
5103:
5048:
5010:
4990:
4928:
4899:
4870:
4850:
4830:
4768:
4742:
4696:
4657:
4637:
4617:
4575:
4539:
4519:
4469:
4401:
4362:
4304:
4232:
4153:
4073:
3859:
3829:characteristic space
3694:
3413:
3186:
3174:into the product of
3053:Spectrum of a matrix
2889:
2835:
2696:
2616:
2448:
2435:into the product of
2278:
2218:is the zero vector.
2132:
2019:
1843:
1523:
1487:
1432:
1394:
1256:
1212:Matrix (mathematics)
1056:Pierre-Simon Laplace
953:characteristic space
919:decompose the matrix
874:
783:
745:
631:
552:is adopted from the
470:
440:
406:
381:
359:
332:
310:
278:
208:
198:characteristic value
160:
140:
116:
94:
25367:Mathematics and art
25277:Operations research
25032:Functional analysis
24783:Numerical stability
24663:Multilinear algebra
24638:Inner product space
24488:Linear independence
24256:footnote references
24118:2000JCoAM.123...35G
23987:Weisstein, Eric W.
23948:2007OptL...32.2309V
23884:1958JG.....66..114S
23829:2017RvMP...89a5005R
23761:Press, William H.;
23666:1968mhse.book.....K
23638:1998CG.....24..243K
23522:2000ESPL...25.1473G
23490:Matrix computations
23202:2020PhRvL.125p5901B
23065:, pp. 365–382.
23053:, pp. 103–107.
23010:, pp. 114–150.
22774:, pp. 115–116.
22609:, pp. 265–271.
22585:, p. 1063, p..
22274:formal power series
20387:probability density
20056:is the (imaginary)
19774:natural frequencies
19723:spectral clustering
19364:{\displaystyle D-A}
19221:covariance matrices
16684:Hyperbolic rotation
16659:
16581:conjugate transpose
15513:matrix is a sum of
15186:
15152:
14528:associative algebra
14483:functional analysis
14424:is diagonalizable.
14345:The eigenspaces of
14321:) of an eigenvalue
13830:, consider the set
11641:
10992:
10960:
10652:
8276:is diagonalizable.
6833:{\displaystyle I+A}
6560:conjugate transpose
6380:
6356:
5649:can be formed from
4129:), also called the
2108:eigenvalue equation
1152:a few years later.
1097:orthogonal matrices
674:eigenvalue equation
672:referred to as the
202:characteristic root
25312:Numerical analysis
24921:Mathematical logic
24916:Information theory
24493:Linear combination
24380:Henk van der Vorst
24333:
23872:Journal of Geology
23763:Teukolsky, Saul A.
23652:Korn, Granino A.;
23363:10.1007/BF00178324
23274:Numerical Analysis
23256:Funk & Wagnall
22984:Bender et al. 2020
22861:, Section 14.3.5a.
22823:Denton et al. 2022
22678:2015-11-01 at the
22619:Kublanovskaya 1962
22438:www.mathsisfun.com
22322:
22286:
22043:Jordan normal form
21981:facial recognition
21977:linear combination
21941:
21914:
21887:
21860:
21833:
21806:
21765:
21712:
21659:
21606:
21579:
21552:
21517:
21488:
21459:
21430:
21377:
21248:Roothaan equations
21209:molecular orbitals
21183:Molecular orbitals
21169:
21136:
21102:
21068:
21048:
21014:
20965:
20930:
20910:
20865:
20845:
20825:
20801:
20763:
20698:
20656:
20636:
20593:
20555:
20531:
20496:
20473:
20415:
20403:
20371:
20330:as well as of the
20211:
20143:
20074:
20046:
20026:
19999:
19950:
19926:
19906:
19883:
19861:
19817:
19770:degrees of freedom
19760:
19750:Vibration analysis
19683:
19663:
19643:
19623:
19572:
19552:
19514:
19487:
19457:
19433:
19361:
19345:, which is either
19327:
19284:hypothesis testing
19249:variance explained
19237:correlation matrix
19195:eigendecomposition
19175:
19167:standard deviation
19151:
19119:
19064:
19019:
18973:quadratic equation
18956:
18954:
18946:
18894:
18833:
18827:
18774:
18772:
18764:
18712:
18648:
18646:
18638:
18589:
18520:
18518:
18450:
18416:
18414:
18346:
18344:
18276:
18242:
18184:
18182:
18114:
18080:
18078:
18010:
18008:
17940:
17906:
17848:
17846:
17699:
17652:
17650:
17491:
17489:
17407:
17360:
17329:
17271:
17228:
17170:
17108:
17058:
17052:
16980:
16974:
16926:
16920:
16845:
16839:
16777:
16771:
16720:
16657:
16595:
16569:
16537:
16466:
16446:
16420:
16389:
16324:
16304:
16290:
16248:
16213:
16187:
16164:
16138:
16118:
16098:
16084:
16045:
16014:
16009:
15922:
15917:
15828:
15822:
15787:
15761:
15714:
15680:
15674:
15601:
15577:
15553:
15541:algebraic formulas
15529:{\displaystyle n!}
15526:
15503:
15457:
15431:
15377:
15190:
15172:
15138:
15095:
15040:
15034:
14964:
14840:
14691:
14404:, then a basis of
14378:invariant subspace
14231:
14229:
14054:
14052:
13889:
13741:
13665:
13605:General definition
13572:
13491:
13426:
13424:
13374:
13258:{\textstyle M_{j}}
13255:
13228:
13190:
13125:
13031:
13017:
12971:
12957:
12878:
12822:
12663:
12654:
12527:
12518:
12459:
12397:
12282:
12187:
12178:
12074:
12065:
12006:
11947:
11842:
11747:
11738:
11645:
11618:
11565:
11551:
11474:
11460:
11377:
11368:
11305:
11255:
11194:
11185:
11122:
11072:
11009:
10978:
10946:
10897:
10888:
10846:
10810:
10752:
10711:
10691:
10689:
10638:
10534:, whose roots are
10518:
10509:
10416:
10402:
10358:
10344:
10298:
10284:
10236:
10234:
10101:
10012:
9943:
9826:
9817:
9701:Thus, the vectors
9678:
9672:
9643:
9555:
9553:
9453:
9424:
9384:
9260:
9254:
9222:
9131:
9080:
9074:
9045:
9005:
8886:
8884:
8798:
8742:
8700:
8585:
8576:
8531:
8495:
8489:
8488:
8419:
8397:
8331:
8297:Jordan normal form
8289:Jordan normal form
8176:eigendecomposition
8158:
8103:
8044:
7989:
7980:
7843:
7834:
7666:
7646:
7626:
7597:
7568:
7540:
7474:
7454:
7430:
7410:
7384:
7364:
7342:
7303:
7273:
7228:
7208:
7154:
7084:
7055:
7035:
6971:
6942:
6914:
6850:
6830:
6804:
6752:
6726:
6703:
6656:
6629:
6598:
6578:
6548:
6521:
6461:
6431:
6404:
6381:
6366:
6342:
6325:
6305:
6278:
6258:
6234:
6130:
6102:
5949:
5921:
5878:
5835:
5789:
5763:
5734:
5714:
5679:
5659:
5639:
5607:
5578:
5554:
5521:
5501:
5477:
5475:
5352:
5332:
5289:
5262:
5216:
5190:
5167:
5109:
5089:
5044:contains a factor
5034:
4996:
4976:
4914:
4885:
4866:. In other words,
4856:
4836:
4816:
4754:
4728:
4682:
4643:
4623:
4603:
4561:
4525:
4505:
4455:
4384:
4348:
4288:
4211:
4095:
3903:belong to the set
3874:
3759:
3680:to be all vectors
3548:
3546:
3375:
3038:complex conjugates
3015:irrational numbers
2996:
2987:
2936:
2873:
2802:
2765:
2664:
2655:
2547:
2308:
2173:
2052:
2002:If it occurs that
1990:
1829:
1823:
1759:
1698:
1509:
1454:
1416:
1378:of each other, or
1362:
1353:
1309:
1297:
1242:
1196:Vera Kublanovskaya
1192:John G. F. Francis
1161:integral operators
1150:Poisson's equation
1142:Laplace's equation
1121:defective matrices
1093:Francesco Brioschi
1086:Hermitian matrices
1078:symmetric matrices
899:
836:
765:
763:
719:
710:
662:
598:facial recognition
590:vibration analysis
586:stability analysis
492:
448:
414:
392:
367:
344:{\displaystyle n.}
341:
318:
292:
214:
182:
146:
122:
102:
25438:
25437:
25037:Harmonic analysis
24824:
24823:
24691:Geometric algebra
24648:Kronecker product
24483:Linear projection
24468:Vector projection
24284:
24283:
24276:
24144:"λ – Eigenvalues"
23942:(16): 2309–2311.
23794:978-0-387-72828-5
23734:978-0-89871-454-8
23590:Topics In Algebra
23516:(13): 1473–1477,
23499:978-0-8018-5414-9
23323:10.1090/bull/1722
22986:, p. 165901.
22692:Press et al. 2007
22458:Press et al. 2007
22289:{\displaystyle n}
22058:Normal eigenvalue
21963:covariance matrix
21311:
21310:
21303:
21233:quantum chemistry
21221:Koopmans' theorem
21197:molecular physics
21189:quantum mechanics
21000:disordered system
20933:{\displaystyle H}
20868:{\displaystyle H}
20848:{\displaystyle E}
20828:{\displaystyle H}
20659:{\displaystyle H}
20605:square integrable
20558:{\displaystyle E}
20499:{\displaystyle H}
20431:quantum mechanics
20418:{\displaystyle T}
20125:
20107:
20077:{\displaystyle k}
20058:angular frequency
19953:{\displaystyle k}
19929:{\displaystyle m}
19909:{\displaystyle n}
19886:{\displaystyle x}
19846:
19799:
19686:{\displaystyle k}
19666:{\displaystyle k}
19646:{\displaystyle k}
19621:
19575:{\displaystyle i}
19460:{\displaystyle D}
19330:{\displaystyle A}
19233:covariance matrix
19163:covariance matrix
18969:
18968:
17248:
17085:
16630:Lanczos algorithm
16535:
16469:{\displaystyle A}
16327:{\displaystyle b}
16300:
16190:{\displaystyle A}
16141:{\displaystyle A}
16121:{\displaystyle a}
16094:
15613:numerical methods
15604:{\displaystyle n}
15580:{\displaystyle n}
15556:{\displaystyle n}
15434:{\displaystyle A}
15353:
15314:
15249:
14913:
14904:
14567:Dynamic equations
13771:, and the scalar
13765:
13764:
13594:) is a constant.
13456:
13423:
13223:
13175:
13116:
13027:
12967:
12393:
12098:triangular matrix
11665:diagonal matrices
11575:with eigenvalues
11561:
11470:
10714:{\displaystyle i}
10685:
10681:
10667:
10615:
10611:
10597:
10412:
10354:
10294:
9694:corresponding to
9274:corresponding to
8388:
8361:
8334:{\displaystyle H}
8240:. Each column of
7669:{\displaystyle A}
7649:{\displaystyle A}
7622:
7593:
7571:{\displaystyle A}
7533:
7513:
7499:
7457:{\displaystyle A}
7433:{\displaystyle u}
7387:{\displaystyle u}
7306:{\displaystyle A}
7231:{\displaystyle A}
7182:right eigenvector
7058:{\displaystyle P}
6853:{\displaystyle I}
6729:{\displaystyle A}
6659:{\displaystyle A}
6641:positive-definite
6632:{\displaystyle A}
6601:{\displaystyle A}
6551:{\displaystyle A}
6519:
6493:
6434:{\displaystyle A}
6407:{\displaystyle A}
6328:{\displaystyle k}
6281:{\displaystyle A}
6261:{\displaystyle k}
6133:{\displaystyle A}
5952:{\displaystyle A}
5766:{\displaystyle A}
5737:{\displaystyle A}
5682:{\displaystyle A}
5662:{\displaystyle n}
5581:{\displaystyle A}
5524:{\displaystyle A}
5504:{\displaystyle A}
5355:{\displaystyle A}
5193:{\displaystyle A}
4999:{\displaystyle D}
4859:{\displaystyle V}
4839:{\displaystyle I}
4646:{\displaystyle V}
4626:{\displaystyle A}
4528:{\displaystyle V}
3589:simple eigenvalue
3148:Suppose a matrix
3145:that polynomial.
3031:algebraic numbers
2571:
2570:
2332:
2331:
2197:
2196:
2076:
2075:
1452:
1308:
1180:Richard von Mises
1126:In the meantime,
970:of the domain of
799:
762:
256:quantum mechanics
125:{\displaystyle T}
16:(Redirected from
25478:
25451:Abstract algebra
25426:
25425:
25414:
25413:
25402:
25401:
25391:
25390:
25322:Computer algebra
25297:Computer science
25017:Complex analysis
24851:
24844:
24837:
24828:
24827:
24814:
24813:
24696:Exterior algebra
24633:Hadamard product
24550:
24538:Linear equations
24409:
24402:
24395:
24386:
24385:
24351:
24342:
24340:
24339:
24334:
24279:
24272:
24268:
24265:
24259:
24227:
24226:
24219:
24209:
24191:
24173:
24171:
24159:
24138:
24129:
24103:
24084:
24082:
24057:
24055:
24043:
24041:
24039:
24023:
24021:
24019:
24003:
24001:
23999:
23983:
23926:
23924:
23911:
23902:
23866:
23848:
23822:
23797:
23779:
23757:
23737:
23719:
23698:
23678:
23654:Korn, Theresa M.
23648:
23620:
23602:
23584:
23573:
23562:
23561:
23540:
23502:
23477:
23459:
23458:
23437:
23436:
23413:
23391:
23374:
23341:
23339:
23316:
23298:
23288:
23277:
23266:
23246:
23237:
23195:
23170:
23159:
23144:
23133:
23111:
23090:
23084:
23078:
23072:
23066:
23060:
23054:
23048:
23042:
23041:
23029:
23023:
23017:
23011:
23005:
22999:
22993:
22987:
22981:
22975:
22974:, p. 15005.
22969:
22960:
22954:
22948:
22942:
22929:
22923:
22917:
22911:
22905:
22892:
22886:
22880:
22874:
22868:
22862:
22856:
22850:
22847:Van Mieghem 2024
22844:
22838:
22835:Van Mieghem 2014
22832:
22826:
22820:
22811:
22805:
22799:
22793:
22787:
22781:
22775:
22769:
22763:
22757:
22751:
22745:
22736:
22730:
22724:
22718:
22707:
22701:
22695:
22689:
22683:
22667:
22661:
22652:
22646:
22640:
22634:
22628:
22622:
22616:
22610:
22604:
22598:
22592:
22586:
22580:
22574:
22568:
22562:
22556:
22550:
22544:
22535:
22529:
22523:
22517:
22504:
22498:
22492:
22486:
22473:
22467:
22461:
22455:
22449:
22448:
22446:
22444:
22430:
22421:
22415:
22409:
22403:
22394:
22388:
22379:
22378:
22376:
22365:
22359:
22353:
22337:
22331:
22329:
22328:
22323:
22318:
22317:
22295:
22293:
22292:
22287:
22266:
22260:
22238:
22232:
22222:
22212:
22191:
22185:
22160:
22154:
22102:, pp. 176–227.
22089:
21989:data compression
21951:image processing
21923:
21921:
21920:
21915:
21913:
21912:
21896:
21894:
21893:
21888:
21886:
21885:
21869:
21867:
21866:
21861:
21859:
21858:
21842:
21840:
21839:
21834:
21832:
21831:
21815:
21813:
21812:
21807:
21805:
21804:
21774:
21772:
21771:
21766:
21764:
21763:
21751:
21750:
21738:
21737:
21721:
21719:
21718:
21713:
21711:
21710:
21698:
21697:
21685:
21684:
21668:
21666:
21665:
21660:
21658:
21657:
21645:
21644:
21632:
21631:
21615:
21613:
21612:
21607:
21605:
21604:
21588:
21586:
21585:
21580:
21578:
21577:
21561:
21559:
21558:
21553:
21551:
21550:
21526:
21524:
21523:
21518:
21516:
21515:
21510:
21497:
21495:
21494:
21489:
21487:
21486:
21481:
21468:
21466:
21465:
21460:
21458:
21457:
21452:
21439:
21437:
21436:
21431:
21429:
21428:
21416:
21415:
21403:
21402:
21386:
21384:
21383:
21378:
21376:
21375:
21370:
21361:
21360:
21355:
21346:
21345:
21340:
21306:
21299:
21295:
21292:
21286:
21266:
21265:
21258:
21178:
21176:
21175:
21170:
21162:
21161:
21145:
21143:
21142:
21137:
21129:
21128:
21111:
21109:
21108:
21103:
21101:
21096:
21095:
21090:
21077:
21075:
21074:
21069:
21057:
21055:
21054:
21049:
21047:
21042:
21041:
21036:
21023:
21021:
21020:
21015:
21013:
20974:
20972:
20971:
20966:
20961:
20960:
20951:
20939:
20937:
20936:
20931:
20919:
20917:
20916:
20911:
20906:
20905:
20896:
20874:
20872:
20871:
20866:
20854:
20852:
20851:
20846:
20834:
20832:
20831:
20826:
20810:
20808:
20807:
20802:
20797:
20796:
20787:
20772:
20770:
20769:
20764:
20759:
20758:
20749:
20735:
20734:
20725:
20707:
20705:
20704:
20699:
20694:
20693:
20684:
20671:bra–ket notation
20665:
20663:
20662:
20657:
20645:
20643:
20642:
20637:
20635:
20634:
20602:
20600:
20599:
20594:
20592:
20591:
20564:
20562:
20561:
20556:
20540:
20538:
20537:
20532:
20530:
20529:
20505:
20503:
20502:
20497:
20482:
20480:
20479:
20474:
20471:
20470:
20455:
20454:
20424:
20422:
20421:
20416:
20383:angular momentum
20380:
20378:
20377:
20372:
20220:
20218:
20217:
20212:
20201:
20197:
20178:
20177:
20152:
20150:
20149:
20144:
20127:
20126:
20118:
20109:
20108:
20100:
20086:damped vibration
20083:
20081:
20080:
20075:
20060:. The principal
20055:
20053:
20052:
20047:
20035:
20033:
20032:
20027:
20025:
20024:
20008:
20006:
20005:
20000:
19992:
19991:
19962:stiffness matrix
19959:
19957:
19956:
19951:
19935:
19933:
19932:
19927:
19915:
19913:
19912:
19907:
19892:
19890:
19889:
19884:
19870:
19868:
19867:
19862:
19848:
19847:
19839:
19826:
19824:
19823:
19818:
19801:
19800:
19792:
19778:eigenfrequencies
19710:adjacency matrix
19692:
19690:
19689:
19684:
19672:
19670:
19669:
19664:
19652:
19650:
19649:
19644:
19632:
19630:
19629:
19624:
19622:
19617:
19616:
19598:
19596:
19581:
19579:
19578:
19573:
19561:
19559:
19558:
19553:
19551:
19550:
19546:
19523:
19521:
19520:
19515:
19513:
19512:
19496:
19494:
19493:
19488:
19486:
19485:
19466:
19464:
19463:
19458:
19442:
19440:
19439:
19434:
19432:
19431:
19427:
19408:
19407:
19403:
19370:
19368:
19367:
19362:
19339:Laplacian matrix
19336:
19334:
19333:
19328:
19316:adjacency matrix
19253:orthogonal basis
19229:linear relations
19210:orthogonal basis
19160:
19158:
19157:
19152:
19128:
19126:
19125:
19120:
19073:
19071:
19070:
19065:
19032:
19028:
19026:
19025:
19020:
19018:
19017:
18965:
18963:
18962:
18957:
18955:
18951:
18950:
18915:
18914:
18909:
18899:
18898:
18866:
18865:
18860:
18842:
18840:
18839:
18834:
18832:
18831:
18803:
18802:
18797:
18783:
18781:
18780:
18775:
18773:
18769:
18768:
18733:
18732:
18727:
18717:
18716:
18681:
18680:
18675:
18657:
18655:
18654:
18649:
18647:
18643:
18642:
18610:
18609:
18604:
18594:
18593:
18561:
18560:
18555:
18529:
18527:
18526:
18521:
18519:
18505:
18504:
18481:
18480:
18459:
18457:
18456:
18451:
18443:
18442:
18425:
18423:
18422:
18417:
18415:
18401:
18400:
18377:
18376:
18355:
18353:
18352:
18347:
18345:
18331:
18330:
18307:
18306:
18285:
18283:
18282:
18277:
18269:
18268:
18251:
18249:
18248:
18243:
18238:
18237:
18219:
18218:
18201:
18193:
18191:
18190:
18185:
18183:
18169:
18168:
18145:
18144:
18123:
18121:
18120:
18115:
18107:
18106:
18089:
18087:
18086:
18081:
18079:
18065:
18064:
18041:
18040:
18019:
18017:
18016:
18011:
18009:
17995:
17994:
17971:
17970:
17949:
17947:
17946:
17941:
17933:
17932:
17915:
17913:
17912:
17907:
17902:
17901:
17883:
17882:
17865:
17857:
17855:
17854:
17849:
17847:
17816:
17812:
17811:
17792:
17791:
17751:
17747:
17746:
17730:
17729:
17708:
17706:
17705:
17700:
17692:
17691:
17679:
17678:
17661:
17659:
17658:
17653:
17651:
17617:
17613:
17612:
17590:
17589:
17546:
17542:
17541:
17522:
17521:
17500:
17498:
17497:
17492:
17490:
17486:
17485:
17469:
17468:
17455:
17454:
17438:
17437:
17416:
17414:
17413:
17408:
17400:
17399:
17387:
17386:
17369:
17367:
17366:
17361:
17359:
17358:
17338:
17336:
17335:
17330:
17298:
17297:
17280:
17278:
17277:
17272:
17270:
17269:
17246:
17237:
17235:
17234:
17229:
17197:
17196:
17179:
17177:
17176:
17171:
17166:
17165:
17144:
17143:
17117:
17115:
17114:
17109:
17107:
17106:
17083:
17067:
17065:
17064:
17059:
17057:
17056:
16989:
16987:
16986:
16981:
16979:
16978:
16935:
16933:
16932:
16927:
16925:
16924:
16854:
16852:
16851:
16846:
16844:
16843:
16836:
16835:
16812:
16811:
16786:
16784:
16783:
16778:
16776:
16775:
16727:
16711:
16704:
16697:
16679:Horizontal shear
16670:Unequal scaling
16660:
16656:
16634:iterative method
16604:
16602:
16601:
16596:
16594:
16578:
16576:
16575:
16570:
16568:
16567:
16562:
16546:
16544:
16543:
16538:
16536:
16534:
16533:
16528:
16527:
16522:
16515:
16514:
16506:
16505:
16500:
16493:
16475:
16473:
16472:
16467:
16455:
16453:
16452:
16447:
16445:
16431:
16429:
16427:
16426:
16421:
16419:
16400:
16398:
16396:
16395:
16390:
16388:
16387:
16333:
16331:
16330:
16325:
16313:
16311:
16310:
16305:
16303:
16302:
16301:
16295:
16294:
16257:
16255:
16254:
16249:
16222:
16220:
16219:
16214:
16196:
16194:
16193:
16188:
16173:
16171:
16170:
16165:
16148:with eigenvalue
16147:
16145:
16144:
16139:
16127:
16125:
16124:
16119:
16107:
16105:
16104:
16099:
16097:
16096:
16095:
16089:
16088:
16054:
16052:
16051:
16046:
16023:
16021:
16020:
16015:
16013:
16010:
15938:
15934:
15931:
15929:
15928:
15923:
15921:
15918:
15842:linear equations
15837:
15835:
15834:
15829:
15827:
15826:
15792:
15791:
15766:
15765:
15723:
15721:
15720:
15715:
15689:
15687:
15686:
15681:
15679:
15678:
15610:
15608:
15607:
15602:
15589:companion matrix
15586:
15584:
15583:
15578:
15562:
15560:
15559:
15554:
15535:
15533:
15532:
15527:
15512:
15510:
15509:
15504:
15477:round-off errors
15466:
15464:
15463:
15458:
15440:
15438:
15437:
15432:
15405:Classical method
15386:
15384:
15383:
15378:
15367:
15366:
15354:
15352:
15344:
15336:
15334:
15333:
15315:
15313:
15312:
15311:
15292:
15288:
15287:
15271:
15269:
15268:
15250:
15248:
15247:
15246:
15233:
15229:
15228:
15218:
15199:
15197:
15196:
15191:
15185:
15180:
15171:
15170:
15151:
15146:
15137:
15136:
15124:
15123:
15104:
15102:
15101:
15096:
15091:
15090:
15070:
15069:
15049:
15047:
15046:
15041:
15039:
15038:
15031:
15030:
15002:
15001:
14973:
14971:
14970:
14965:
14960:
14959:
14935:
14934:
14911:
14902:
14898:
14897:
14879:
14878:
14849:
14847:
14846:
14841:
14830:
14829:
14814:
14813:
14789:
14788:
14773:
14772:
14760:
14759:
14744:
14743:
14731:
14730:
14700:
14698:
14697:
14692:
14687:
14686:
14671:
14670:
14652:
14651:
14636:
14635:
14623:
14622:
14607:
14606:
14594:
14593:
14557:Hecke eigensheaf
14290:associated with
14259:associated with
14240:
14238:
14237:
14232:
14230:
14220:
14196:
14172:
14164:
14143:
14135:
14086:. Therefore, if
14063:
14061:
14060:
14055:
14053:
14043:
14019:
13995:
13978:
13957:
13949:
13898:
13896:
13895:
13890:
13885:
13881:
13880:
13866:
13852:
13759:
13750:
13748:
13747:
13742:
13737:
13723:
13704:
13674:
13672:
13671:
13666:
13581:
13579:
13578:
13573:
13568:
13567:
13500:
13498:
13497:
13492:
13457:
13455:
13444:
13435:
13433:
13432:
13427:
13425:
13422:
13411:
13383:
13381:
13380:
13375:
13264:
13262:
13261:
13256:
13254:
13253:
13237:
13235:
13234:
13229:
13224:
13222:
13221:
13217:
13216:
13204:
13203:
13189:
13173:
13172:
13165:
13164:
13152:
13151:
13139:
13138:
13124:
13114:
13109:
13108:
13103:
13097:
13096:
13081:
13060:Hermitian matrix
13040:
13038:
13037:
13032:
13030:
13029:
13028:
13022:
13021:
12980:
12978:
12977:
12972:
12970:
12969:
12968:
12962:
12961:
12887:
12885:
12884:
12879:
12874:
12873:
12852:
12851:
12827:
12826:
12672:
12670:
12669:
12664:
12659:
12658:
12536:
12534:
12533:
12528:
12523:
12522:
12487:
12486:
12485:
12484:
12474:
12464:
12463:
12425:
12424:
12423:
12422:
12412:
12402:
12401:
12394:
12386:
12356:
12355:
12354:
12353:
12343:
12323:
12313:
12303:
12291:
12289:
12288:
12283:
12196:
12194:
12193:
12188:
12183:
12182:
12083:
12081:
12080:
12075:
12070:
12069:
12034:
12033:
12032:
12031:
12021:
12011:
12010:
11975:
11974:
11973:
11972:
11962:
11952:
11951:
11916:
11915:
11914:
11913:
11903:
11883:
11873:
11863:
11851:
11849:
11848:
11843:
11756:
11754:
11753:
11748:
11743:
11742:
11654:
11652:
11651:
11646:
11640:
11635:
11634:
11633:
11623:
11614:
11613:
11612:
11611:
11601:
11574:
11572:
11571:
11566:
11564:
11563:
11562:
11556:
11555:
11548:
11547:
11536:
11535:
11509:
11508:
11507:
11506:
11496:
11483:
11481:
11480:
11475:
11473:
11472:
11471:
11465:
11464:
11457:
11456:
11445:
11444:
11418:
11417:
11416:
11415:
11405:
11386:
11384:
11383:
11378:
11373:
11372:
11365:
11364:
11351:
11350:
11323:
11322:
11310:
11309:
11295:
11294:
11281:
11280:
11260:
11259:
11252:
11251:
11238:
11237:
11203:
11201:
11200:
11195:
11190:
11189:
11182:
11181:
11168:
11167:
11140:
11139:
11127:
11126:
11112:
11111:
11098:
11097:
11077:
11076:
11069:
11068:
11055:
11054:
11018:
11016:
11015:
11010:
11005:
11004:
10991:
10986:
10973:
10972:
10959:
10954:
10935:
10934:
10925:
10924:
10906:
10904:
10903:
10898:
10893:
10892:
10851:
10850:
10815:
10814:
10763:
10761:
10759:
10758:
10753:
10742:
10741:
10720:
10718:
10717:
10712:
10700:
10698:
10697:
10692:
10690:
10686:
10677:
10676:
10668:
10660:
10651:
10646:
10630:
10629:
10616:
10607:
10606:
10598:
10590:
10578:
10577:
10554:
10553:
10527:
10525:
10524:
10519:
10514:
10513:
10427:
10425:
10423:
10422:
10417:
10415:
10414:
10413:
10407:
10406:
10369:
10367:
10365:
10364:
10359:
10357:
10356:
10355:
10349:
10348:
10309:
10307:
10305:
10304:
10299:
10297:
10296:
10295:
10289:
10288:
10245:
10243:
10242:
10237:
10235:
10216:
10215:
10200:
10199:
10184:
10183:
10141:
10140:
10113:
10106:
10105:
10022:
10018:
10017:
10016:
9948:
9947:
9835:
9833:
9832:
9827:
9822:
9821:
9739:, respectively.
9738:
9731:
9687:
9685:
9684:
9679:
9677:
9676:
9648:
9647:
9640:
9639:
9626:
9625:
9605:
9604:
9593:
9564:
9562:
9561:
9556:
9554:
9540:
9539:
9524:
9523:
9494:
9493:
9478:
9477:
9458:
9457:
9429:
9428:
9421:
9420:
9407:
9406:
9389:
9388:
9340:
9339:
9328:
9287:
9269:
9267:
9266:
9261:
9259:
9258:
9227:
9226:
9219:
9218:
9202:
9201:
9181:
9180:
9169:
9140:
9138:
9137:
9132:
9124:
9123:
9108:
9107:
9089:
9087:
9086:
9081:
9079:
9078:
9050:
9049:
9042:
9041:
9028:
9027:
9010:
9009:
8971:
8970:
8959:
8925:
8911:
8904:
8895:
8893:
8892:
8887:
8885:
8842:
8838:
8837:
8807:
8803:
8802:
8752:
8748:
8747:
8746:
8705:
8704:
8594:
8592:
8591:
8586:
8581:
8580:
8504:
8502:
8501:
8496:
8494:
8490:
8428:
8426:
8425:
8420:
8418:
8406:
8404:
8403:
8398:
8396:
8391:
8390:
8389:
8383:
8377:
8372:
8364:
8363:
8362:
8356:
8340:
8338:
8337:
8332:
8239:
8182:. Such a matrix
8167:
8165:
8164:
8159:
8142:
8141:
8112:
8110:
8109:
8104:
8099:
8098:
8053:
8051:
8050:
8045:
7998:
7996:
7995:
7990:
7985:
7984:
7977:
7976:
7971:
7965:
7964:
7948:
7947:
7942:
7936:
7935:
7924:
7923:
7918:
7912:
7911:
7852:
7850:
7849:
7844:
7839:
7838:
7831:
7830:
7825:
7812:
7811:
7806:
7798:
7797:
7792:
7675:
7673:
7672:
7667:
7655:
7653:
7652:
7647:
7635:
7633:
7632:
7627:
7625:
7624:
7623:
7606:
7604:
7603:
7598:
7596:
7595:
7594:
7577:
7575:
7574:
7569:
7549:
7547:
7546:
7541:
7536:
7535:
7534:
7528:
7516:
7515:
7514:
7508:
7502:
7501:
7500:
7483:
7481:
7480:
7475:
7463:
7461:
7460:
7455:
7442:left eigenvector
7439:
7437:
7436:
7431:
7419:
7417:
7416:
7411:
7393:
7391:
7390:
7385:
7374:is a scalar and
7373:
7371:
7370:
7365:
7351:
7349:
7348:
7343:
7338:
7324:
7312:
7310:
7309:
7304:
7293:multiply matrix
7282:
7280:
7279:
7274:
7269:
7258:
7237:
7235:
7234:
7229:
7217:
7215:
7214:
7209:
7163:
7161:
7160:
7155:
7147:
7146:
7119:
7118:
7093:
7091:
7090:
7085:
7064:
7062:
7061:
7056:
7044:
7042:
7041:
7036:
7025:
7024:
7000:
6999:
6980:
6978:
6977:
6972:
6951:
6949:
6948:
6943:
6941:
6923:
6921:
6920:
6915:
6904:
6903:
6879:
6878:
6859:
6857:
6856:
6851:
6839:
6837:
6836:
6831:
6813:
6811:
6810:
6805:
6800:
6799:
6781:
6780:
6761:
6759:
6758:
6753:
6735:
6733:
6732:
6727:
6712:
6710:
6709:
6704:
6696:
6691:
6690:
6681:
6665:
6663:
6662:
6657:
6638:
6636:
6635:
6630:
6607:
6605:
6604:
6599:
6587:
6585:
6584:
6579:
6577:
6576:
6558:is equal to its
6557:
6555:
6554:
6549:
6530:
6528:
6527:
6522:
6520:
6518:
6517:
6505:
6494:
6492:
6491:
6479:
6470:
6468:
6467:
6462:
6460:
6459:
6440:
6438:
6437:
6432:
6413:
6411:
6410:
6405:
6390:
6388:
6387:
6382:
6379:
6374:
6355:
6350:
6334:
6332:
6331:
6326:
6314:
6312:
6311:
6306:
6304:
6303:
6287:
6285:
6284:
6279:
6267:
6265:
6264:
6259:
6243:
6241:
6240:
6235:
6230:
6229:
6217:
6216:
6207:
6206:
6194:
6193:
6183:
6178:
6139:
6137:
6136:
6131:
6111:
6109:
6108:
6103:
6098:
6097:
6079:
6078:
6066:
6065:
6053:
6052:
6042:
6037:
6019:
6018:
6005:
6000:
5958:
5956:
5955:
5950:
5930:
5928:
5927:
5922:
5917:
5916:
5904:
5903:
5887:
5885:
5884:
5879:
5874:
5873:
5861:
5860:
5844:
5842:
5841:
5836:
5834:
5833:
5815:
5814:
5798:
5796:
5795:
5790:
5773:be an arbitrary
5772:
5770:
5769:
5764:
5743:
5741:
5740:
5735:
5723:
5721:
5720:
5715:
5713:
5712:
5707:
5688:
5686:
5685:
5680:
5668:
5666:
5665:
5660:
5648:
5646:
5645:
5640:
5638:
5637:
5632:
5616:
5614:
5613:
5608:
5606:
5605:
5600:
5587:
5585:
5584:
5579:
5563:
5561:
5560:
5555:
5547:
5546:
5530:
5528:
5527:
5522:
5510:
5508:
5507:
5502:
5486:
5484:
5483:
5478:
5476:
5463:
5462:
5433:
5432:
5420:
5419:
5409:
5404:
5382:
5381:
5361:
5359:
5358:
5353:
5341:
5339:
5338:
5333:
5328:
5327:
5315:
5314:
5298:
5296:
5295:
5290:
5288:
5287:
5271:
5269:
5268:
5263:
5261:
5260:
5242:
5241:
5225:
5223:
5222:
5217:
5199:
5197:
5196:
5191:
5176:
5174:
5173:
5168:
5157:
5156:
5135:
5134:
5118:
5116:
5115:
5110:
5098:
5096:
5095:
5090:
5088:
5087:
5077:
5076:
5043:
5041:
5040:
5035:
5005:
5003:
5002:
4997:
4985:
4983:
4982:
4977:
4923:
4921:
4920:
4915:
4894:
4892:
4891:
4886:
4865:
4863:
4862:
4857:
4845:
4843:
4842:
4837:
4825:
4823:
4822:
4817:
4763:
4761:
4760:
4755:
4737:
4735:
4734:
4729:
4727:
4726:
4716:
4715:
4691:
4689:
4688:
4683:
4675:
4674:
4652:
4650:
4649:
4644:
4632:
4630:
4629:
4624:
4612:
4610:
4609:
4604:
4593:
4592:
4570:
4568:
4567:
4562:
4551:
4550:
4534:
4532:
4531:
4526:
4514:
4512:
4511:
4506:
4504:
4503:
4498:
4486:
4485:
4480:
4464:
4462:
4461:
4456:
4454:
4453:
4443:
4442:
4432:
4415:
4414:
4409:
4393:
4391:
4390:
4385:
4374:
4373:
4357:
4355:
4354:
4349:
4338:
4337:
4316:
4315:
4297:
4295:
4294:
4289:
4272:
4271:
4250:
4249:
4220:
4218:
4217:
4212:
4165:
4164:
4104:
4102:
4101:
4096:
4085:
4084:
4058:associated with
4047:associated with
4023:
3999:or equivalently
3998:
3980:
3962:
3936:or equivalently
3935:
3920:
3883:
3881:
3880:
3875:
3873:
3872:
3867:
3835:associated with
3803:associated with
3787:associated with
3768:
3766:
3765:
3760:
3755:
3751:
3750:
3742:
3737:
3733:
3712:
3645:is said to be a
3587:is said to be a
3557:
3555:
3554:
3549:
3547:
3534:
3530:
3529:
3516:
3515:
3505:
3500:
3478:
3477:
3452:
3451:
3439:
3438:
3384:
3382:
3381:
3376:
3371:
3370:
3366:
3365:
3353:
3352:
3332:
3331:
3316:
3315:
3311:
3310:
3298:
3297:
3277:
3276:
3264:
3263:
3259:
3258:
3246:
3245:
3225:
3224:
3023:rational numbers
3005:
3003:
3002:
2997:
2992:
2991:
2963:
2962:
2951:
2941:
2940:
2909:
2908:
2897:
2884:
2882:
2880:
2879:
2874:
2872:
2864:
2859:
2855:
2830:in the equation
2821:
2817:
2811:
2809:
2808:
2803:
2798:
2797:
2770:
2769:
2687:
2673:
2671:
2670:
2665:
2660:
2659:
2565:
2556:
2554:
2553:
2548:
2534:
2533:
2509:
2508:
2487:
2486:
2442:
2402:secular equation
2396:) is called the
2326:
2317:
2315:
2314:
2309:
2272:
2260:
2210:
2206:
2202:
2191:
2182:
2180:
2179:
2174:
2169:
2161:
2156:
2152:
2126:
2113:
2095:
2091:
2083:
2070:
2061:
2059:
2058:
2053:
2048:
2037:
2029:
2013:
2009:
2005:
1999:
1997:
1996:
1991:
1986:
1985:
1976:
1975:
1962:
1957:
1939:
1938:
1929:
1928:
1907:
1906:
1897:
1896:
1881:
1880:
1871:
1870:
1855:
1854:
1838:
1836:
1835:
1830:
1828:
1827:
1820:
1819:
1799:
1798:
1785:
1784:
1764:
1763:
1756:
1755:
1735:
1734:
1721:
1720:
1703:
1702:
1695:
1694:
1675:
1674:
1660:
1659:
1621:
1620:
1601:
1600:
1589:
1588:
1575:
1574:
1555:
1554:
1543:
1542:
1518:
1516:
1515:
1510:
1505:
1497:
1482:
1478:
1474:
1470:
1463:
1461:
1460:
1455:
1453:
1445:
1425:
1423:
1422:
1417:
1412:
1401:
1389:
1376:scalar multiples
1371:
1369:
1368:
1363:
1358:
1357:
1316:
1310:
1306:
1302:
1301:
1263:
1251:
1247:
1208:Euclidean vector
1128:Joseph Liouville
1117:stability theory
1113:Karl Weierstrass
1031:quadric surfaces
921:—for example by
908:
906:
905:
900:
895:
884:
845:
843:
842:
837:
832:
831:
813:
812:
800:
798:
787:
774:
772:
771:
766:
764:
761:
750:
671:
669:
668:
663:
658:
644:
501:
499:
498:
493:
491:
480:
465:
461:
457:
455:
454:
449:
447:
431:
427:
423:
421:
420:
415:
413:
402:) simply scales
401:
399:
398:
393:
391:
376:
374:
373:
368:
366:
354:
350:
348:
347:
342:
327:
325:
324:
319:
317:
305:
301:
299:
298:
293:
288:
223:
221:
220:
215:
191:
189:
188:
183:
181:
170:
155:
153:
152:
147:
131:
129:
128:
123:
111:
109:
108:
103:
101:
74:
69:
68:
65:
64:
61:
58:
55:
21:
25486:
25485:
25481:
25480:
25479:
25477:
25476:
25475:
25441:
25440:
25439:
25434:
25385:
25376:
25326:
25283:
25262:Systems science
25193:
25189:Homotopy theory
25155:
25122:
25074:
25046:
24993:
24940:
24911:Category theory
24897:
24862:
24855:
24825:
24820:
24802:
24764:
24720:
24657:
24609:
24551:
24542:
24508:Change of basis
24498:Multilinear map
24436:
24418:
24413:
24363:
24322:
24319:
24318:
24280:
24269:
24263:
24260:
24241:
24232:This article's
24228:
24224:
24217:
24212:
24207:
24189:
24169:
24141:
24101:
24092:
24090:Further reading
24087:
24053:
24037:
24035:
24033:Quanta Magazine
24017:
24015:
23997:
23995:
23864:
23795:
23777:
23735:
23716:
23715:978-007139880-0
23676:
23618:
23600:
23500:
23475:
23440:
23411:
23337:
23296:
23286:
23157:
23131:
23098:
23093:
23085:
23081:
23073:
23069:
23061:
23057:
23049:
23045:
23030:
23026:
23018:
23014:
23006:
23002:
22994:
22990:
22982:
22978:
22970:
22963:
22955:
22951:
22943:
22932:
22924:
22920:
22912:
22908:
22897:, p. 107;
22893:
22889:
22881:
22877:
22869:
22865:
22857:
22853:
22845:
22841:
22833:
22829:
22821:
22814:
22806:
22802:
22794:
22790:
22782:
22778:
22770:
22766:
22758:
22754:
22746:
22739:
22731:
22727:
22719:
22710:
22702:
22698:
22690:
22686:
22680:Wayback Machine
22668:
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22653:
22649:
22641:
22637:
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22601:
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22589:
22581:
22577:
22569:
22565:
22557:
22553:
22545:
22538:
22530:
22526:
22518:
22507:
22499:
22495:
22487:
22476:
22468:
22464:
22456:
22452:
22442:
22440:
22432:
22431:
22424:
22418:Betteridge 1965
22416:
22412:
22404:
22397:
22389:
22382:
22374:
22366:
22362:
22354:
22350:
22346:
22341:
22340:
22313:
22309:
22301:
22298:
22297:
22281:
22278:
22277:
22267:
22263:
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22188:
22161:
22157:
22090:
22086:
22081:
22009:
21947:
21930:
21908:
21904:
21902:
21899:
21898:
21881:
21877:
21875:
21872:
21871:
21854:
21850:
21848:
21845:
21844:
21827:
21823:
21821:
21818:
21817:
21800:
21796:
21794:
21791:
21790:
21787:
21781:
21759:
21755:
21746:
21742:
21733:
21729:
21727:
21724:
21723:
21706:
21702:
21693:
21689:
21680:
21676:
21674:
21671:
21670:
21653:
21649:
21640:
21636:
21627:
21623:
21621:
21618:
21617:
21600:
21596:
21594:
21591:
21590:
21573:
21569:
21567:
21564:
21563:
21546:
21542:
21540:
21537:
21536:
21511:
21506:
21505:
21503:
21500:
21499:
21482:
21477:
21476:
21474:
21471:
21470:
21453:
21448:
21447:
21445:
21442:
21441:
21424:
21420:
21411:
21407:
21398:
21394:
21392:
21389:
21388:
21371:
21366:
21365:
21356:
21351:
21350:
21341:
21336:
21335:
21333:
21330:
21329:
21307:
21296:
21290:
21287:
21279:help improve it
21276:
21267:
21263:
21256:
21185:
21157:
21153:
21151:
21148:
21147:
21124:
21120:
21118:
21115:
21114:
21097:
21091:
21086:
21085:
21083:
21080:
21079:
21063:
21060:
21059:
21043:
21037:
21032:
21031:
21029:
21026:
21025:
21009:
21007:
21004:
21003:
20981:
20956:
20952:
20947:
20945:
20942:
20941:
20925:
20922:
20921:
20901:
20897:
20892:
20887:
20884:
20883:
20860:
20857:
20856:
20840:
20837:
20836:
20820:
20817:
20816:
20792:
20788:
20783:
20781:
20778:
20777:
20754:
20750:
20745:
20730:
20726:
20721:
20716:
20713:
20712:
20689:
20685:
20680:
20678:
20675:
20674:
20651:
20648:
20647:
20630:
20626:
20624:
20621:
20620:
20587:
20583:
20581:
20578:
20577:
20550:
20547:
20546:
20525:
20521:
20519:
20516:
20515:
20491:
20488:
20487:
20466:
20462:
20450:
20446:
20441:
20438:
20437:
20410:
20407:
20406:
20389:for a position
20339:
20336:
20335:
20304:
20284:solid mechanics
20280:
20244:
20173:
20169:
20168:
20164:
20162:
20159:
20158:
20117:
20116:
20099:
20098:
20093:
20090:
20089:
20069:
20066:
20065:
20062:vibration modes
20041:
20038:
20037:
20020:
20016:
20014:
20011:
20010:
19987:
19983:
19972:
19969:
19968:
19945:
19942:
19941:
19921:
19918:
19917:
19901:
19898:
19897:
19878:
19875:
19874:
19838:
19837:
19832:
19829:
19828:
19791:
19790:
19785:
19782:
19781:
19766:
19752:
19731:
19678:
19675:
19674:
19658:
19655:
19654:
19638:
19635:
19634:
19612:
19608:
19597:
19592:
19587:
19584:
19583:
19567:
19564:
19563:
19542:
19535:
19531:
19529:
19526:
19525:
19508:
19504:
19502:
19499:
19498:
19478:
19474:
19472:
19469:
19468:
19452:
19449:
19448:
19423:
19416:
19412:
19399:
19392:
19388:
19380:
19377:
19376:
19350:
19347:
19346:
19322:
19319:
19318:
19304:
19292:factor analysis
19241:sample variance
19191:
19189:Factor analysis
19181:
19134:
19131:
19130:
19102:
19099:
19098:
19087:
19079:squeeze mapping
19038:
19035:
19034:
19030:
19013:
19009:
18983:
18980:
18979:
18953:
18952:
18945:
18944:
18935:
18934:
18924:
18923:
18916:
18910:
18905:
18904:
18901:
18900:
18893:
18892:
18886:
18885:
18875:
18874:
18867:
18861:
18856:
18855:
18851:
18849:
18846:
18845:
18826:
18825:
18819:
18818:
18808:
18807:
18798:
18793:
18792:
18790:
18787:
18786:
18771:
18770:
18763:
18762:
18753:
18752:
18742:
18741:
18734:
18728:
18723:
18722:
18719:
18718:
18711:
18710:
18701:
18700:
18690:
18689:
18682:
18676:
18671:
18670:
18666:
18664:
18661:
18660:
18645:
18644:
18637:
18636:
18630:
18629:
18619:
18618:
18611:
18605:
18600:
18599:
18596:
18595:
18588:
18587:
18581:
18580:
18570:
18569:
18562:
18556:
18551:
18550:
18546:
18544:
18541:
18540:
18517:
18516:
18506:
18500:
18496:
18493:
18492:
18482:
18476:
18472:
18468:
18466:
18463:
18462:
18438:
18434:
18432:
18429:
18428:
18413:
18412:
18402:
18396:
18392:
18389:
18388:
18378:
18372:
18368:
18364:
18362:
18359:
18358:
18343:
18342:
18332:
18326:
18322:
18319:
18318:
18308:
18302:
18298:
18294:
18292:
18289:
18288:
18264:
18260:
18258:
18255:
18254:
18233:
18229:
18214:
18210:
18208:
18205:
18204:
18203:
18181:
18180:
18170:
18164:
18160:
18157:
18156:
18146:
18140:
18136:
18132:
18130:
18127:
18126:
18102:
18098:
18096:
18093:
18092:
18077:
18076:
18066:
18060:
18056:
18053:
18052:
18042:
18036:
18032:
18028:
18026:
18023:
18022:
18007:
18006:
17996:
17990:
17986:
17983:
17982:
17972:
17966:
17962:
17958:
17956:
17953:
17952:
17928:
17924:
17922:
17919:
17918:
17897:
17893:
17878:
17874:
17872:
17869:
17868:
17867:
17845:
17844:
17814:
17813:
17804:
17800:
17793:
17787:
17783:
17780:
17779:
17749:
17748:
17742:
17738:
17731:
17725:
17721:
17717:
17715:
17712:
17711:
17687:
17683:
17674:
17670:
17668:
17665:
17664:
17649:
17648:
17615:
17614:
17602:
17598:
17591:
17585:
17581:
17578:
17577:
17544:
17543:
17534:
17530:
17523:
17517:
17513:
17509:
17507:
17504:
17503:
17488:
17487:
17481:
17477:
17470:
17464:
17460:
17457:
17456:
17450:
17446:
17439:
17433:
17429:
17425:
17423:
17420:
17419:
17395:
17391:
17382:
17378:
17376:
17373:
17372:
17354:
17350:
17348:
17345:
17344:
17293:
17289:
17287:
17284:
17283:
17265:
17261:
17244:
17241:
17240:
17192:
17188:
17186:
17183:
17182:
17161:
17157:
17139:
17135:
17124:
17121:
17120:
17102:
17098:
17081:
17078:
17077:
17073:
17051:
17050:
17039:
17027:
17026:
17015:
16999:
16998:
16996:
16993:
16992:
16973:
16972:
16967:
16961:
16960:
16955:
16945:
16944:
16942:
16939:
16938:
16919:
16918:
16907:
16895:
16894:
16880:
16864:
16863:
16861:
16858:
16857:
16838:
16837:
16831:
16827:
16825:
16819:
16818:
16813:
16807:
16803:
16796:
16795:
16793:
16790:
16789:
16770:
16769:
16764:
16758:
16757:
16752:
16742:
16741:
16739:
16736:
16735:
16651:
16646:
16626:sparse matrices
16611:
16590:
16588:
16585:
16584:
16563:
16558:
16557:
16555:
16552:
16551:
16529:
16523:
16518:
16517:
16516:
16510:
16501:
16496:
16495:
16494:
16492:
16484:
16481:
16480:
16461:
16458:
16457:
16441:
16439:
16436:
16435:
16415:
16407:
16404:
16403:
16402:
16380:
16376:
16359:
16356:
16355:
16354:
16346:
16344:Power iteration
16340:
16319:
16316:
16315:
16297:
16296:
16289:
16288:
16277:
16267:
16266:
16265:
16263:
16260:
16259:
16228:
16225:
16224:
16202:
16199:
16198:
16182:
16179:
16178:
16153:
16150:
16149:
16133:
16130:
16129:
16113:
16110:
16109:
16091:
16090:
16083:
16082:
16074:
16064:
16063:
16062:
16060:
16057:
16056:
16031:
16028:
16027:
16008:
16007:
15997:
15979:
15978:
15968:
15949:
15945:
15943:
15940:
15939:
15936:
15932:
15916:
15915:
15902:
15884:
15883:
15870:
15854:
15850:
15848:
15845:
15844:
15821:
15820:
15814:
15813:
15803:
15802:
15786:
15785:
15779:
15778:
15768:
15767:
15760:
15759:
15754:
15748:
15747:
15742:
15732:
15731:
15729:
15726:
15725:
15697:
15694:
15693:
15673:
15672:
15667:
15661:
15660:
15655:
15645:
15644:
15636:
15633:
15632:
15625:
15596:
15593:
15592:
15572:
15569:
15568:
15548:
15545:
15544:
15518:
15515:
15514:
15492:
15489:
15488:
15487:, which for an
15446:
15443:
15442:
15426:
15423:
15422:
15419:
15407:
15399:
15393:
15362:
15358:
15345:
15337:
15335:
15329:
15325:
15301:
15297:
15293:
15277:
15273:
15272:
15270:
15258:
15254:
15242:
15238:
15234:
15224:
15220:
15219:
15217:
15215:
15212:
15211:
15181:
15176:
15166:
15162:
15147:
15142:
15132:
15128:
15119:
15115:
15113:
15110:
15109:
15086:
15082:
15065:
15061:
15059:
15056:
15055:
15033:
15032:
15014:
15010:
15008:
15003:
14997:
14993:
14986:
14985:
14983:
14980:
14979:
14943:
14939:
14918:
14914:
14887:
14883:
14868:
14864:
14862:
14859:
14858:
14825:
14821:
14803:
14799:
14778:
14774:
14768:
14764:
14749:
14745:
14739:
14735:
14726:
14722:
14720:
14717:
14716:
14676:
14672:
14666:
14662:
14641:
14637:
14631:
14627:
14612:
14608:
14602:
14598:
14589:
14585:
14583:
14580:
14579:
14569:
14520:
14514:
14436:
14434:Spectral theory
14430:
14428:Spectral theory
14337:
14316:
14228:
14227:
14216:
14200:
14192:
14180:
14179:
14168:
14160:
14147:
14139:
14131:
14121:
14119:
14116:
14115:
14051:
14050:
14039:
14023:
14015:
14003:
14002:
13991:
13974:
13961:
13953:
13945:
13935:
13933:
13930:
13929:
13876:
13862:
13848:
13847:
13843:
13835:
13832:
13831:
13824:
13733:
13719:
13711:
13708:
13707:
13645:
13642:
13641:
13607:
13560:
13556:
13527:
13524:
13523:
13448:
13443:
13441:
13438:
13437:
13415:
13409:
13407:
13404:
13403:
13400:
13339:
13336:
13335:
13309:function spaces
13289:
13283:
13249:
13245:
13243:
13240:
13239:
13212:
13208:
13199:
13195:
13191:
13179:
13174:
13160:
13156:
13147:
13143:
13134:
13130:
13126:
13120:
13115:
13113:
13104:
13099:
13098:
13086:
13082:
13077:
13075:
13072:
13071:
13056:
13049:
13024:
13023:
13016:
13015:
13010:
13005:
13000:
12990:
12989:
12988:
12986:
12983:
12982:
12964:
12963:
12956:
12955:
12950:
12942:
12937:
12927:
12926:
12925:
12923:
12920:
12919:
12902:
12869:
12865:
12847:
12843:
12821:
12820:
12809:
12804:
12799:
12793:
12792:
12787:
12776:
12771:
12765:
12764:
12759:
12754:
12743:
12737:
12736:
12731:
12726:
12721:
12705:
12704:
12678:
12675:
12674:
12653:
12652:
12647:
12642:
12637:
12631:
12630:
12625:
12620:
12615:
12609:
12608:
12603:
12598:
12593:
12587:
12586:
12581:
12576:
12571:
12561:
12560:
12552:
12549:
12548:
12545:
12517:
12516:
12510:
12509:
12503:
12502:
12492:
12491:
12480:
12476:
12475:
12470:
12469:
12458:
12457:
12448:
12447:
12441:
12440:
12430:
12429:
12418:
12414:
12413:
12408:
12407:
12396:
12395:
12385:
12382:
12381:
12372:
12371:
12361:
12360:
12349:
12345:
12344:
12339:
12338:
12336:
12333:
12332:
12321:
12315:
12311:
12305:
12301:
12295:
12208:
12205:
12204:
12177:
12176:
12171:
12166:
12160:
12159:
12154:
12149:
12143:
12142:
12137:
12132:
12122:
12121:
12113:
12110:
12109:
12092:
12064:
12063:
12057:
12056:
12050:
12049:
12039:
12038:
12027:
12023:
12022:
12017:
12016:
12005:
12004:
11998:
11997:
11991:
11990:
11980:
11979:
11968:
11964:
11963:
11958:
11957:
11946:
11945:
11939:
11938:
11932:
11931:
11921:
11920:
11909:
11905:
11904:
11899:
11898:
11896:
11893:
11892:
11881:
11875:
11871:
11865:
11861:
11855:
11768:
11765:
11764:
11737:
11736:
11731:
11726:
11720:
11719:
11714:
11709:
11703:
11702:
11697:
11692:
11682:
11681:
11673:
11670:
11669:
11660:
11636:
11629:
11625:
11624:
11619:
11607:
11603:
11602:
11597:
11596:
11594:
11591:
11590:
11588:
11581:
11558:
11557:
11550:
11549:
11543:
11539:
11537:
11531:
11527:
11525:
11515:
11514:
11513:
11502:
11498:
11497:
11492:
11491:
11489:
11486:
11485:
11467:
11466:
11459:
11458:
11452:
11448:
11446:
11440:
11436:
11434:
11424:
11423:
11422:
11411:
11407:
11406:
11401:
11400:
11398:
11395:
11394:
11367:
11366:
11360:
11356:
11353:
11352:
11346:
11342:
11339:
11338:
11328:
11327:
11318:
11314:
11304:
11303:
11297:
11296:
11290:
11286:
11283:
11282:
11276:
11272:
11265:
11264:
11254:
11253:
11247:
11243:
11240:
11239:
11233:
11229:
11226:
11225:
11215:
11214:
11209:
11206:
11205:
11184:
11183:
11177:
11173:
11170:
11169:
11163:
11159:
11156:
11155:
11145:
11144:
11135:
11131:
11121:
11120:
11114:
11113:
11107:
11103:
11100:
11099:
11093:
11089:
11082:
11081:
11071:
11070:
11064:
11060:
11057:
11056:
11050:
11046:
11043:
11042:
11032:
11031:
11026:
11023:
11022:
11000:
10996:
10987:
10982:
10968:
10964:
10955:
10950:
10930:
10926:
10920:
10916:
10914:
10911:
10910:
10887:
10886:
10880:
10879:
10873:
10872:
10862:
10861:
10845:
10844:
10838:
10837:
10831:
10830:
10820:
10819:
10809:
10808:
10802:
10801:
10795:
10794:
10784:
10783:
10778:
10775:
10774:
10772:
10737:
10733:
10731:
10728:
10727:
10726:
10706:
10703:
10702:
10688:
10687:
10675:
10659:
10647:
10642:
10631:
10625:
10621:
10618:
10617:
10605:
10589:
10579:
10573:
10569:
10566:
10565:
10555:
10549:
10545:
10541:
10539:
10536:
10535:
10508:
10507:
10502:
10497:
10491:
10490:
10485:
10480:
10474:
10473:
10468:
10463:
10453:
10452:
10444:
10441:
10440:
10434:
10409:
10408:
10401:
10400:
10395:
10390:
10380:
10379:
10378:
10376:
10373:
10372:
10371:
10351:
10350:
10343:
10342:
10337:
10329:
10319:
10318:
10317:
10315:
10312:
10311:
10310:
10291:
10290:
10283:
10282:
10277:
10272:
10262:
10261:
10260:
10258:
10255:
10254:
10253:
10233:
10232:
10211:
10207:
10195:
10191:
10179:
10178:
10136:
10135:
10111:
10110:
10100:
10099:
10088:
10083:
10077:
10076:
10071:
10060:
10054:
10053:
10048:
10043:
10027:
10026:
10011:
10010:
10005:
10000:
9994:
9993:
9988:
9983:
9977:
9976:
9971:
9966:
9956:
9955:
9942:
9941:
9936:
9931:
9925:
9924:
9919:
9914:
9908:
9907:
9902:
9897:
9887:
9886:
9885:
9881:
9874:
9849:
9847:
9844:
9843:
9816:
9815:
9810:
9805:
9799:
9798:
9793:
9788:
9782:
9781:
9776:
9771:
9761:
9760:
9752:
9749:
9748:
9745:
9733:
9726:
9720:
9710:
9671:
9670:
9664:
9663:
9653:
9652:
9642:
9641:
9635:
9631:
9628:
9627:
9621:
9617:
9610:
9609:
9594:
9589:
9588:
9586:
9583:
9582:
9580:
9573:
9552:
9551:
9541:
9535:
9531:
9519:
9515:
9509:
9508:
9495:
9489:
9485:
9473:
9469:
9460:
9459:
9452:
9451:
9445:
9444:
9434:
9433:
9423:
9422:
9416:
9412:
9409:
9408:
9402:
9398:
9391:
9390:
9383:
9382:
9374:
9368:
9367:
9362:
9349:
9348:
9341:
9329:
9324:
9323:
9301:
9299:
9296:
9295:
9282:
9253:
9252:
9243:
9242:
9232:
9231:
9221:
9220:
9214:
9210:
9204:
9203:
9197:
9193:
9186:
9185:
9170:
9165:
9164:
9162:
9159:
9158:
9156:
9149:
9119:
9115:
9103:
9099:
9094:
9091:
9090:
9073:
9072:
9066:
9065:
9055:
9054:
9044:
9043:
9037:
9033:
9030:
9029:
9023:
9019:
9012:
9011:
9004:
9003:
8998:
8992:
8991:
8986:
8976:
8975:
8960:
8955:
8954:
8937:
8934:
8933:
8920:
8906:
8899:
8883:
8882:
8840:
8839:
8833:
8829:
8805:
8804:
8797:
8796:
8785:
8779:
8778:
8773:
8757:
8756:
8741:
8740:
8735:
8729:
8728:
8723:
8713:
8712:
8699:
8698:
8693:
8687:
8686:
8681:
8671:
8670:
8669:
8665:
8658:
8633:
8631:
8628:
8627:
8575:
8574:
8569:
8563:
8562:
8557:
8547:
8546:
8538:
8535:
8534:
8524:
8514:
8487:
8486:
8481:
8475:
8474:
8469:
8461:
8457:
8455:
8452:
8451:
8440:
8435:
8433:Matrix examples
8414:
8412:
8409:
8408:
8392:
8385:
8384:
8379:
8378:
8373:
8368:
8358:
8357:
8352:
8351:
8349:
8346:
8345:
8326:
8323:
8322:
8315:
8313:Min-max theorem
8309:
8231:
8134:
8130:
8128:
8125:
8124:
8091:
8087:
8073:
8070:
8069:
8024:
8021:
8020:
8007:
7979:
7978:
7972:
7967:
7966:
7960:
7956:
7954:
7949:
7943:
7938:
7937:
7931:
7927:
7925:
7919:
7914:
7913:
7907:
7903:
7896:
7895:
7884:
7881:
7880:
7833:
7832:
7826:
7821:
7820:
7818:
7813:
7807:
7802:
7801:
7799:
7793:
7788:
7787:
7780:
7779:
7771:
7768:
7767:
7747:
7738:
7731:
7724:
7715:
7708:
7688:
7682:
7661:
7658:
7657:
7641:
7638:
7637:
7619:
7618:
7614:
7612:
7609:
7608:
7590:
7589:
7585:
7583:
7580:
7579:
7563:
7560:
7559:
7530:
7529:
7524:
7523:
7510:
7509:
7504:
7503:
7496:
7495:
7491:
7489:
7486:
7485:
7469:
7466:
7465:
7449:
7446:
7445:
7425:
7422:
7421:
7399:
7396:
7395:
7379:
7376:
7375:
7359:
7356:
7355:
7334:
7320:
7318:
7315:
7314:
7298:
7295:
7294:
7265:
7254:
7249:
7246:
7245:
7223:
7220:
7219:
7197:
7194:
7193:
7192:multiplies the
7177:
7171:
7142:
7138:
7114:
7110:
7099:
7096:
7095:
7070:
7067:
7066:
7050:
7047:
7046:
7020:
7016:
6995:
6991:
6986:
6983:
6982:
6957:
6954:
6953:
6937:
6929:
6926:
6925:
6924:. Moreover, if
6899:
6895:
6874:
6870:
6865:
6862:
6861:
6845:
6842:
6841:
6819:
6816:
6815:
6795:
6791:
6776:
6772:
6767:
6764:
6763:
6741:
6738:
6737:
6721:
6718:
6717:
6692:
6686:
6682:
6677:
6675:
6672:
6671:
6651:
6648:
6647:
6624:
6621:
6620:
6593:
6590:
6589:
6572:
6568:
6566:
6563:
6562:
6543:
6540:
6539:
6513:
6509:
6504:
6487:
6483:
6478:
6476:
6473:
6472:
6452:
6448:
6446:
6443:
6442:
6426:
6423:
6422:
6399:
6396:
6395:
6375:
6370:
6351:
6346:
6340:
6337:
6336:
6320:
6317:
6316:
6299:
6295:
6293:
6290:
6289:
6273:
6270:
6269:
6253:
6250:
6249:
6225:
6221:
6212:
6208:
6202:
6198:
6189:
6185:
6179:
6168:
6147:
6144:
6143:
6125:
6122:
6121:
6093:
6089:
6074:
6070:
6061:
6057:
6048:
6044:
6038:
6027:
6011:
6007:
6001:
5990:
5966:
5963:
5962:
5944:
5941:
5940:
5912:
5908:
5899:
5895:
5893:
5890:
5889:
5869:
5865:
5856:
5852:
5850:
5847:
5846:
5829:
5825:
5810:
5806:
5804:
5801:
5800:
5778:
5775:
5774:
5758:
5755:
5754:
5751:
5729:
5726:
5725:
5708:
5703:
5702:
5700:
5697:
5696:
5674:
5671:
5670:
5654:
5651:
5650:
5633:
5628:
5627:
5625:
5622:
5621:
5601:
5596:
5595:
5593:
5590:
5589:
5573:
5570:
5569:
5542:
5538:
5536:
5533:
5532:
5516:
5513:
5512:
5496:
5493:
5492:
5474:
5473:
5458:
5454:
5447:
5441:
5440:
5428:
5424:
5415:
5411:
5405:
5394:
5383:
5377:
5373:
5369:
5367:
5364:
5363:
5347:
5344:
5343:
5323:
5319:
5310:
5306:
5304:
5301:
5300:
5283:
5279:
5277:
5274:
5273:
5256:
5252:
5237:
5233:
5231:
5228:
5227:
5205:
5202:
5201:
5185:
5182:
5181:
5152:
5148:
5130:
5126:
5124:
5121:
5120:
5104:
5101:
5100:
5072:
5068:
5067:
5063:
5049:
5046:
5045:
5011:
5008:
5007:
5006:, we know that
4991:
4988:
4987:
4929:
4926:
4925:
4900:
4897:
4896:
4871:
4868:
4867:
4851:
4848:
4847:
4831:
4828:
4827:
4769:
4766:
4765:
4743:
4740:
4739:
4711:
4707:
4706:
4702:
4697:
4694:
4693:
4670:
4666:
4658:
4655:
4654:
4638:
4635:
4634:
4618:
4615:
4614:
4588:
4584:
4576:
4573:
4572:
4546:
4542:
4540:
4537:
4536:
4520:
4517:
4516:
4499:
4494:
4493:
4481:
4476:
4475:
4470:
4467:
4466:
4438:
4434:
4433:
4428:
4427:
4410:
4405:
4404:
4402:
4399:
4398:
4369:
4365:
4363:
4360:
4359:
4333:
4329:
4311:
4307:
4305:
4302:
4301:
4267:
4263:
4245:
4241:
4233:
4230:
4229:
4160:
4156:
4154:
4151:
4150:
4080:
4076:
4074:
4071:
4070:
4000:
3986:
3972:
3937:
3922:
3908:
3868:
3863:
3862:
3860:
3857:
3856:
3849:linear subspace
3746:
3738:
3720:
3716:
3708:
3707:
3703:
3695:
3692:
3691:
3655:
3644:
3635:
3626:
3617:
3608:
3599:
3586:
3577:
3568:
3545:
3544:
3525:
3521:
3517:
3511:
3507:
3501:
3490:
3479:
3473:
3469:
3466:
3465:
3447:
3443:
3434:
3430:
3423:
3416:
3414:
3411:
3410:
3361:
3357:
3348:
3344:
3343:
3339:
3327:
3323:
3306:
3302:
3293:
3289:
3288:
3284:
3272:
3268:
3254:
3250:
3241:
3237:
3236:
3232:
3220:
3216:
3187:
3184:
3183:
3140:
3119:
3110:
3086:
3076:
3070:of the matrix.
3068:spectral radius
3055:
2986:
2985:
2979:
2978:
2968:
2967:
2952:
2947:
2946:
2935:
2934:
2925:
2924:
2914:
2913:
2898:
2893:
2892:
2890:
2887:
2886:
2868:
2860:
2842:
2838:
2836:
2833:
2832:
2831:
2819:
2815:
2793:
2789:
2764:
2763:
2752:
2746:
2745:
2740:
2724:
2723:
2697:
2694:
2693:
2677:
2654:
2653:
2648:
2642:
2641:
2636:
2626:
2625:
2617:
2614:
2613:
2604:
2595:
2588:
2581:
2529:
2525:
2504:
2500:
2482:
2478:
2449:
2446:
2445:
2279:
2276:
2275:
2250:
2230:
2224:
2212:identity matrix
2208:
2204:
2200:
2165:
2157:
2139:
2135:
2133:
2130:
2129:
2111:
2110:for the matrix
2093:
2089:
2079:
2044:
2033:
2025:
2020:
2017:
2016:
2007:
2003:
1981:
1977:
1968:
1964:
1958:
1947:
1934:
1930:
1921:
1917:
1902:
1898:
1889:
1885:
1876:
1872:
1863:
1859:
1850:
1846:
1844:
1841:
1840:
1822:
1821:
1815:
1811:
1808:
1807:
1801:
1800:
1794:
1790:
1787:
1786:
1780:
1776:
1769:
1768:
1758:
1757:
1751:
1747:
1744:
1743:
1737:
1736:
1730:
1726:
1723:
1722:
1716:
1712:
1705:
1704:
1697:
1696:
1687:
1683:
1681:
1676:
1667:
1663:
1661:
1652:
1648:
1645:
1644:
1639:
1634:
1629:
1623:
1622:
1613:
1609:
1607:
1602:
1596:
1592:
1590:
1584:
1580:
1577:
1576:
1567:
1563:
1561:
1556:
1550:
1546:
1544:
1538:
1534:
1527:
1526:
1524:
1521:
1520:
1501:
1493:
1488:
1485:
1484:
1480:
1476:
1472:
1468:
1444:
1433:
1430:
1429:
1408:
1397:
1395:
1392:
1391:
1387:
1352:
1351:
1342:
1341:
1335:
1334:
1321:
1320:
1312:
1304:
1296:
1295:
1289:
1288:
1279:
1278:
1268:
1267:
1259:
1257:
1254:
1253:
1249:
1245:
1214:
1204:
1082:Charles Hermite
997:quadratic forms
985:
891:
880:
875:
872:
871:
824:
820:
805:
801:
791:
786:
784:
781:
780:
754:
748:
746:
743:
742:
688:. For example,
654:
640:
632:
629:
628:
594:atomic orbitals
544:
514:square matrices
487:
476:
471:
468:
467:
463:
459:
443:
441:
438:
437:
429:
425:
424:by a factor of
409:
407:
404:
403:
387:
382:
379:
378:
362:
360:
357:
356:
352:
351:If multiplying
333:
330:
329:
313:
311:
308:
307:
303:
284:
279:
276:
275:
272:
266:of the system.
209:
206:
205:
177:
166:
161:
158:
157:
141:
138:
137:
117:
114:
113:
97:
95:
92:
91:
72:
52:
48:
35:
28:
23:
22:
15:
12:
11:
5:
25484:
25474:
25473:
25468:
25463:
25458:
25456:Linear algebra
25453:
25436:
25435:
25433:
25432:
25420:
25408:
25396:
25381:
25378:
25377:
25375:
25374:
25369:
25364:
25359:
25354:
25349:
25348:
25347:
25340:Mathematicians
25336:
25334:
25332:Related topics
25328:
25327:
25325:
25324:
25319:
25314:
25309:
25304:
25299:
25293:
25291:
25285:
25284:
25282:
25281:
25280:
25279:
25274:
25269:
25267:Control theory
25259:
25254:
25249:
25244:
25239:
25234:
25229:
25224:
25219:
25214:
25209:
25203:
25201:
25195:
25194:
25192:
25191:
25186:
25181:
25176:
25171:
25165:
25163:
25157:
25156:
25154:
25153:
25148:
25143:
25138:
25132:
25130:
25124:
25123:
25121:
25120:
25115:
25110:
25105:
25100:
25095:
25090:
25084:
25082:
25076:
25075:
25073:
25072:
25067:
25062:
25056:
25054:
25048:
25047:
25045:
25044:
25042:Measure theory
25039:
25034:
25029:
25024:
25019:
25014:
25009:
25003:
25001:
24995:
24994:
24992:
24991:
24986:
24981:
24976:
24971:
24966:
24961:
24956:
24950:
24948:
24942:
24941:
24939:
24938:
24933:
24928:
24923:
24918:
24913:
24907:
24905:
24899:
24898:
24896:
24895:
24890:
24885:
24884:
24883:
24878:
24867:
24864:
24863:
24854:
24853:
24846:
24839:
24831:
24822:
24821:
24819:
24818:
24807:
24804:
24803:
24801:
24800:
24795:
24790:
24785:
24780:
24778:Floating-point
24774:
24772:
24766:
24765:
24763:
24762:
24760:Tensor product
24757:
24752:
24747:
24745:Function space
24742:
24737:
24731:
24729:
24722:
24721:
24719:
24718:
24713:
24708:
24703:
24698:
24693:
24688:
24683:
24681:Triple product
24678:
24673:
24667:
24665:
24659:
24658:
24656:
24655:
24650:
24645:
24640:
24635:
24630:
24625:
24619:
24617:
24611:
24610:
24608:
24607:
24602:
24597:
24595:Transformation
24592:
24587:
24585:Multiplication
24582:
24577:
24572:
24567:
24561:
24559:
24553:
24552:
24545:
24543:
24541:
24540:
24535:
24530:
24525:
24520:
24515:
24510:
24505:
24500:
24495:
24490:
24485:
24480:
24475:
24470:
24465:
24460:
24455:
24450:
24444:
24442:
24441:Basic concepts
24438:
24437:
24435:
24434:
24429:
24423:
24420:
24419:
24416:Linear algebra
24412:
24411:
24404:
24397:
24389:
24383:
24382:
24369:
24362:
24359:
24345:
24344:
24332:
24329:
24326:
24312:
24303:
24297:
24291:
24282:
24281:
24236:external links
24231:
24229:
24222:
24216:
24215:External links
24213:
24211:
24210:
24205:
24192:
24187:
24174:
24160:
24139:
24112:(1–2): 35–65,
24093:
24091:
24088:
24086:
24085:
24058:
24044:
24024:
24004:
23984:
23936:Optics Letters
23927:
23912:
23903:
23892:10.1086/626490
23878:(2): 114–150,
23867:
23862:
23854:Linear algebra
23849:
23798:
23793:
23780:
23776:978-0521880688
23775:
23758:
23738:
23733:
23720:
23714:
23699:
23690:(3): 637–657,
23679:
23674:
23649:
23621:
23616:
23603:
23599:978-1114541016
23598:
23585:
23580:Linear Algebra
23574:
23563:
23541:
23503:
23498:
23482:Golub, Gene H.
23478:
23473:
23465:Linear algebra
23460:
23449:(4): 332–345,
23438:
23427:(3): 265–271,
23414:
23409:
23401:Addison-Wesley
23392:
23357:(4): 365–382,
23342:
23289:
23284:
23267:
23247:
23238:
23186:(16): 165901.
23171:
23160:
23155:
23134:
23129:
23112:
23099:
23097:
23094:
23092:
23091:
23079:
23067:
23055:
23043:
23024:
23022:, p. 243.
23012:
23000:
22988:
22976:
22961:
22949:
22930:
22918:
22916:, p. 111.
22906:
22901:, p. 109
22887:
22875:
22863:
22851:
22839:
22827:
22812:
22808:Wolchover 2019
22800:
22798:, p. 116.
22788:
22786:, p. 290.
22776:
22764:
22762:, p. 272.
22752:
22750:, p. 307.
22737:
22725:
22723:, p. 316.
22708:
22706:, p. 358.
22696:
22684:
22662:
22647:
22635:
22623:
22611:
22599:
22587:
22575:
22573:, pp. 706–707.
22563:
22561:, pp. 715–716.
22551:
22549:, pp. 807–808.
22536:
22524:
22505:
22493:
22491:, p. 107.
22474:
22462:
22460:, p. 536.
22450:
22422:
22410:
22395:
22380:
22360:
22358:, p. 401.
22347:
22345:
22342:
22339:
22338:
22321:
22316:
22312:
22308:
22305:
22285:
22261:
22233:
22231:
22230:
22224:
22215:eigenfunctions
22186:
22167:Comptes rendus
22155:
22153:
22152:
22134:
22111:
22083:
22082:
22080:
22077:
22076:
22075:
22070:
22068:Singular value
22065:
22060:
22055:
22050:
22045:
22040:
22038:Quantum states
22035:
22030:
22025:
22020:
22015:
22008:
22005:
21993:identification
21943:Main article:
21929:
21926:
21911:
21907:
21884:
21880:
21857:
21853:
21830:
21826:
21803:
21799:
21783:Main article:
21780:
21777:
21762:
21758:
21754:
21749:
21745:
21741:
21736:
21732:
21709:
21705:
21701:
21696:
21692:
21688:
21683:
21679:
21656:
21652:
21648:
21643:
21639:
21635:
21630:
21626:
21603:
21599:
21576:
21572:
21549:
21545:
21514:
21509:
21485:
21480:
21456:
21451:
21427:
21423:
21419:
21414:
21410:
21406:
21401:
21397:
21374:
21369:
21364:
21359:
21354:
21349:
21344:
21339:
21309:
21308:
21270:
21268:
21261:
21255:
21252:
21184:
21181:
21168:
21165:
21160:
21156:
21135:
21132:
21127:
21123:
21100:
21094:
21089:
21067:
21046:
21040:
21035:
21012:
20988:acoustic waves
20980:
20979:Wave transport
20977:
20964:
20959:
20955:
20950:
20929:
20909:
20904:
20900:
20895:
20891:
20864:
20844:
20824:
20800:
20795:
20791:
20786:
20774:
20773:
20762:
20757:
20753:
20748:
20744:
20741:
20738:
20733:
20729:
20724:
20720:
20697:
20692:
20688:
20683:
20655:
20633:
20629:
20613:scalar product
20590:
20586:
20554:
20528:
20524:
20495:
20484:
20483:
20469:
20465:
20461:
20458:
20453:
20449:
20445:
20414:
20395:atomic nucleus
20370:
20366:
20363:
20359:
20356:
20352:
20349:
20346:
20343:
20303:
20300:
20279:
20276:
20272:center of mass
20256:principal axes
20243:
20240:
20210:
20207:
20204:
20200:
20196:
20193:
20190:
20187:
20184:
20181:
20176:
20172:
20167:
20142:
20139:
20136:
20133:
20130:
20124:
20121:
20115:
20112:
20106:
20103:
20097:
20088:, governed by
20073:
20045:
20023:
20019:
19998:
19995:
19990:
19986:
19982:
19979:
19976:
19949:
19925:
19905:
19882:
19860:
19857:
19854:
19851:
19845:
19842:
19836:
19816:
19813:
19810:
19807:
19804:
19798:
19795:
19789:
19762:Main article:
19751:
19748:
19730:
19727:
19682:
19673:th largest or
19662:
19642:
19620:
19615:
19611:
19607:
19604:
19601:
19595:
19591:
19571:
19549:
19545:
19541:
19538:
19534:
19511:
19507:
19484:
19481:
19477:
19456:
19430:
19426:
19422:
19419:
19415:
19411:
19406:
19402:
19398:
19395:
19391:
19387:
19384:
19360:
19357:
19354:
19326:
19303:
19300:
19268:bioinformatics
19177:Main article:
19150:
19147:
19144:
19141:
19138:
19118:
19115:
19112:
19109:
19106:
19086:
19083:
19063:
19060:
19057:
19054:
19051:
19048:
19045:
19042:
19016:
19012:
19008:
19005:
19002:
18999:
18996:
18993:
18990:
18987:
18967:
18966:
18949:
18943:
18940:
18937:
18936:
18933:
18930:
18929:
18927:
18922:
18919:
18917:
18913:
18908:
18903:
18902:
18897:
18891:
18888:
18887:
18884:
18881:
18880:
18878:
18873:
18870:
18868:
18864:
18859:
18854:
18853:
18843:
18830:
18824:
18821:
18820:
18817:
18814:
18813:
18811:
18806:
18801:
18796:
18784:
18767:
18761:
18758:
18755:
18754:
18751:
18748:
18747:
18745:
18740:
18737:
18735:
18731:
18726:
18721:
18720:
18715:
18709:
18706:
18703:
18702:
18699:
18696:
18695:
18693:
18688:
18685:
18683:
18679:
18674:
18669:
18668:
18658:
18641:
18635:
18632:
18631:
18628:
18625:
18624:
18622:
18617:
18614:
18612:
18608:
18603:
18598:
18597:
18592:
18586:
18583:
18582:
18579:
18576:
18575:
18573:
18568:
18565:
18563:
18559:
18554:
18549:
18548:
18538:
18535:
18531:
18530:
18515:
18512:
18509:
18507:
18503:
18499:
18495:
18494:
18491:
18488:
18485:
18483:
18479:
18475:
18471:
18470:
18460:
18449:
18446:
18441:
18437:
18426:
18411:
18408:
18405:
18403:
18399:
18395:
18391:
18390:
18387:
18384:
18381:
18379:
18375:
18371:
18367:
18366:
18356:
18341:
18338:
18335:
18333:
18329:
18325:
18321:
18320:
18317:
18314:
18311:
18309:
18305:
18301:
18297:
18296:
18286:
18275:
18272:
18267:
18263:
18252:
18241:
18236:
18232:
18228:
18225:
18222:
18217:
18213:
18195:
18194:
18179:
18176:
18173:
18171:
18167:
18163:
18159:
18158:
18155:
18152:
18149:
18147:
18143:
18139:
18135:
18134:
18124:
18113:
18110:
18105:
18101:
18090:
18075:
18072:
18069:
18067:
18063:
18059:
18055:
18054:
18051:
18048:
18045:
18043:
18039:
18035:
18031:
18030:
18020:
18005:
18002:
17999:
17997:
17993:
17989:
17985:
17984:
17981:
17978:
17975:
17973:
17969:
17965:
17961:
17960:
17950:
17939:
17936:
17931:
17927:
17916:
17905:
17900:
17896:
17892:
17889:
17886:
17881:
17877:
17859:
17858:
17843:
17840:
17837:
17834:
17831:
17828:
17825:
17822:
17819:
17817:
17815:
17810:
17807:
17803:
17799:
17796:
17794:
17790:
17786:
17782:
17781:
17778:
17775:
17772:
17769:
17766:
17763:
17760:
17757:
17754:
17752:
17750:
17745:
17741:
17737:
17734:
17732:
17728:
17724:
17720:
17719:
17709:
17698:
17695:
17690:
17686:
17682:
17677:
17673:
17662:
17647:
17644:
17641:
17638:
17635:
17632:
17629:
17626:
17623:
17620:
17618:
17616:
17611:
17608:
17605:
17601:
17597:
17594:
17592:
17588:
17584:
17580:
17579:
17576:
17573:
17570:
17567:
17564:
17561:
17558:
17555:
17552:
17549:
17547:
17545:
17540:
17537:
17533:
17529:
17526:
17524:
17520:
17516:
17512:
17511:
17501:
17484:
17480:
17476:
17473:
17471:
17467:
17463:
17459:
17458:
17453:
17449:
17445:
17442:
17440:
17436:
17432:
17428:
17427:
17417:
17406:
17403:
17398:
17394:
17390:
17385:
17381:
17370:
17357:
17353:
17340:
17339:
17328:
17325:
17322:
17319:
17316:
17313:
17310:
17307:
17304:
17301:
17296:
17292:
17281:
17268:
17264:
17260:
17257:
17254:
17251:
17238:
17227:
17224:
17221:
17218:
17215:
17212:
17209:
17206:
17203:
17200:
17195:
17191:
17180:
17169:
17164:
17160:
17156:
17153:
17150:
17147:
17142:
17138:
17134:
17131:
17128:
17118:
17105:
17101:
17097:
17094:
17091:
17088:
17075:
17072:Characteristic
17069:
17068:
17055:
17049:
17046:
17043:
17040:
17038:
17035:
17032:
17029:
17028:
17025:
17022:
17019:
17016:
17014:
17011:
17008:
17005:
17004:
17002:
16990:
16977:
16971:
16968:
16966:
16963:
16962:
16959:
16956:
16954:
16951:
16950:
16948:
16936:
16923:
16917:
16914:
16911:
16908:
16906:
16903:
16900:
16897:
16896:
16893:
16890:
16887:
16884:
16881:
16879:
16876:
16873:
16870:
16869:
16867:
16855:
16842:
16834:
16830:
16826:
16824:
16821:
16820:
16817:
16814:
16810:
16806:
16802:
16801:
16799:
16787:
16774:
16768:
16765:
16763:
16760:
16759:
16756:
16753:
16751:
16748:
16747:
16745:
16733:
16729:
16728:
16721:
16712:
16705:
16698:
16691:
16687:
16686:
16681:
16676:
16671:
16668:
16663:
16650:
16647:
16645:
16642:
16610:
16609:Modern methods
16607:
16593:
16566:
16561:
16548:
16547:
16532:
16526:
16521:
16513:
16509:
16504:
16499:
16491:
16488:
16465:
16444:
16418:
16414:
16411:
16386:
16383:
16379:
16375:
16372:
16369:
16366:
16363:
16342:Main article:
16339:
16336:
16323:
16293:
16287:
16284:
16281:
16278:
16276:
16273:
16272:
16270:
16247:
16244:
16241:
16238:
16235:
16232:
16212:
16209:
16206:
16186:
16163:
16160:
16157:
16137:
16117:
16087:
16081:
16078:
16075:
16073:
16070:
16069:
16067:
16044:
16041:
16038:
16035:
16012:
16006:
16003:
16000:
15998:
15996:
15993:
15990:
15987:
15984:
15981:
15980:
15977:
15974:
15971:
15969:
15967:
15964:
15961:
15958:
15955:
15952:
15951:
15948:
15920:
15914:
15911:
15908:
15905:
15903:
15901:
15898:
15895:
15892:
15889:
15886:
15885:
15882:
15879:
15876:
15873:
15871:
15869:
15866:
15863:
15860:
15857:
15856:
15853:
15825:
15819:
15816:
15815:
15812:
15809:
15808:
15806:
15801:
15798:
15795:
15790:
15784:
15781:
15780:
15777:
15774:
15773:
15771:
15764:
15758:
15755:
15753:
15750:
15749:
15746:
15743:
15741:
15738:
15737:
15735:
15713:
15710:
15707:
15704:
15701:
15677:
15671:
15668:
15666:
15663:
15662:
15659:
15656:
15654:
15651:
15650:
15648:
15643:
15640:
15624:
15621:
15600:
15576:
15552:
15525:
15522:
15502:
15499:
15496:
15456:
15453:
15450:
15430:
15418:
15415:
15411:floating-point
15406:
15403:
15395:Main article:
15392:
15389:
15388:
15387:
15376:
15373:
15370:
15365:
15361:
15357:
15351:
15348:
15343:
15340:
15332:
15328:
15324:
15321:
15318:
15310:
15307:
15304:
15300:
15296:
15291:
15286:
15283:
15280:
15276:
15267:
15264:
15261:
15257:
15253:
15245:
15241:
15237:
15232:
15227:
15223:
15201:
15200:
15189:
15184:
15179:
15175:
15169:
15165:
15161:
15158:
15155:
15150:
15145:
15141:
15135:
15131:
15127:
15122:
15118:
15094:
15089:
15085:
15080:
15077:
15073:
15068:
15064:
15037:
15029:
15026:
15023:
15020:
15017:
15013:
15009:
15007:
15004:
15000:
14996:
14992:
14991:
14989:
14963:
14958:
14955:
14952:
14949:
14946:
14942:
14938:
14933:
14930:
14927:
14924:
14921:
14917:
14910:
14907:
14901:
14896:
14893:
14890:
14886:
14882:
14877:
14874:
14871:
14867:
14851:
14850:
14839:
14836:
14833:
14828:
14824:
14820:
14817:
14812:
14809:
14806:
14802:
14798:
14795:
14792:
14787:
14784:
14781:
14777:
14771:
14767:
14763:
14758:
14755:
14752:
14748:
14742:
14738:
14734:
14729:
14725:
14702:
14701:
14690:
14685:
14682:
14679:
14675:
14669:
14665:
14661:
14658:
14655:
14650:
14647:
14644:
14640:
14634:
14630:
14626:
14621:
14618:
14615:
14611:
14605:
14601:
14597:
14592:
14588:
14575:have the form
14568:
14565:
14547:weight vectors
14516:Main article:
14513:
14510:
14432:Main article:
14429:
14426:
14349:always form a
14333:
14312:
14226:
14223:
14219:
14215:
14212:
14209:
14206:
14203:
14201:
14199:
14195:
14191:
14188:
14185:
14182:
14181:
14178:
14175:
14171:
14167:
14163:
14159:
14156:
14153:
14150:
14148:
14146:
14142:
14138:
14134:
14130:
14127:
14124:
14123:
14049:
14046:
14042:
14038:
14035:
14032:
14029:
14026:
14024:
14022:
14018:
14014:
14011:
14008:
14005:
14004:
14001:
13998:
13994:
13990:
13987:
13984:
13981:
13977:
13973:
13970:
13967:
13964:
13962:
13960:
13956:
13952:
13948:
13944:
13941:
13938:
13937:
13909:is called the
13888:
13884:
13879:
13875:
13872:
13869:
13865:
13861:
13858:
13855:
13851:
13846:
13842:
13839:
13823:
13820:
13799:to the vector
13763:
13762:
13753:
13751:
13740:
13736:
13732:
13729:
13726:
13722:
13718:
13715:
13664:
13661:
13658:
13655:
13652:
13649:
13606:
13603:
13571:
13566:
13563:
13559:
13555:
13552:
13549:
13546:
13543:
13540:
13537:
13534:
13531:
13490:
13487:
13484:
13481:
13478:
13475:
13472:
13469:
13466:
13463:
13460:
13454:
13451:
13447:
13421:
13418:
13414:
13399:
13396:
13392:eigenfunctions
13373:
13370:
13367:
13364:
13361:
13358:
13355:
13352:
13349:
13346:
13343:
13321:differentiable
13319:of infinitely
13285:Main article:
13282:
13279:
13252:
13248:
13227:
13220:
13215:
13211:
13207:
13202:
13198:
13194:
13188:
13185:
13182:
13178:
13171:
13168:
13163:
13159:
13155:
13150:
13146:
13142:
13137:
13133:
13129:
13123:
13119:
13112:
13107:
13102:
13095:
13092:
13089:
13085:
13080:
13055:
13052:
13045:
13020:
13014:
13011:
13009:
13006:
13004:
13001:
12999:
12996:
12995:
12993:
12960:
12954:
12951:
12949:
12946:
12943:
12941:
12938:
12936:
12933:
12932:
12930:
12898:
12877:
12872:
12868:
12864:
12861:
12858:
12855:
12850:
12846:
12842:
12839:
12836:
12833:
12830:
12825:
12819:
12816:
12813:
12810:
12808:
12805:
12803:
12800:
12798:
12795:
12794:
12791:
12788:
12786:
12783:
12780:
12777:
12775:
12772:
12770:
12767:
12766:
12763:
12760:
12758:
12755:
12753:
12750:
12747:
12744:
12742:
12739:
12738:
12735:
12732:
12730:
12727:
12725:
12722:
12720:
12717:
12714:
12711:
12710:
12708:
12703:
12700:
12697:
12694:
12691:
12688:
12685:
12682:
12662:
12657:
12651:
12648:
12646:
12643:
12641:
12638:
12636:
12633:
12632:
12629:
12626:
12624:
12621:
12619:
12616:
12614:
12611:
12610:
12607:
12604:
12602:
12599:
12597:
12594:
12592:
12589:
12588:
12585:
12582:
12580:
12577:
12575:
12572:
12570:
12567:
12566:
12564:
12559:
12556:
12544:
12541:
12526:
12521:
12515:
12512:
12511:
12508:
12505:
12504:
12501:
12498:
12497:
12495:
12490:
12483:
12479:
12473:
12467:
12462:
12456:
12453:
12450:
12449:
12446:
12443:
12442:
12439:
12436:
12435:
12433:
12428:
12421:
12417:
12411:
12405:
12400:
12392:
12389:
12384:
12383:
12380:
12377:
12374:
12373:
12370:
12367:
12366:
12364:
12359:
12352:
12348:
12342:
12319:
12309:
12299:
12281:
12278:
12275:
12272:
12269:
12266:
12263:
12260:
12257:
12254:
12251:
12248:
12245:
12242:
12239:
12236:
12233:
12230:
12227:
12224:
12221:
12218:
12215:
12212:
12186:
12181:
12175:
12172:
12170:
12167:
12165:
12162:
12161:
12158:
12155:
12153:
12150:
12148:
12145:
12144:
12141:
12138:
12136:
12133:
12131:
12128:
12127:
12125:
12120:
12117:
12091:
12088:
12073:
12068:
12062:
12059:
12058:
12055:
12052:
12051:
12048:
12045:
12044:
12042:
12037:
12030:
12026:
12020:
12014:
12009:
12003:
12000:
11999:
11996:
11993:
11992:
11989:
11986:
11985:
11983:
11978:
11971:
11967:
11961:
11955:
11950:
11944:
11941:
11940:
11937:
11934:
11933:
11930:
11927:
11926:
11924:
11919:
11912:
11908:
11902:
11879:
11869:
11859:
11841:
11838:
11835:
11832:
11829:
11826:
11823:
11820:
11817:
11814:
11811:
11808:
11805:
11802:
11799:
11796:
11793:
11790:
11787:
11784:
11781:
11778:
11775:
11772:
11746:
11741:
11735:
11732:
11730:
11727:
11725:
11722:
11721:
11718:
11715:
11713:
11710:
11708:
11705:
11704:
11701:
11698:
11696:
11693:
11691:
11688:
11687:
11685:
11680:
11677:
11659:
11656:
11644:
11639:
11632:
11628:
11622:
11617:
11610:
11606:
11600:
11586:
11579:
11554:
11546:
11542:
11538:
11534:
11530:
11526:
11524:
11521:
11520:
11518:
11512:
11505:
11501:
11495:
11463:
11455:
11451:
11447:
11443:
11439:
11435:
11433:
11430:
11429:
11427:
11421:
11414:
11410:
11404:
11376:
11371:
11363:
11359:
11355:
11354:
11349:
11345:
11341:
11340:
11337:
11334:
11333:
11331:
11326:
11321:
11317:
11313:
11308:
11302:
11299:
11298:
11293:
11289:
11285:
11284:
11279:
11275:
11271:
11270:
11268:
11263:
11258:
11250:
11246:
11242:
11241:
11236:
11232:
11228:
11227:
11224:
11221:
11220:
11218:
11213:
11193:
11188:
11180:
11176:
11172:
11171:
11166:
11162:
11158:
11157:
11154:
11151:
11150:
11148:
11143:
11138:
11134:
11130:
11125:
11119:
11116:
11115:
11110:
11106:
11102:
11101:
11096:
11092:
11088:
11087:
11085:
11080:
11075:
11067:
11063:
11059:
11058:
11053:
11049:
11045:
11044:
11041:
11038:
11037:
11035:
11030:
11008:
11003:
10999:
10995:
10990:
10985:
10981:
10976:
10971:
10967:
10963:
10958:
10953:
10949:
10944:
10941:
10938:
10933:
10929:
10923:
10919:
10896:
10891:
10885:
10882:
10881:
10878:
10875:
10874:
10871:
10868:
10867:
10865:
10860:
10857:
10854:
10849:
10843:
10840:
10839:
10836:
10833:
10832:
10829:
10826:
10825:
10823:
10818:
10813:
10807:
10804:
10803:
10800:
10797:
10796:
10793:
10790:
10789:
10787:
10782:
10770:
10751:
10748:
10745:
10740:
10736:
10723:imaginary unit
10710:
10684:
10680:
10674:
10671:
10666:
10663:
10658:
10655:
10650:
10645:
10641:
10637:
10634:
10632:
10628:
10624:
10620:
10619:
10614:
10610:
10604:
10601:
10596:
10593:
10588:
10585:
10582:
10580:
10576:
10572:
10568:
10567:
10564:
10561:
10558:
10556:
10552:
10548:
10544:
10543:
10517:
10512:
10506:
10503:
10501:
10498:
10496:
10493:
10492:
10489:
10486:
10484:
10481:
10479:
10476:
10475:
10472:
10469:
10467:
10464:
10462:
10459:
10458:
10456:
10451:
10448:
10433:
10430:
10405:
10399:
10396:
10394:
10391:
10389:
10386:
10385:
10383:
10347:
10341:
10338:
10336:
10333:
10330:
10328:
10325:
10324:
10322:
10287:
10281:
10278:
10276:
10273:
10271:
10268:
10267:
10265:
10231:
10228:
10225:
10222:
10219:
10214:
10210:
10206:
10203:
10198:
10194:
10190:
10187:
10182:
10177:
10174:
10171:
10168:
10165:
10162:
10159:
10156:
10153:
10150:
10147:
10144:
10139:
10134:
10131:
10128:
10125:
10122:
10119:
10116:
10114:
10112:
10109:
10104:
10098:
10095:
10092:
10089:
10087:
10084:
10082:
10079:
10078:
10075:
10072:
10070:
10067:
10064:
10061:
10059:
10056:
10055:
10052:
10049:
10047:
10044:
10042:
10039:
10036:
10033:
10032:
10030:
10025:
10021:
10015:
10009:
10006:
10004:
10001:
9999:
9996:
9995:
9992:
9989:
9987:
9984:
9982:
9979:
9978:
9975:
9972:
9970:
9967:
9965:
9962:
9961:
9959:
9954:
9951:
9946:
9940:
9937:
9935:
9932:
9930:
9927:
9926:
9923:
9920:
9918:
9915:
9913:
9910:
9909:
9906:
9903:
9901:
9898:
9896:
9893:
9892:
9890:
9884:
9880:
9877:
9875:
9873:
9870:
9867:
9864:
9861:
9858:
9855:
9852:
9851:
9825:
9820:
9814:
9811:
9809:
9806:
9804:
9801:
9800:
9797:
9794:
9792:
9789:
9787:
9784:
9783:
9780:
9777:
9775:
9772:
9770:
9767:
9766:
9764:
9759:
9756:
9744:
9741:
9715:
9705:
9675:
9669:
9666:
9665:
9662:
9659:
9658:
9656:
9651:
9646:
9638:
9634:
9630:
9629:
9624:
9620:
9616:
9615:
9613:
9608:
9603:
9600:
9597:
9592:
9578:
9571:
9550:
9547:
9544:
9542:
9538:
9534:
9530:
9527:
9522:
9518:
9514:
9511:
9510:
9507:
9504:
9501:
9498:
9496:
9492:
9488:
9484:
9481:
9476:
9472:
9468:
9465:
9462:
9461:
9456:
9450:
9447:
9446:
9443:
9440:
9439:
9437:
9432:
9427:
9419:
9415:
9411:
9410:
9405:
9401:
9397:
9396:
9394:
9387:
9381:
9378:
9375:
9373:
9370:
9369:
9366:
9363:
9361:
9358:
9355:
9354:
9352:
9347:
9344:
9342:
9338:
9335:
9332:
9327:
9322:
9319:
9316:
9313:
9310:
9307:
9304:
9303:
9257:
9251:
9248:
9245:
9244:
9241:
9238:
9237:
9235:
9230:
9225:
9217:
9213:
9209:
9206:
9205:
9200:
9196:
9192:
9191:
9189:
9184:
9179:
9176:
9173:
9168:
9154:
9147:
9130:
9127:
9122:
9118:
9114:
9111:
9106:
9102:
9098:
9077:
9071:
9068:
9067:
9064:
9061:
9060:
9058:
9053:
9048:
9040:
9036:
9032:
9031:
9026:
9022:
9018:
9017:
9015:
9008:
9002:
8999:
8997:
8994:
8993:
8990:
8987:
8985:
8982:
8981:
8979:
8974:
8969:
8966:
8963:
8958:
8953:
8950:
8947:
8944:
8941:
8881:
8878:
8875:
8872:
8869:
8866:
8863:
8860:
8857:
8854:
8851:
8848:
8845:
8843:
8841:
8836:
8832:
8828:
8825:
8822:
8819:
8816:
8813:
8810:
8808:
8806:
8801:
8795:
8792:
8789:
8786:
8784:
8781:
8780:
8777:
8774:
8772:
8769:
8766:
8763:
8762:
8760:
8755:
8751:
8745:
8739:
8736:
8734:
8731:
8730:
8727:
8724:
8722:
8719:
8718:
8716:
8711:
8708:
8703:
8697:
8694:
8692:
8689:
8688:
8685:
8682:
8680:
8677:
8676:
8674:
8668:
8664:
8661:
8659:
8657:
8654:
8651:
8648:
8645:
8642:
8639:
8636:
8635:
8584:
8579:
8573:
8570:
8568:
8565:
8564:
8561:
8558:
8556:
8553:
8552:
8550:
8545:
8542:
8519:
8509:
8493:
8485:
8482:
8480:
8477:
8476:
8473:
8470:
8468:
8465:
8464:
8460:
8439:
8436:
8434:
8431:
8417:
8395:
8382:
8376:
8371:
8367:
8355:
8343:quadratic form
8330:
8311:Main article:
8308:
8305:
8193:diagonalizable
8186:is said to be
8169:
8168:
8157:
8154:
8151:
8148:
8145:
8140:
8137:
8133:
8114:
8113:
8102:
8097:
8094:
8090:
8086:
8083:
8080:
8077:
8055:
8054:
8043:
8040:
8037:
8034:
8031:
8028:
8003:
8000:
7999:
7988:
7983:
7975:
7970:
7963:
7959:
7955:
7953:
7950:
7946:
7941:
7934:
7930:
7926:
7922:
7917:
7910:
7906:
7902:
7901:
7899:
7894:
7891:
7888:
7854:
7853:
7842:
7837:
7829:
7824:
7819:
7817:
7814:
7810:
7805:
7800:
7796:
7791:
7786:
7785:
7783:
7778:
7775:
7743:
7736:
7729:
7720:
7713:
7706:
7684:Main article:
7681:
7678:
7665:
7645:
7617:
7588:
7567:
7539:
7527:
7522:
7519:
7507:
7494:
7473:
7453:
7429:
7409:
7406:
7403:
7383:
7363:
7341:
7337:
7333:
7330:
7327:
7323:
7302:
7272:
7268:
7264:
7261:
7257:
7253:
7227:
7207:
7204:
7201:
7170:
7167:
7166:
7165:
7153:
7150:
7145:
7141:
7137:
7134:
7131:
7128:
7125:
7122:
7117:
7113:
7109:
7106:
7103:
7083:
7080:
7077:
7074:
7054:
7034:
7031:
7028:
7023:
7019:
7015:
7012:
7009:
7006:
7003:
6998:
6994:
6990:
6970:
6967:
6964:
6961:
6940:
6936:
6933:
6913:
6910:
6907:
6902:
6898:
6894:
6891:
6888:
6885:
6882:
6877:
6873:
6869:
6849:
6829:
6826:
6823:
6803:
6798:
6794:
6790:
6787:
6784:
6779:
6775:
6771:
6751:
6748:
6745:
6725:
6714:
6702:
6699:
6695:
6689:
6685:
6680:
6655:
6644:
6628:
6617:
6597:
6575:
6571:
6547:
6536:
6516:
6512:
6508:
6503:
6500:
6497:
6490:
6486:
6482:
6458:
6455:
6451:
6430:
6419:
6403:
6392:
6378:
6373:
6369:
6365:
6362:
6359:
6354:
6349:
6345:
6324:
6302:
6298:
6277:
6257:
6246:
6245:
6244:
6233:
6228:
6224:
6220:
6215:
6211:
6205:
6201:
6197:
6192:
6188:
6182:
6177:
6174:
6171:
6167:
6163:
6160:
6157:
6154:
6151:
6129:
6114:
6113:
6112:
6101:
6096:
6092:
6088:
6085:
6082:
6077:
6073:
6069:
6064:
6060:
6056:
6051:
6047:
6041:
6036:
6033:
6030:
6026:
6022:
6017:
6014:
6010:
6004:
5999:
5996:
5993:
5989:
5985:
5982:
5979:
5976:
5973:
5970:
5948:
5920:
5915:
5911:
5907:
5902:
5898:
5877:
5872:
5868:
5864:
5859:
5855:
5832:
5828:
5824:
5821:
5818:
5813:
5809:
5788:
5785:
5782:
5762:
5750:
5747:
5746:
5745:
5733:
5711:
5706:
5695:Any vector in
5693:
5678:
5658:
5636:
5631:
5618:
5604:
5599:
5577:
5553:
5550:
5545:
5541:
5520:
5500:
5472:
5469:
5466:
5461:
5457:
5453:
5450:
5448:
5446:
5443:
5442:
5439:
5436:
5431:
5427:
5423:
5418:
5414:
5408:
5403:
5400:
5397:
5393:
5389:
5386:
5384:
5380:
5376:
5372:
5371:
5351:
5331:
5326:
5322:
5318:
5313:
5309:
5286:
5282:
5259:
5255:
5251:
5248:
5245:
5240:
5236:
5215:
5212:
5209:
5189:
5166:
5163:
5160:
5155:
5151:
5147:
5144:
5141:
5138:
5133:
5129:
5108:
5086:
5083:
5080:
5075:
5071:
5066:
5062:
5059:
5056:
5053:
5033:
5030:
5027:
5024:
5021:
5018:
5015:
4995:
4975:
4972:
4969:
4966:
4963:
4960:
4957:
4954:
4951:
4948:
4945:
4942:
4939:
4936:
4933:
4913:
4910:
4907:
4904:
4895:is similar to
4884:
4881:
4878:
4875:
4855:
4846:commutes with
4835:
4815:
4812:
4809:
4806:
4803:
4800:
4797:
4794:
4791:
4788:
4785:
4782:
4779:
4776:
4773:
4753:
4750:
4747:
4725:
4722:
4719:
4714:
4710:
4705:
4701:
4681:
4678:
4673:
4669:
4665:
4662:
4642:
4622:
4602:
4599:
4596:
4591:
4587:
4583:
4580:
4560:
4557:
4554:
4549:
4545:
4524:
4502:
4497:
4492:
4489:
4484:
4479:
4474:
4452:
4449:
4446:
4441:
4437:
4431:
4425:
4422:
4418:
4413:
4408:
4383:
4380:
4377:
4372:
4368:
4347:
4344:
4341:
4336:
4332:
4328:
4325:
4322:
4319:
4314:
4310:
4287:
4284:
4281:
4278:
4275:
4270:
4266:
4262:
4259:
4256:
4253:
4248:
4244:
4240:
4237:
4210:
4207:
4204:
4201:
4198:
4195:
4192:
4189:
4186:
4183:
4180:
4177:
4174:
4171:
4168:
4163:
4159:
4094:
4091:
4088:
4083:
4079:
3871:
3866:
3823:is called the
3791:. So, the set
3758:
3754:
3749:
3745:
3741:
3736:
3732:
3729:
3726:
3723:
3719:
3715:
3711:
3706:
3702:
3699:
3654:
3651:
3640:
3631:
3622:
3613:
3604:
3595:
3582:
3573:
3564:
3543:
3540:
3537:
3533:
3528:
3524:
3520:
3514:
3510:
3504:
3499:
3496:
3493:
3489:
3485:
3482:
3480:
3476:
3472:
3468:
3467:
3464:
3461:
3458:
3455:
3450:
3446:
3442:
3437:
3433:
3429:
3426:
3424:
3422:
3419:
3418:
3374:
3369:
3364:
3360:
3356:
3351:
3347:
3342:
3338:
3335:
3330:
3326:
3322:
3319:
3314:
3309:
3305:
3301:
3296:
3292:
3287:
3283:
3280:
3275:
3271:
3267:
3262:
3257:
3253:
3249:
3244:
3240:
3235:
3231:
3228:
3223:
3219:
3215:
3212:
3209:
3206:
3203:
3200:
3197:
3194:
3191:
3152:has dimension
3143:divides evenly
3136:
3115:
3106:
3082:
3075:
3072:
3054:
3051:
2995:
2990:
2984:
2981:
2980:
2977:
2974:
2973:
2971:
2966:
2961:
2958:
2955:
2950:
2944:
2939:
2933:
2930:
2927:
2926:
2923:
2920:
2919:
2917:
2912:
2907:
2904:
2901:
2896:
2871:
2867:
2863:
2858:
2854:
2851:
2848:
2845:
2841:
2801:
2796:
2792:
2788:
2785:
2782:
2779:
2776:
2773:
2768:
2762:
2759:
2756:
2753:
2751:
2748:
2747:
2744:
2741:
2739:
2736:
2733:
2730:
2729:
2727:
2722:
2719:
2716:
2713:
2710:
2707:
2704:
2701:
2663:
2658:
2652:
2649:
2647:
2644:
2643:
2640:
2637:
2635:
2632:
2631:
2629:
2624:
2621:
2600:
2593:
2586:
2577:
2569:
2568:
2559:
2557:
2546:
2543:
2540:
2537:
2532:
2528:
2524:
2521:
2518:
2515:
2512:
2507:
2503:
2499:
2496:
2493:
2490:
2485:
2481:
2477:
2474:
2471:
2468:
2465:
2462:
2459:
2456:
2453:
2439:linear terms,
2376:is always (−1)
2330:
2329:
2320:
2318:
2307:
2304:
2301:
2298:
2295:
2292:
2289:
2286:
2283:
2265:are values of
2249:of the matrix
2243:if and only if
2226:Main article:
2223:
2220:
2195:
2194:
2185:
2183:
2172:
2168:
2164:
2160:
2155:
2151:
2148:
2145:
2142:
2138:
2074:
2073:
2064:
2062:
2051:
2047:
2043:
2040:
2036:
2032:
2028:
2024:
1989:
1984:
1980:
1974:
1971:
1967:
1961:
1956:
1953:
1950:
1946:
1942:
1937:
1933:
1927:
1924:
1920:
1916:
1913:
1910:
1905:
1901:
1895:
1892:
1888:
1884:
1879:
1875:
1869:
1866:
1862:
1858:
1853:
1849:
1826:
1818:
1814:
1810:
1809:
1806:
1803:
1802:
1797:
1793:
1789:
1788:
1783:
1779:
1775:
1774:
1772:
1767:
1762:
1754:
1750:
1746:
1745:
1742:
1739:
1738:
1733:
1729:
1725:
1724:
1719:
1715:
1711:
1710:
1708:
1701:
1693:
1690:
1686:
1682:
1680:
1677:
1673:
1670:
1666:
1662:
1658:
1655:
1651:
1647:
1646:
1643:
1640:
1638:
1635:
1633:
1630:
1628:
1625:
1624:
1619:
1616:
1612:
1608:
1606:
1603:
1599:
1595:
1591:
1587:
1583:
1579:
1578:
1573:
1570:
1566:
1562:
1560:
1557:
1553:
1549:
1545:
1541:
1537:
1533:
1532:
1530:
1508:
1504:
1500:
1496:
1492:
1451:
1448:
1443:
1440:
1437:
1428:In this case,
1415:
1411:
1407:
1404:
1400:
1361:
1356:
1350:
1347:
1344:
1343:
1340:
1337:
1336:
1333:
1330:
1327:
1326:
1324:
1319:
1315:
1300:
1294:
1291:
1290:
1287:
1284:
1281:
1280:
1277:
1274:
1273:
1271:
1266:
1262:
1217:applications.
1203:
1200:
1182:published the
1105:Alfred Clebsch
1052:Joseph Fourier
1016:principal axes
1008:Leonhard Euler
989:linear algebra
984:
981:
980:
979:
960:
941:
898:
894:
890:
887:
883:
879:
835:
830:
827:
823:
819:
816:
811:
808:
804:
797:
794:
790:
777:eigenfunctions
760:
757:
753:
680:. In general,
661:
657:
653:
650:
647:
643:
639:
636:
578:principal axes
543:
540:
490:
486:
483:
479:
475:
446:
412:
390:
386:
365:
340:
337:
316:
291:
287:
283:
271:
268:
213:
180:
176:
173:
169:
165:
145:
121:
100:
41:linear algebra
26:
9:
6:
4:
3:
2:
25483:
25472:
25469:
25467:
25466:Matrix theory
25464:
25462:
25459:
25457:
25454:
25452:
25449:
25448:
25446:
25431:
25430:
25421:
25419:
25418:
25409:
25407:
25406:
25397:
25395:
25394:
25389:
25383:
25382:
25379:
25373:
25370:
25368:
25365:
25363:
25360:
25358:
25355:
25353:
25350:
25346:
25343:
25342:
25341:
25338:
25337:
25335:
25333:
25329:
25323:
25320:
25318:
25315:
25313:
25310:
25308:
25305:
25303:
25300:
25298:
25295:
25294:
25292:
25290:
25289:Computational
25286:
25278:
25275:
25273:
25270:
25268:
25265:
25264:
25263:
25260:
25258:
25255:
25253:
25250:
25248:
25245:
25243:
25240:
25238:
25235:
25233:
25230:
25228:
25225:
25223:
25220:
25218:
25215:
25213:
25210:
25208:
25205:
25204:
25202:
25200:
25196:
25190:
25187:
25185:
25182:
25180:
25177:
25175:
25172:
25170:
25167:
25166:
25164:
25162:
25158:
25152:
25149:
25147:
25144:
25142:
25139:
25137:
25134:
25133:
25131:
25129:
25128:Number theory
25125:
25119:
25116:
25114:
25111:
25109:
25106:
25104:
25101:
25099:
25096:
25094:
25091:
25089:
25086:
25085:
25083:
25081:
25077:
25071:
25068:
25066:
25063:
25061:
25060:Combinatorics
25058:
25057:
25055:
25053:
25049:
25043:
25040:
25038:
25035:
25033:
25030:
25028:
25025:
25023:
25020:
25018:
25015:
25013:
25012:Real analysis
25010:
25008:
25005:
25004:
25002:
25000:
24996:
24990:
24987:
24985:
24982:
24980:
24977:
24975:
24972:
24970:
24967:
24965:
24962:
24960:
24957:
24955:
24952:
24951:
24949:
24947:
24943:
24937:
24934:
24932:
24929:
24927:
24924:
24922:
24919:
24917:
24914:
24912:
24909:
24908:
24906:
24904:
24900:
24894:
24891:
24889:
24886:
24882:
24879:
24877:
24874:
24873:
24872:
24869:
24868:
24865:
24860:
24852:
24847:
24845:
24840:
24838:
24833:
24832:
24829:
24817:
24809:
24808:
24805:
24799:
24796:
24794:
24793:Sparse matrix
24791:
24789:
24786:
24784:
24781:
24779:
24776:
24775:
24773:
24771:
24767:
24761:
24758:
24756:
24753:
24751:
24748:
24746:
24743:
24741:
24738:
24736:
24733:
24732:
24730:
24728:constructions
24727:
24723:
24717:
24716:Outermorphism
24714:
24712:
24709:
24707:
24704:
24702:
24699:
24697:
24694:
24692:
24689:
24687:
24684:
24682:
24679:
24677:
24676:Cross product
24674:
24672:
24669:
24668:
24666:
24664:
24660:
24654:
24651:
24649:
24646:
24644:
24643:Outer product
24641:
24639:
24636:
24634:
24631:
24629:
24626:
24624:
24623:Orthogonality
24621:
24620:
24618:
24616:
24612:
24606:
24603:
24601:
24600:Cramer's rule
24598:
24596:
24593:
24591:
24588:
24586:
24583:
24581:
24578:
24576:
24573:
24571:
24570:Decomposition
24568:
24566:
24563:
24562:
24560:
24558:
24554:
24549:
24539:
24536:
24534:
24531:
24529:
24526:
24524:
24521:
24519:
24516:
24514:
24511:
24509:
24506:
24504:
24501:
24499:
24496:
24494:
24491:
24489:
24486:
24484:
24481:
24479:
24476:
24474:
24471:
24469:
24466:
24464:
24461:
24459:
24456:
24454:
24451:
24449:
24446:
24445:
24443:
24439:
24433:
24430:
24428:
24425:
24424:
24421:
24417:
24410:
24405:
24403:
24398:
24396:
24391:
24390:
24387:
24381:
24377:
24373:
24370:
24368:
24365:
24364:
24358:
24357:
24356:
24350:
24330:
24327:
24324:
24316:
24313:
24311:
24307:
24304:
24301:
24298:
24295:
24292:
24289:
24286:
24285:
24278:
24275:
24267:
24264:December 2019
24257:
24253:
24252:inappropriate
24249:
24245:
24239:
24237:
24230:
24221:
24220:
24208:
24206:0-03-010567-6
24202:
24198:
24193:
24190:
24188:0-9614088-5-5
24184:
24180:
24175:
24168:
24167:
24161:
24157:
24153:
24149:
24148:Sixty Symbols
24145:
24140:
24137:
24133:
24128:
24123:
24119:
24115:
24111:
24107:
24100:
24095:
24094:
24081:
24076:
24072:
24068:
24064:
24059:
24052:
24051:
24045:
24034:
24030:
24025:
24014:
24010:
24005:
23994:
23990:
23989:"Eigenvector"
23985:
23981:
23977:
23973:
23969:
23965:
23961:
23957:
23953:
23949:
23945:
23941:
23937:
23933:
23928:
23923:
23918:
23913:
23909:
23904:
23901:
23897:
23893:
23889:
23885:
23881:
23877:
23873:
23868:
23865:
23863:0-486-63518-X
23859:
23855:
23850:
23846:
23842:
23838:
23834:
23830:
23826:
23821:
23816:
23813:(1): 015005.
23812:
23808:
23804:
23799:
23796:
23790:
23786:
23781:
23778:
23772:
23768:
23764:
23759:
23756:
23752:
23748:
23744:
23739:
23736:
23730:
23726:
23721:
23717:
23711:
23707:
23706:
23700:
23697:
23693:
23689:
23685:
23680:
23677:
23675:0-486-41147-8
23671:
23667:
23663:
23659:
23655:
23650:
23647:
23643:
23639:
23635:
23631:
23627:
23622:
23619:
23617:0-19-501496-0
23613:
23609:
23604:
23601:
23595:
23591:
23586:
23582:
23581:
23575:
23571:
23570:
23564:
23560:
23555:
23551:
23547:
23542:
23539:
23535:
23531:
23527:
23523:
23519:
23515:
23511:
23510:
23504:
23501:
23495:
23491:
23487:
23483:
23479:
23476:
23474:0-13-537102-3
23470:
23466:
23461:
23457:
23452:
23448:
23444:
23439:
23435:
23430:
23426:
23422:
23421:
23415:
23412:
23410:0-201-01984-1
23406:
23402:
23398:
23393:
23390:
23386:
23382:
23378:
23373:
23368:
23364:
23360:
23356:
23352:
23348:
23343:
23336:
23332:
23328:
23324:
23320:
23315:
23310:
23306:
23302:
23295:
23290:
23287:
23285:0-534-93219-3
23281:
23276:
23275:
23268:
23265:
23261:
23257:
23253:
23248:
23244:
23239:
23235:
23231:
23227:
23223:
23219:
23215:
23211:
23207:
23203:
23199:
23194:
23189:
23185:
23181:
23177:
23172:
23168:
23167:
23161:
23158:
23156:0-395-14017-X
23152:
23148:
23143:
23142:
23135:
23132:
23130:0-471-84819-0
23126:
23122:
23118:
23113:
23110:
23106:
23101:
23100:
23088:
23083:
23076:
23071:
23064:
23059:
23052:
23047:
23039:
23035:
23028:
23021:
23016:
23009:
23004:
22997:
22992:
22985:
22980:
22973:
22968:
22966:
22958:
22953:
22946:
22941:
22939:
22937:
22935:
22927:
22922:
22915:
22910:
22904:
22900:
22896:
22891:
22884:
22879:
22872:
22867:
22860:
22855:
22848:
22843:
22836:
22831:
22824:
22819:
22817:
22809:
22804:
22797:
22792:
22785:
22784:Herstein 1964
22780:
22773:
22768:
22761:
22760:Herstein 1964
22756:
22749:
22744:
22742:
22734:
22729:
22722:
22717:
22715:
22713:
22705:
22704:Fraleigh 1976
22700:
22694:, p. 38.
22693:
22688:
22681:
22677:
22674:
22673:
22666:
22659:
22658:
22651:
22644:
22639:
22632:
22627:
22620:
22615:
22608:
22603:
22596:
22591:
22584:
22579:
22572:
22567:
22560:
22555:
22548:
22543:
22541:
22533:
22528:
22521:
22516:
22514:
22512:
22510:
22502:
22497:
22490:
22485:
22483:
22481:
22479:
22471:
22466:
22459:
22454:
22439:
22435:
22429:
22427:
22419:
22414:
22408:, p. 38.
22407:
22402:
22400:
22392:
22391:Herstein 1964
22387:
22385:
22373:
22372:
22364:
22357:
22352:
22348:
22336:into account.
22335:
22314:
22310:
22303:
22283:
22276:truncated to
22275:
22271:
22265:
22259:
22255:
22251:
22250:Hefferon 2001
22247:
22243:
22237:
22229:
22225:
22221:
22216:
22211:
22207:
22203:
22199:
22195:
22194:
22190:
22183:
22179:
22176:
22172:
22168:
22164:
22159:
22151:
22147:
22143:
22139:
22138:Arthur Cayley
22135:
22132:
22128:
22124:
22120:
22116:
22112:
22109:
22105:
22101:
22097:
22093:
22092:
22088:
22084:
22074:
22071:
22069:
22066:
22064:
22061:
22059:
22056:
22054:
22051:
22049:
22046:
22044:
22041:
22039:
22036:
22034:
22031:
22029:
22026:
22024:
22021:
22019:
22018:Eigenoperator
22016:
22014:
22011:
22010:
22004:
22001:
21996:
21994:
21991:to faces for
21990:
21986:
21982:
21978:
21974:
21970:
21969:
21964:
21960:
21956:
21952:
21946:
21938:
21934:
21925:
21909:
21905:
21882:
21878:
21855:
21851:
21828:
21824:
21801:
21797:
21786:
21776:
21760:
21756:
21752:
21747:
21743:
21739:
21734:
21730:
21707:
21703:
21699:
21694:
21690:
21686:
21681:
21677:
21654:
21650:
21646:
21641:
21637:
21633:
21628:
21624:
21601:
21597:
21574:
21570:
21547:
21543:
21534:
21530:
21512:
21483:
21454:
21425:
21421:
21417:
21412:
21408:
21404:
21399:
21395:
21372:
21362:
21357:
21347:
21342:
21326:
21324:
21320:
21316:
21305:
21302:
21294:
21291:December 2023
21284:
21280:
21274:
21271:This section
21269:
21260:
21259:
21251:
21249:
21245:
21241:
21238:
21234:
21230:
21226:
21222:
21218:
21214:
21213:Fock operator
21210:
21206:
21202:
21199:, within the
21198:
21194:
21190:
21180:
21166:
21163:
21154:
21133:
21130:
21121:
21092:
21065:
21038:
21001:
20997:
20994:are randomly
20993:
20989:
20985:
20976:
20957:
20927:
20902:
20889:
20881:
20878:
20862:
20842:
20822:
20814:
20793:
20755:
20742:
20739:
20731:
20718:
20711:
20710:
20709:
20690:
20672:
20667:
20653:
20631:
20627:
20618:
20614:
20610:
20609:Hilbert space
20606:
20588:
20584:
20575:
20570:
20568:
20552:
20544:
20526:
20522:
20513:
20509:
20493:
20467:
20463:
20459:
20456:
20451:
20447:
20443:
20436:
20435:
20434:
20432:
20428:
20412:
20400:
20396:
20392:
20388:
20384:
20368:
20364:
20361:
20357:
20354:
20350:
20347:
20344:
20341:
20333:
20329:
20325:
20324:hydrogen atom
20321:
20317:
20313:
20312:wavefunctions
20308:
20299:
20297:
20293:
20289:
20285:
20278:Stress tensor
20275:
20273:
20269:
20266:of moment of
20265:
20261:
20257:
20253:
20249:
20239:
20237:
20233:
20228:
20226:
20221:
20208:
20205:
20202:
20198:
20194:
20191:
20188:
20185:
20182:
20179:
20174:
20170:
20165:
20156:
20140:
20137:
20134:
20131:
20128:
20122:
20119:
20113:
20110:
20104:
20101:
20095:
20087:
20071:
20063:
20059:
20043:
20021:
20017:
19996:
19993:
19988:
19984:
19980:
19977:
19974:
19967:
19963:
19947:
19939:
19923:
19903:
19894:
19880:
19871:
19858:
19855:
19852:
19849:
19843:
19840:
19834:
19814:
19811:
19808:
19805:
19802:
19796:
19793:
19787:
19779:
19775:
19771:
19765:
19756:
19747:
19745:
19740:
19736:
19729:Markov chains
19726:
19724:
19719:
19715:
19711:
19707:
19703:
19699:
19694:
19680:
19660:
19640:
19613:
19609:
19602:
19599:
19593:
19589:
19569:
19547:
19543:
19539:
19536:
19532:
19509:
19505:
19482:
19479:
19475:
19454:
19446:
19428:
19424:
19420:
19417:
19413:
19409:
19404:
19400:
19396:
19393:
19389:
19385:
19382:
19374:
19358:
19355:
19352:
19344:
19340:
19324:
19317:
19313:
19309:
19299:
19297:
19293:
19289:
19285:
19281:
19277:
19273:
19272:Q methodology
19269:
19265:
19261:
19256:
19254:
19250:
19246:
19242:
19238:
19234:
19230:
19226:
19222:
19219:
19215:
19211:
19207:
19203:
19200:
19196:
19190:
19186:
19180:
19172:
19168:
19164:
19145:
19142:
19139:
19113:
19110:
19107:
19096:
19091:
19082:
19080:
19075:
19061:
19058:
19055:
19052:
19049:
19046:
19043:
19040:
19014:
19006:
19003:
19000:
18994:
18991:
18988:
18985:
18978:
18974:
18947:
18941:
18938:
18931:
18925:
18920:
18918:
18911:
18895:
18889:
18882:
18876:
18871:
18869:
18862:
18844:
18828:
18822:
18815:
18809:
18804:
18799:
18785:
18765:
18759:
18756:
18749:
18743:
18738:
18736:
18729:
18713:
18707:
18704:
18697:
18691:
18686:
18684:
18677:
18659:
18639:
18633:
18626:
18620:
18615:
18613:
18606:
18590:
18584:
18577:
18571:
18566:
18564:
18557:
18539:
18536:
18534:Eigenvectors
18532:
18513:
18510:
18508:
18501:
18497:
18489:
18486:
18484:
18477:
18473:
18461:
18447:
18444:
18439:
18435:
18427:
18409:
18406:
18404:
18397:
18393:
18385:
18382:
18380:
18373:
18369:
18357:
18339:
18336:
18334:
18327:
18323:
18315:
18312:
18310:
18303:
18299:
18287:
18273:
18270:
18265:
18261:
18253:
18234:
18230:
18223:
18220:
18215:
18211:
18196:
18177:
18174:
18172:
18165:
18161:
18153:
18150:
18148:
18141:
18137:
18125:
18111:
18108:
18103:
18099:
18091:
18073:
18070:
18068:
18061:
18057:
18049:
18046:
18044:
18037:
18033:
18021:
18003:
18000:
17998:
17991:
17987:
17979:
17976:
17974:
17967:
17963:
17951:
17937:
17934:
17929:
17925:
17917:
17898:
17894:
17887:
17884:
17879:
17875:
17860:
17841:
17838:
17835:
17832:
17829:
17826:
17823:
17820:
17818:
17808:
17805:
17801:
17797:
17795:
17788:
17784:
17776:
17773:
17770:
17767:
17764:
17761:
17758:
17755:
17753:
17743:
17739:
17735:
17733:
17726:
17722:
17710:
17696:
17693:
17688:
17684:
17680:
17675:
17671:
17663:
17645:
17642:
17639:
17636:
17633:
17630:
17627:
17624:
17621:
17619:
17609:
17606:
17603:
17599:
17595:
17593:
17586:
17582:
17574:
17571:
17568:
17565:
17562:
17559:
17556:
17553:
17550:
17548:
17538:
17535:
17531:
17527:
17525:
17518:
17514:
17502:
17482:
17478:
17474:
17472:
17465:
17461:
17451:
17447:
17443:
17441:
17434:
17430:
17418:
17404:
17401:
17396:
17392:
17388:
17383:
17379:
17371:
17355:
17351:
17343:Eigenvalues,
17341:
17326:
17323:
17320:
17314:
17308:
17305:
17302:
17299:
17294:
17290:
17282:
17266:
17258:
17255:
17252:
17239:
17225:
17222:
17219:
17213:
17207:
17204:
17201:
17198:
17193:
17189:
17181:
17162:
17158:
17154:
17151:
17140:
17136:
17132:
17129:
17119:
17103:
17095:
17092:
17089:
17076:
17070:
17053:
17047:
17044:
17041:
17036:
17033:
17030:
17023:
17020:
17017:
17012:
17009:
17006:
17000:
16991:
16975:
16969:
16964:
16957:
16952:
16946:
16937:
16921:
16915:
16912:
16909:
16904:
16901:
16898:
16891:
16888:
16885:
16882:
16877:
16874:
16871:
16865:
16856:
16840:
16832:
16828:
16822:
16815:
16808:
16804:
16797:
16788:
16772:
16766:
16761:
16754:
16749:
16743:
16734:
16730:
16726:
16722:
16717:
16713:
16710:
16706:
16703:
16699:
16696:
16692:
16690:Illustration
16688:
16685:
16680:
16675:
16667:
16662:
16661:
16655:
16641:
16637:
16635:
16631:
16627:
16624:
16620:
16616:
16606:
16582:
16564:
16524:
16507:
16502:
16489:
16486:
16479:
16478:
16477:
16463:
16432:
16412:
16409:
16384:
16381:
16373:
16370:
16367:
16364:
16352:
16345:
16335:
16321:
16291:
16285:
16282:
16279:
16274:
16268:
16245:
16242:
16239:
16236:
16233:
16230:
16210:
16207:
16204:
16184:
16175:
16161:
16158:
16155:
16135:
16115:
16085:
16079:
16076:
16071:
16065:
16042:
16039:
16036:
16033:
16024:
16004:
16001:
15999:
15994:
15991:
15988:
15985:
15982:
15975:
15972:
15970:
15965:
15962:
15959:
15956:
15953:
15946:
15912:
15909:
15906:
15904:
15899:
15896:
15893:
15890:
15887:
15880:
15877:
15874:
15872:
15867:
15864:
15861:
15858:
15851:
15843:
15838:
15823:
15817:
15810:
15804:
15799:
15796:
15793:
15788:
15782:
15775:
15769:
15762:
15756:
15751:
15744:
15739:
15733:
15711:
15708:
15705:
15702:
15699:
15690:
15675:
15669:
15664:
15657:
15652:
15646:
15641:
15638:
15630:
15620:
15618:
15617:exact formula
15614:
15598:
15590:
15574:
15566:
15550:
15542:
15537:
15523:
15520:
15500:
15497:
15494:
15486:
15482:
15478:
15474:
15468:
15454:
15451:
15448:
15428:
15414:
15412:
15402:
15398:
15374:
15371:
15368:
15363:
15359:
15355:
15349:
15346:
15341:
15338:
15330:
15326:
15322:
15319:
15316:
15308:
15305:
15302:
15298:
15294:
15289:
15284:
15281:
15278:
15274:
15265:
15262:
15259:
15255:
15251:
15243:
15239:
15235:
15230:
15225:
15221:
15210:
15209:
15208:
15206:
15187:
15182:
15177:
15173:
15167:
15163:
15159:
15156:
15153:
15148:
15143:
15139:
15133:
15129:
15125:
15120:
15116:
15108:
15107:
15106:
15092:
15087:
15083:
15078:
15075:
15071:
15066:
15062:
15053:
15035:
15027:
15024:
15021:
15018:
15015:
15011:
15005:
14998:
14994:
14987:
14977:
14961:
14956:
14953:
14950:
14947:
14944:
14940:
14936:
14931:
14928:
14925:
14922:
14919:
14915:
14908:
14905:
14899:
14894:
14891:
14888:
14884:
14880:
14875:
14872:
14869:
14865:
14856:
14837:
14834:
14831:
14826:
14822:
14818:
14815:
14810:
14807:
14804:
14800:
14796:
14793:
14790:
14785:
14782:
14779:
14775:
14769:
14765:
14761:
14756:
14753:
14750:
14746:
14740:
14736:
14732:
14727:
14723:
14715:
14714:
14713:
14711:
14707:
14688:
14683:
14680:
14677:
14673:
14667:
14663:
14659:
14656:
14653:
14648:
14645:
14642:
14638:
14632:
14628:
14624:
14619:
14616:
14613:
14609:
14603:
14599:
14595:
14590:
14586:
14578:
14577:
14576:
14574:
14571:The simplest
14564:
14562:
14558:
14554:
14552:
14551:weight spaces
14548:
14544:
14539:
14537:
14533:
14529:
14525:
14519:
14509:
14507:
14503:
14499:
14495:
14491:
14488:
14484:
14479:
14477:
14473:
14469:
14465:
14461:
14457:
14453:
14449:
14445:
14441:
14435:
14425:
14423:
14419:
14415:
14411:
14407:
14403:
14399:
14395:
14391:
14387:
14383:
14379:
14375:
14370:
14368:
14364:
14360:
14356:
14352:
14348:
14343:
14341:
14336:
14332:
14328:
14324:
14320:
14315:
14311:
14308:
14303:
14301:
14297:
14293:
14289:
14285:
14281:
14277:
14274:
14270:
14266:
14262:
14258:
14254:
14250:
14246:
14241:
14224:
14213:
14207:
14204:
14202:
14189:
14183:
14176:
14165:
14154:
14151:
14149:
14136:
14125:
14113:
14109:
14105:
14101:
14097:
14093:
14089:
14085:
14081:
14077:
14073:
14069:
14064:
14047:
14033:
14030:
14027:
14025:
14012:
14006:
13999:
13985:
13982:
13968:
13965:
13963:
13950:
13939:
13926:
13924:
13920:
13916:
13912:
13908:
13904:
13899:
13886:
13882:
13873:
13870:
13856:
13853:
13844:
13840:
13837:
13829:
13819:
13817:
13813:
13809:
13806:
13802:
13798:
13794:
13790:
13786:
13782:
13778:
13774:
13770:
13761:
13754:
13752:
13738:
13730:
13727:
13713:
13706:
13705:
13702:
13700:
13696:
13692:
13688:
13684:
13680:
13675:
13662:
13659:
13653:
13650:
13647:
13639:
13635:
13631:
13627:
13623:
13620:
13616:
13612:
13602:
13600:
13599:eigenfunction
13595:
13593:
13589:
13585:
13569:
13564:
13561:
13557:
13550:
13544:
13541:
13535:
13529:
13522:
13518:
13514:
13510:
13506:
13501:
13488:
13482:
13476:
13473:
13470:
13464:
13458:
13452:
13449:
13445:
13419:
13416:
13412:
13395:
13393:
13389:
13384:
13368:
13362:
13359:
13356:
13350:
13344:
13341:
13334:
13330:
13326:
13322:
13318:
13314:
13310:
13306:
13302:
13298:
13294:
13288:
13287:Eigenfunction
13278:
13276:
13272:
13268:
13250:
13246:
13225:
13213:
13209:
13205:
13200:
13196:
13186:
13183:
13180:
13176:
13161:
13157:
13148:
13144:
13140:
13135:
13131:
13121:
13117:
13110:
13105:
13093:
13090:
13087:
13083:
13069:
13065:
13061:
13051:
13048:
13044:
13018:
13012:
13007:
13002:
12997:
12991:
12958:
12952:
12947:
12944:
12939:
12934:
12928:
12917:
12912:
12910:
12906:
12901:
12897:
12893:
12888:
12875:
12870:
12862:
12859:
12856:
12848:
12840:
12837:
12834:
12828:
12823:
12817:
12814:
12811:
12806:
12801:
12796:
12789:
12784:
12781:
12778:
12773:
12768:
12761:
12756:
12751:
12748:
12745:
12740:
12733:
12728:
12723:
12718:
12715:
12712:
12706:
12701:
12695:
12692:
12689:
12686:
12660:
12655:
12649:
12644:
12639:
12634:
12627:
12622:
12617:
12612:
12605:
12600:
12595:
12590:
12583:
12578:
12573:
12568:
12562:
12557:
12554:
12540:
12537:
12524:
12519:
12513:
12506:
12499:
12493:
12488:
12481:
12477:
12465:
12460:
12454:
12451:
12444:
12437:
12431:
12426:
12419:
12415:
12403:
12398:
12390:
12387:
12378:
12375:
12368:
12362:
12357:
12350:
12346:
12329:
12327:
12318:
12308:
12298:
12292:
12279:
12273:
12270:
12267:
12258:
12255:
12252:
12243:
12240:
12237:
12231:
12225:
12222:
12219:
12216:
12202:
12197:
12184:
12179:
12173:
12168:
12163:
12156:
12151:
12146:
12139:
12134:
12129:
12123:
12118:
12115:
12106:
12104:
12100:
12099:
12087:
12084:
12071:
12066:
12060:
12053:
12046:
12040:
12035:
12028:
12024:
12012:
12007:
12001:
11994:
11987:
11981:
11976:
11969:
11965:
11953:
11948:
11942:
11935:
11928:
11922:
11917:
11910:
11906:
11889:
11887:
11878:
11868:
11858:
11852:
11839:
11833:
11830:
11827:
11818:
11815:
11812:
11803:
11800:
11797:
11791:
11785:
11782:
11779:
11776:
11762:
11757:
11744:
11739:
11733:
11728:
11723:
11716:
11711:
11706:
11699:
11694:
11689:
11683:
11678:
11675:
11667:
11666:
11655:
11642:
11637:
11630:
11626:
11615:
11608:
11604:
11585:
11578:
11552:
11544:
11540:
11532:
11528:
11522:
11516:
11510:
11503:
11499:
11461:
11453:
11449:
11441:
11437:
11431:
11425:
11419:
11412:
11408:
11392:
11387:
11374:
11369:
11361:
11357:
11347:
11343:
11335:
11329:
11324:
11319:
11315:
11311:
11306:
11300:
11291:
11287:
11277:
11273:
11266:
11261:
11256:
11248:
11244:
11234:
11230:
11222:
11216:
11211:
11191:
11186:
11178:
11174:
11164:
11160:
11152:
11146:
11141:
11136:
11132:
11128:
11123:
11117:
11108:
11104:
11094:
11090:
11083:
11078:
11073:
11065:
11061:
11051:
11047:
11039:
11033:
11028:
11019:
11006:
11001:
10997:
10993:
10988:
10983:
10979:
10974:
10969:
10965:
10961:
10956:
10951:
10947:
10942:
10939:
10936:
10931:
10927:
10921:
10917:
10907:
10894:
10889:
10883:
10876:
10869:
10863:
10858:
10855:
10852:
10847:
10841:
10834:
10827:
10821:
10816:
10811:
10805:
10798:
10791:
10785:
10780:
10769:
10764:
10749:
10746:
10743:
10738:
10734:
10724:
10708:
10682:
10678:
10672:
10669:
10664:
10661:
10656:
10653:
10648:
10643:
10639:
10635:
10633:
10626:
10622:
10612:
10608:
10602:
10599:
10594:
10591:
10586:
10583:
10581:
10574:
10570:
10562:
10559:
10557:
10550:
10546:
10533:
10528:
10515:
10510:
10504:
10499:
10494:
10487:
10482:
10477:
10470:
10465:
10460:
10454:
10449:
10446:
10439:
10436:Consider the
10429:
10403:
10397:
10392:
10387:
10381:
10345:
10339:
10334:
10331:
10326:
10320:
10285:
10279:
10274:
10269:
10263:
10251:
10246:
10229:
10226:
10223:
10220:
10217:
10212:
10208:
10204:
10201:
10196:
10192:
10188:
10185:
10175:
10172:
10166:
10163:
10160:
10151:
10148:
10145:
10129:
10126:
10123:
10117:
10115:
10107:
10102:
10096:
10093:
10090:
10085:
10080:
10073:
10068:
10065:
10062:
10057:
10050:
10045:
10040:
10037:
10034:
10028:
10023:
10019:
10013:
10007:
10002:
9997:
9990:
9985:
9980:
9973:
9968:
9963:
9957:
9952:
9949:
9944:
9938:
9933:
9928:
9921:
9916:
9911:
9904:
9899:
9894:
9888:
9882:
9878:
9876:
9868:
9865:
9862:
9859:
9841:
9836:
9823:
9818:
9812:
9807:
9802:
9795:
9790:
9785:
9778:
9773:
9768:
9762:
9757:
9754:
9740:
9736:
9729:
9724:
9718:
9714:
9708:
9704:
9699:
9697:
9693:
9688:
9673:
9667:
9660:
9654:
9649:
9644:
9636:
9632:
9622:
9618:
9611:
9606:
9601:
9598:
9595:
9577:
9570:
9565:
9548:
9545:
9543:
9536:
9532:
9528:
9525:
9520:
9516:
9512:
9505:
9502:
9499:
9497:
9490:
9486:
9482:
9479:
9474:
9470:
9466:
9463:
9454:
9448:
9441:
9435:
9430:
9425:
9417:
9413:
9403:
9399:
9392:
9385:
9379:
9376:
9371:
9364:
9359:
9356:
9350:
9345:
9343:
9336:
9333:
9330:
9317:
9314:
9311:
9308:
9293:
9292:
9285:
9279:
9277:
9273:
9255:
9249:
9246:
9239:
9233:
9228:
9223:
9215:
9211:
9207:
9198:
9194:
9187:
9182:
9177:
9174:
9171:
9153:
9146:
9141:
9128:
9125:
9120:
9116:
9112:
9109:
9104:
9100:
9096:
9075:
9069:
9062:
9056:
9051:
9046:
9038:
9034:
9024:
9020:
9013:
9006:
9000:
8995:
8988:
8983:
8977:
8972:
8967:
8964:
8961:
8948:
8945:
8942:
8931:
8930:
8923:
8917:
8915:
8909:
8902:
8896:
8879:
8873:
8870:
8867:
8858:
8855:
8852:
8846:
8844:
8834:
8830:
8826:
8823:
8820:
8817:
8814:
8811:
8809:
8799:
8793:
8790:
8787:
8782:
8775:
8770:
8767:
8764:
8758:
8753:
8749:
8743:
8737:
8732:
8725:
8720:
8714:
8709:
8706:
8701:
8695:
8690:
8683:
8678:
8672:
8666:
8662:
8660:
8652:
8649:
8646:
8643:
8625:
8620:
8618:
8615: −
8614:
8610:
8606:
8605:
8600:
8595:
8582:
8577:
8571:
8566:
8559:
8554:
8548:
8543:
8540:
8528:
8522:
8518:
8512:
8508:
8491:
8483:
8478:
8471:
8466:
8458:
8449:
8444:
8430:
8407:. A value of
8374:
8365:
8344:
8328:
8320:
8314:
8304:
8302:
8298:
8294:
8290:
8286:
8282:
8277:
8275:
8271:
8267:
8263:
8259:
8255:
8251:
8247:
8243:
8238:
8234:
8229:
8225:
8221:
8218:
8214:
8210:
8205:
8203:
8199:
8196:. The matrix
8195:
8194:
8189:
8185:
8181:
8177:
8173:
8155:
8149:
8146:
8143:
8138:
8135:
8131:
8123:
8122:
8121:
8119:
8100:
8095:
8092:
8088:
8081:
8078:
8075:
8068:
8067:
8066:
8064:
8060:
8041:
8035:
8032:
8029:
8026:
8019:
8018:
8017:
8015:
8012:th column of
8011:
8006:
7986:
7981:
7973:
7961:
7957:
7951:
7944:
7932:
7928:
7920:
7908:
7904:
7897:
7892:
7889:
7886:
7879:
7878:
7877:
7875:
7871:
7867:
7863:
7859:
7840:
7835:
7827:
7815:
7808:
7794:
7781:
7776:
7773:
7766:
7765:
7764:
7762:
7758:
7754:
7751:
7750:square matrix
7746:
7742:
7735:
7728:
7723:
7719:
7712:
7705:
7701:
7697:
7693:
7687:
7677:
7663:
7643:
7615:
7586:
7565:
7557:
7556:
7550:
7537:
7520:
7517:
7492:
7471:
7451:
7443:
7427:
7407:
7404:
7401:
7381:
7361:
7352:
7339:
7331:
7328:
7325:
7300:
7292:
7289:vectors that
7288:
7283:
7270:
7262:
7259:
7251:
7243:
7242:
7225:
7205:
7202:
7199:
7191:
7187:
7183:
7176:
7143:
7139:
7132:
7129:
7126:
7123:
7115:
7111:
7104:
7078:
7072:
7052:
7029:
7026:
7021:
7017:
7013:
7010:
7007:
7004:
7001:
6996:
6992:
6968:
6965:
6962:
6959:
6934:
6931:
6908:
6905:
6900:
6896:
6892:
6889:
6886:
6883:
6880:
6875:
6871:
6847:
6827:
6824:
6821:
6796:
6792:
6788:
6785:
6782:
6777:
6773:
6749:
6746:
6743:
6723:
6715:
6700:
6697:
6687:
6683:
6669:
6653:
6645:
6642:
6626:
6618:
6615:
6611:
6595:
6573:
6569:
6561:
6545:
6537:
6534:
6514:
6510:
6506:
6501:
6498:
6495:
6488:
6484:
6480:
6456:
6453:
6449:
6428:
6420:
6417:
6401:
6393:
6376:
6371:
6367:
6363:
6360:
6357:
6352:
6347:
6343:
6322:
6300:
6296:
6275:
6255:
6247:
6231:
6226:
6222:
6218:
6213:
6209:
6203:
6199:
6195:
6190:
6186:
6180:
6175:
6172:
6169:
6165:
6161:
6155:
6142:
6141:
6127:
6119:
6115:
6099:
6094:
6090:
6086:
6083:
6080:
6075:
6071:
6067:
6062:
6058:
6054:
6049:
6045:
6039:
6034:
6031:
6028:
6024:
6020:
6015:
6012:
6008:
6002:
5997:
5994:
5991:
5987:
5983:
5977:
5971:
5968:
5961:
5960:
5946:
5938:
5934:
5933:
5932:
5913:
5909:
5900:
5896:
5870:
5866:
5857:
5853:
5830:
5826:
5822:
5819:
5816:
5811:
5807:
5786:
5783:
5780:
5760:
5731:
5709:
5694:
5692:
5676:
5656:
5634:
5619:
5602:
5575:
5567:
5566:
5565:
5551:
5548:
5543:
5539:
5518:
5498:
5490:
5470:
5467:
5464:
5459:
5455:
5451:
5449:
5444:
5437:
5429:
5425:
5416:
5412:
5406:
5401:
5398:
5395:
5391:
5387:
5385:
5378:
5374:
5349:
5324:
5320:
5311:
5307:
5284:
5280:
5257:
5253:
5249:
5246:
5243:
5238:
5234:
5213:
5210:
5207:
5187:
5178:
5161:
5153:
5149:
5145:
5139:
5131:
5127:
5119:must satisfy
5106:
5081:
5073:
5069:
5060:
5057:
5054:
5028:
5025:
5022:
5019:
4993:
4970:
4967:
4964:
4961:
4952:
4946:
4943:
4940:
4937:
4911:
4908:
4905:
4902:
4882:
4879:
4876:
4873:
4853:
4833:
4810:
4807:
4804:
4801:
4795:
4792:
4789:
4783:
4780:
4777:
4774:
4751:
4748:
4745:
4720:
4712:
4708:
4703:
4699:
4679:
4676:
4671:
4667:
4663:
4660:
4640:
4620:
4597:
4589:
4585:
4581:
4578:
4555:
4547:
4543:
4522:
4500:
4490:
4487:
4482:
4472:
4447:
4439:
4435:
4423:
4420:
4416:
4411:
4397:eigenvectors
4396:
4378:
4370:
4366:
4342:
4334:
4330:
4326:
4320:
4312:
4308:
4298:
4285:
4282:
4276:
4268:
4264:
4260:
4254:
4246:
4242:
4238:
4235:
4227:
4221:
4208:
4202:
4199:
4196:
4193:
4187:
4184:
4181:
4178:
4175:
4169:
4161:
4157:
4148:
4144:
4140:
4136:
4132:
4128:
4124:
4120:
4116:
4112:
4108:
4089:
4081:
4077:
4069:
4065:
4061:
4057:
4052:
4050:
4046:
4042:
4039:
4035:
4031:
4028:. As long as
4027:
4021:
4018:
4014:
4010:
4007:
4003:
3997:
3993:
3990:
3984:
3979:
3975:
3970:
3966:
3960:
3956:
3952:
3948:
3944:
3940:
3934:
3930:
3926:
3919:
3915:
3911:
3906:
3902:
3898:
3894:
3890:
3885:
3869:
3854:
3850:
3846:
3842:
3839:. In general
3838:
3834:
3830:
3826:
3822:
3818:
3814:
3810:
3806:
3802:
3798:
3794:
3790:
3786:
3782:
3778:
3774:
3769:
3756:
3752:
3743:
3734:
3730:
3727:
3724:
3721:
3717:
3713:
3704:
3700:
3697:
3689:
3688:
3683:
3679:
3676:
3673:, define the
3672:
3668:
3664:
3660:
3650:
3648:
3643:
3639:
3634:
3630:
3625:
3621:
3616:
3612:
3607:
3603:
3598:
3594:
3590:
3585:
3581:
3576:
3572:
3567:
3563:
3558:
3541:
3538:
3535:
3531:
3526:
3522:
3518:
3512:
3508:
3502:
3497:
3494:
3491:
3487:
3483:
3481:
3474:
3470:
3462:
3459:
3456:
3448:
3444:
3435:
3431:
3427:
3425:
3420:
3408:
3404:
3403:
3398:
3394:
3390:
3385:
3372:
3362:
3358:
3349:
3345:
3336:
3333:
3328:
3324:
3317:
3307:
3303:
3294:
3290:
3281:
3278:
3273:
3269:
3255:
3251:
3242:
3238:
3229:
3226:
3221:
3217:
3210:
3204:
3201:
3198:
3195:
3181:
3177:
3173:
3169:
3168:
3163:
3159:
3155:
3151:
3146:
3144:
3139:
3135:
3131:
3127:
3123:
3118:
3114:
3109:
3105:
3102:
3098:
3094:
3090:
3085:
3081:
3071:
3069:
3064:
3062:
3061:
3050:
3047:
3043:
3039:
3034:
3032:
3028:
3024:
3020:
3016:
3011:
3006:
2993:
2988:
2982:
2975:
2969:
2964:
2959:
2956:
2953:
2942:
2937:
2931:
2928:
2921:
2915:
2910:
2905:
2902:
2899:
2865:
2856:
2852:
2849:
2846:
2843:
2839:
2829:
2825:
2812:
2799:
2794:
2790:
2786:
2783:
2780:
2777:
2774:
2771:
2766:
2760:
2757:
2754:
2749:
2742:
2737:
2734:
2731:
2725:
2720:
2714:
2711:
2708:
2705:
2691:
2685:
2681:
2674:
2661:
2656:
2650:
2645:
2638:
2633:
2627:
2622:
2619:
2610:
2608:
2603:
2599:
2592:
2585:
2580:
2576:
2567:
2560:
2558:
2544:
2538:
2535:
2530:
2526:
2519:
2513:
2510:
2505:
2501:
2491:
2488:
2483:
2479:
2472:
2466:
2463:
2460:
2457:
2444:
2443:
2440:
2438:
2434:
2430:
2426:
2422:
2418:
2414:
2409:
2407:
2403:
2399:
2395:
2394:
2389:
2385:
2384:
2379:
2375:
2371:
2367:
2363:
2359:
2355:
2351:
2347:
2343:
2342:
2337:
2328:
2321:
2319:
2305:
2302:
2296:
2293:
2290:
2287:
2274:
2273:
2270:
2268:
2264:
2258:
2254:
2248:
2244:
2241:
2237:
2236:
2229:
2219:
2217:
2213:
2193:
2186:
2184:
2170:
2162:
2153:
2149:
2146:
2143:
2140:
2136:
2128:
2127:
2124:
2122:
2121:
2115:
2109:
2105:
2104:
2099:
2087:
2082:
2072:
2065:
2063:
2049:
2041:
2038:
2030:
2022:
2015:
2014:
2011:
2000:
1987:
1982:
1978:
1972:
1969:
1965:
1959:
1954:
1951:
1948:
1944:
1940:
1935:
1931:
1925:
1922:
1918:
1914:
1911:
1908:
1903:
1899:
1893:
1890:
1886:
1882:
1877:
1873:
1867:
1864:
1860:
1856:
1851:
1847:
1824:
1816:
1812:
1804:
1795:
1791:
1781:
1777:
1770:
1765:
1760:
1752:
1748:
1740:
1731:
1727:
1717:
1713:
1706:
1699:
1691:
1688:
1684:
1678:
1671:
1668:
1664:
1656:
1653:
1649:
1641:
1636:
1631:
1626:
1617:
1614:
1610:
1604:
1597:
1593:
1585:
1581:
1571:
1568:
1564:
1558:
1551:
1547:
1539:
1535:
1528:
1506:
1498:
1490:
1465:
1449:
1446:
1441:
1438:
1435:
1426:
1413:
1405:
1402:
1385:
1381:
1377:
1372:
1359:
1354:
1348:
1345:
1338:
1331:
1328:
1322:
1317:
1298:
1292:
1285:
1282:
1275:
1269:
1264:
1239:
1235:
1231:
1227:
1222:
1218:
1213:
1209:
1199:
1197:
1193:
1189:
1185:
1181:
1176:
1174:
1170:
1166:
1162:
1158:
1157:David Hilbert
1153:
1151:
1147:
1143:
1139:
1135:
1134:
1129:
1124:
1122:
1118:
1114:
1110:
1106:
1102:
1098:
1094:
1089:
1087:
1083:
1079:
1075:
1071:
1070:
1065:
1061:
1060:heat equation
1058:to solve the
1057:
1053:
1048:
1046:
1045:
1040:
1036:
1032:
1028:
1023:
1021:
1017:
1013:
1009:
1004:
1002:
998:
994:
993:matrix theory
990:
977:
973:
969:
965:
961:
958:
954:
950:
946:
942:
939:
935:
934:
933:
931:
926:
924:
923:diagonalizing
920:
916:
912:
896:
888:
885:
877:
869:
865:
861:
857:
853:
849:
833:
828:
825:
821:
817:
814:
809:
806:
802:
795:
792:
788:
778:
758:
755:
751:
740:
735:
732:
728:
727:shear mapping
724:
714:
706:
705:shear mapping
701:
697:
695:
691:
687:
683:
679:
678:eigenequation
675:
659:
651:
648:
634:
626:
622:
618:
614:
610:
605:
603:
599:
595:
591:
587:
583:
579:
575:
574:
569:
565:
561:
560:
555:
551:
550:
539:
537:
532:
530:
526:
522:
520:
515:
512:
508:
503:
484:
481:
473:
435:
384:
338:
335:
289:
285:
281:
267:
265:
261:
257:
253:
248:
245:
241:
237:
233:
229:
225:
211:
203:
199:
195:
174:
171:
163:
143:
135:
119:
89:
85:
81:
77:
76:
67:
46:
42:
37:
33:
19:
25427:
25415:
25403:
25384:
25317:Optimization
25179:Differential
25103:Differential
25070:Order theory
25065:Graph theory
24969:Group theory
24726:Vector space
24527:
24458:Vector space
24376:James Demmel
24354:
24346:
24270:
24261:
24246:by removing
24233:
24196:
24178:
24165:
24147:
24109:
24105:
24070:
24066:
24049:
24036:. Retrieved
24032:
24016:. Retrieved
24012:
24009:"Eigenvalue"
23996:. Retrieved
23992:
23939:
23935:
23907:
23875:
23871:
23853:
23810:
23806:
23784:
23766:
23742:
23724:
23704:
23687:
23683:
23657:
23629:
23625:
23607:
23589:
23579:
23568:
23549:
23545:
23513:
23507:
23489:
23464:
23446:
23442:
23424:
23418:
23396:
23354:
23350:
23307:(1): 31–58.
23304:
23300:
23273:
23254:, New York:
23251:
23242:
23183:
23179:
23165:
23140:
23116:
23108:
23082:
23070:
23058:
23046:
23037:
23027:
23015:
23003:
22991:
22979:
22952:
22928:, p. 189 §8.
22921:
22909:
22890:
22878:
22866:
22854:
22842:
22830:
22803:
22791:
22779:
22767:
22755:
22728:
22699:
22687:
22671:
22665:
22656:
22650:
22638:
22626:
22614:
22607:Francis 1961
22602:
22595:Aldrich 2006
22590:
22578:
22566:
22554:
22527:
22520:Hawkins 1975
22501:Hawkins 1975
22496:
22465:
22453:
22441:. Retrieved
22437:
22413:
22370:
22363:
22351:
22264:
22236:
22214:
22201:
22189:
22181:
22177:
22175:From p. 827:
22170:
22166:
22158:
22150:pp. 225–226.
22145:
22141:
22130:
22126:
22118:
22107:
22099:
22087:
22028:Eigenmoments
21999:
21997:
21966:
21955:brightnesses
21948:
21788:
21529:compass rose
21327:
21319:glacial till
21312:
21297:
21288:
21272:
21203:theory, the
21201:Hartree–Fock
21186:
20982:
20812:
20775:
20668:
20571:
20543:wavefunction
20485:
20404:
20316:bound states
20281:
20245:
20229:
20222:
19916:dimensions,
19895:
19872:
19777:
19767:
19735:Markov chain
19732:
19718:Markov chain
19695:
19444:
19372:
19305:
19275:
19257:
19216:, where the
19192:
19097:centered at
19076:
18977:discriminant
18970:
16652:
16644:Applications
16638:
16615:QR algorithm
16612:
16579:denotes the
16549:
16433:
16347:
16176:
16025:
15839:
15691:
15626:
15623:Eigenvectors
15538:
15469:
15420:
15408:
15400:
15207:of the form
15202:
15051:
14975:
14854:
14852:
14709:
14708:in terms of
14705:
14703:
14570:
14555:
14550:
14546:
14540:
14530:acting on a
14521:
14501:
14497:
14493:
14489:
14480:
14475:
14471:
14467:
14463:
14459:
14451:
14447:
14443:
14439:
14437:
14421:
14417:
14413:
14409:
14405:
14401:
14397:
14393:
14389:
14385:
14381:
14373:
14371:
14366:
14362:
14358:
14354:
14346:
14344:
14339:
14334:
14330:
14326:
14322:
14318:
14313:
14309:
14306:
14304:
14299:
14295:
14291:
14287:
14283:
14279:
14275:
14272:
14268:
14264:
14260:
14256:
14252:
14248:
14244:
14242:
14111:
14107:
14103:
14099:
14095:
14091:
14087:
14083:
14079:
14075:
14071:
14067:
14065:
13927:
13922:
13918:
13914:
13910:
13906:
13902:
13900:
13827:
13825:
13815:
13811:
13807:
13804:
13800:
13796:
13792:
13788:
13784:
13780:
13776:
13772:
13768:
13766:
13755:
13698:
13694:
13690:
13686:
13682:
13678:
13676:
13637:
13633:
13629:
13621:
13614:
13608:
13596:
13591:
13587:
13583:
13512:
13508:
13504:
13502:
13401:
13391:
13387:
13385:
13328:
13324:
13316:
13312:
13301:Banach space
13292:
13290:
13270:
13068:minor matrix
13063:
13057:
13046:
13042:
12915:
12913:
12908:
12904:
12899:
12895:
12891:
12889:
12546:
12538:
12330:
12325:
12316:
12306:
12296:
12293:
12200:
12198:
12107:
12102:
12095:
12093:
12085:
11890:
11885:
11876:
11866:
11856:
11853:
11760:
11758:
11663:
11661:
11583:
11576:
11390:
11388:
11020:
10908:
10767:
10765:
10531:
10529:
10435:
10249:
10247:
9839:
9837:
9746:
9734:
9727:
9722:
9716:
9712:
9706:
9702:
9700:
9695:
9691:
9689:
9575:
9568:
9566:
9289:
9288:, equation (
9283:
9280:
9275:
9271:
9151:
9144:
9142:
8927:
8926:, equation (
8921:
8918:
8913:
8907:
8900:
8897:
8623:
8621:
8616:
8612:
8608:
8602:
8598:
8596:
8532:
8520:
8516:
8510:
8506:
8447:
8316:
8292:
8278:
8273:
8269:
8265:
8261:
8257:
8253:
8249:
8245:
8241:
8236:
8232:
8227:
8223:
8219:
8216:
8212:
8208:
8206:
8201:
8197:
8191:
8187:
8183:
8178:and it is a
8171:
8170:
8117:
8115:
8062:
8058:
8056:
8013:
8009:
8004:
8001:
7873:
7869:
7865:
7861:
7857:
7855:
7760:
7756:
7752:
7744:
7740:
7733:
7726:
7721:
7717:
7710:
7703:
7699:
7695:
7691:
7689:
7553:
7551:
7441:
7353:
7290:
7286:
7284:
7239:
7189:
7188:vector that
7185:
7181:
7178:
6616:real matrix.
6268:th power of
5752:
5690:
5179:
4535:whose first
4465:, such that
4299:
4225:
4222:
4146:
4142:
4138:
4134:
4130:
4126:
4122:
4118:
4114:
4110:
4106:
4067:
4063:
4059:
4055:
4053:
4048:
4044:
4040:
4037:
4033:
4029:
4019:
4016:
4012:
4008:
4005:
4001:
3995:
3991:
3988:
3982:
3977:
3973:
3968:
3958:
3954:
3950:
3946:
3942:
3938:
3932:
3928:
3924:
3917:
3913:
3909:
3904:
3900:
3896:
3888:
3886:
3852:
3844:
3840:
3836:
3832:
3828:
3824:
3820:
3816:
3812:
3808:
3804:
3800:
3792:
3788:
3784:
3780:
3776:
3770:
3685:
3681:
3677:
3670:
3666:
3662:
3658:
3656:
3646:
3641:
3637:
3632:
3628:
3623:
3619:
3614:
3610:
3605:
3601:
3596:
3592:
3588:
3583:
3579:
3578:) = 1, then
3574:
3570:
3565:
3561:
3559:
3406:
3400:
3396:
3392:
3388:
3386:
3179:
3175:
3171:
3165:
3161:
3157:
3153:
3149:
3147:
3137:
3133:
3129:
3125:
3116:
3112:
3107:
3103:
3100:
3096:
3092:
3088:
3083:
3079:
3077:
3065:
3058:
3056:
3035:
3026:
3018:
3009:
3007:
2827:
2823:
2813:
2689:
2683:
2679:
2675:
2611:
2606:
2601:
2597:
2590:
2583:
2578:
2574:
2572:
2561:
2436:
2428:
2424:
2420:
2416:
2410:
2405:
2401:
2397:
2391:
2390:. Equation (
2387:
2381:
2377:
2373:
2369:
2366:coefficients
2361:
2357:
2349:
2339:
2333:
2322:
2266:
2262:
2256:
2252:
2239:
2233:
2231:
2215:
2198:
2187:
2118:
2116:
2107:
2101:
2097:
2085:
2080:
2077:
2066:
2001:
1466:
1427:
1373:
1243:
1237:
1233:
1229:
1225:
1215:
1188:QR algorithm
1184:power method
1177:
1168:
1154:
1131:
1125:
1090:
1067:
1049:
1042:
1038:
1034:
1024:
1005:
986:
975:
971:
963:
956:
952:
948:
944:
937:
929:
927:
914:
910:
867:
863:
859:
855:
851:
847:
736:
730:
720:
689:
681:
677:
673:
624:
620:
616:
612:
608:
606:
582:rigid bodies
571:
557:
547:
545:
533:
521:-dimensional
518:
510:
506:
504:
377:(denoted by
274:Consider an
273:
264:steady state
249:
226:
201:
197:
193:
79:
44:
38:
36:
25429:WikiProject
25272:Game theory
25252:Probability
24989:Homological
24979:Multilinear
24959:Commutative
24936:Type theory
24903:Foundations
24859:mathematics
24706:Multivector
24671:Determinant
24628:Dot product
24473:Linear span
24310:3Blue1Brown
24152:Brady Haran
24038:27 November
22899:Shilov 1977
22895:Nering 1970
22885:, p. 186 §8
22796:Nering 1970
22772:Nering 1970
22489:Nering 1970
22406:Nering 1970
22254:Beezer 2006
22246:Shilov 1977
22206:From p. 51:
22000:eigenvoices
21231:method. In
20574:bound state
20508:Hamiltonian
20391:measurement
20254:define the
19938:mass matrix
19341:due to its
19093:PCA of the
17074:polynomial
16351:A variation
16177:The matrix
15615:. Even the
15485:determinant
15417:Eigenvalues
15391:Calculation
13687:eigenvector
13517:integrating
8932:) becomes,
7184:, namely a
6762:matrix and
6394:The matrix
6118:determinant
5620:A basis of
4395:orthonormal
4026:commutative
3128:such that (
3046:real matrix
2573:where each
2247:determinant
2086:eigenvector
1111:. Finally,
1101:unit circle
1099:lie on the
938:eigensystem
684:may be any
232:dimensional
45:eigenvector
18:Eigensystem
25445:Categories
25257:Statistics
25136:Arithmetic
25098:Arithmetic
24964:Elementary
24931:Set theory
24740:Direct sum
24575:Invertible
24478:Linear map
24073:: 91–134.
23820:1702.05395
23632:(3): 243,
23314:1908.03795
23193:2004.12167
23145:, Boston:
22926:Roman 2008
22883:Roman 2008
22733:Anton 1987
22643:Meyer 2000
22583:Kline 1972
22571:Kline 1972
22559:Kline 1972
22547:Kline 1972
22532:Kline 1972
22252:, p. 364;
22248:, p. 109;
22242:Roman 2008
22163:Kline 1972
22023:Eigenplane
21985:biometrics
21983:branch of
21968:eigenfaces
21937:Eigenfaces
21928:Eigenfaces
21237:orthogonal
20992:microwaves
20877:observable
20813:eigenstate
20260:rigid body
19936:becomes a
19698:centrality
19208:yields an
19183:See also:
18198:Geometric
17862:Algebraic
15724:, that is
14456:one-to-one
14410:eigenbasis
14408:called an
14351:direct sum
13911:eigenspace
13777:eigenvalue
13701:such that
13628:, and let
9294:) becomes
7173:See also:
6416:invertible
5691:eigenbasis
4105:. Because
3907:, written
3825:eigenspace
2346:polynomial
2334:Using the
2232:Equation (
2117:Equation (
2098:eigenvalue
1390:such that
1206:See also:
1039:eigenvalue
1012:rigid body
976:eigenbasis
949:eigenspace
328:of length
270:Definition
230:are multi-
194:eigenvalue
25184:Geometric
25174:Algebraic
25113:Euclidean
25088:Algebraic
24984:Universal
24770:Numerical
24533:Transpose
24328:×
24248:excessive
24136:1874/2663
24018:19 August
23964:1539-4794
23922:1401.4580
23900:129658242
23845:119330480
23538:128825838
23372:1874/8051
23331:213918682
23234:216553547
23218:0031-9007
22873:, p. 217.
22534:, p. 673.
22443:19 August
22344:Citations
22268:By doing
22113:In 1755,
22104:On p. 212
21945:Eigenface
21418:≥
21405:≥
21240:basis set
21225:iteration
21155:τ
21122:τ
21093:†
21066:τ
21039:†
20996:scattered
20963:⟩
20954:Ψ
20908:⟩
20899:Ψ
20799:⟩
20790:Ψ
20761:⟩
20752:Ψ
20737:⟩
20728:Ψ
20696:⟩
20687:Ψ
20628:ψ
20619:in which
20617:basis set
20585:ψ
20523:ψ
20464:ψ
20448:ψ
20369:…
20248:mechanics
20186:ω
20171:ω
20123:˙
20105:¨
20044:ω
20018:ω
19985:ω
19853:−
19844:¨
19797:¨
19764:Vibration
19603:
19537:−
19524:, and in
19447:), where
19418:−
19394:−
19386:−
19356:−
19276:practical
19264:data sets
19199:symmetric
19062:θ
19059:
19050:±
19047:θ
19044:
19007:θ
19004:
18992:−
18939:−
18705:−
18498:γ
18474:γ
18436:γ
18394:γ
18370:γ
18324:γ
18300:γ
18262:γ
18231:λ
18224:γ
18212:γ
18162:μ
18138:μ
18100:μ
18058:μ
18034:μ
17988:μ
17964:μ
17926:μ
17895:λ
17888:μ
17876:μ
17842:φ
17839:
17833:−
17830:φ
17827:
17809:φ
17806:−
17785:λ
17777:φ
17774:
17765:φ
17762:
17744:φ
17723:λ
17685:λ
17672:λ
17646:θ
17643:
17634:−
17631:θ
17628:
17610:θ
17604:−
17583:λ
17575:θ
17572:
17560:θ
17557:
17539:θ
17515:λ
17462:λ
17431:λ
17393:λ
17380:λ
17352:λ
17321:λ
17315:φ
17309:
17300:−
17291:λ
17256:−
17253:λ
17220:λ
17214:θ
17208:
17199:−
17190:λ
17155:−
17152:λ
17133:−
17130:λ
17093:−
17090:λ
17048:φ
17045:
17037:φ
17034:
17024:φ
17021:
17013:φ
17010:
16916:θ
16913:
16905:θ
16902:
16892:θ
16889:
16883:−
16878:θ
16875:
16623:Hermitian
16565:∗
16525:∗
16503:∗
16487:λ
16413:∈
16410:μ
16382:−
16371:μ
16368:−
16280:−
16205:λ
16156:λ
15989:−
15954:−
15800:⋅
15591:of order
15539:Explicit
15498:×
15452:×
15320:⋯
15306:−
15282:−
15263:−
15174:λ
15157:⋯
15140:λ
15084:λ
15076:…
15063:λ
15019:−
15006:⋯
14974:giving a
14948:−
14923:−
14906:…
14892:−
14873:−
14819:−
14816:λ
14808:−
14797:−
14794:⋯
14791:−
14783:−
14776:λ
14762:−
14754:−
14747:λ
14733:−
14724:λ
14681:−
14657:⋯
14646:−
14617:−
14504:) has no
14454:) is not
14355:different
14300:eigenline
14263:, namely
14243:So, both
14214:α
14208:λ
14190:α
14155:λ
14102:, namely
14031:α
14013:α
13874:λ
13731:λ
13657:→
13597:The main
13562:λ
13474:λ
13360:λ
13267:submatrix
13210:λ
13206:−
13197:λ
13184:≠
13177:∏
13145:λ
13141:−
13132:λ
13118:∏
12945:−
12863:λ
12860:−
12841:λ
12838:−
12818:λ
12815:−
12785:λ
12782:−
12752:λ
12749:−
12719:λ
12716:−
12693:λ
12690:−
12478:λ
12452:−
12416:λ
12376:−
12347:λ
12274:λ
12271:−
12259:λ
12256:−
12244:λ
12241:−
12223:λ
12220:−
12025:λ
11966:λ
11907:λ
11834:λ
11831:−
11819:λ
11816:−
11804:λ
11801:−
11783:λ
11780:−
11638:∗
11627:λ
11605:λ
11541:λ
11529:λ
11500:λ
11450:λ
11438:λ
11409:λ
11358:λ
11344:λ
11325:⋅
11316:λ
11288:λ
11274:λ
11245:λ
11231:λ
11175:λ
11161:λ
11142:⋅
11133:λ
11105:λ
11091:λ
11062:λ
11048:λ
10998:λ
10980:λ
10966:λ
10948:λ
10928:λ
10918:λ
10859:⋅
10747:−
10670:−
10657:−
10649:∗
10640:λ
10623:λ
10587:−
10571:λ
10547:λ
10332:−
10224:λ
10218:−
10209:λ
10193:λ
10189:−
10173:−
10167:λ
10164:−
10152:λ
10149:−
10130:λ
10127:−
10097:λ
10094:−
10069:λ
10066:−
10041:λ
10038:−
9953:λ
9950:−
9866:λ
9863:−
9596:λ
9526:−
9464:−
9377:−
9357:−
9331:λ
9312:−
9247:−
9208:−
9172:λ
8962:λ
8946:−
8871:−
8868:λ
8856:−
8853:λ
8831:λ
8824:λ
8818:−
8794:λ
8791:−
8771:λ
8768:−
8710:λ
8707:−
8650:λ
8647:−
8319:Hermitian
8281:defective
8153:Λ
8136:−
8093:−
8085:Λ
8039:Λ
7958:λ
7952:⋯
7929:λ
7905:λ
7816:⋯
7521:κ
7472:κ
7405:×
7362:κ
7332:κ
7263:λ
7203:×
7140:λ
7127:…
7112:λ
7030:α
7018:λ
7011:…
7005:α
6993:λ
6960:α
6935:∈
6932:α
6897:λ
6890:…
6872:λ
6793:λ
6786:…
6774:λ
6747:×
6684:λ
6614:symmetric
6610:Hermitian
6574:∗
6511:λ
6499:…
6485:λ
6454:−
6368:λ
6361:…
6344:λ
6223:λ
6219:⋯
6210:λ
6200:λ
6187:λ
6166:∏
6091:λ
6084:⋯
6072:λ
6059:λ
6046:λ
6025:∑
5988:∑
5972:
5910:λ
5897:μ
5867:λ
5854:μ
5827:λ
5820:…
5808:λ
5784:×
5540:γ
5465:≤
5456:γ
5452:≤
5426:λ
5413:γ
5392:∑
5375:γ
5321:λ
5308:γ
5281:λ
5254:λ
5247:…
5235:λ
5211:≤
5162:λ
5150:γ
5146:≥
5140:λ
5128:μ
5107:λ
5082:λ
5070:γ
5061:λ
5058:−
5055:ξ
5026:ξ
5023:−
4968:ξ
4965:−
4944:ξ
4941:−
4909:ξ
4906:−
4880:ξ
4877:−
4808:ξ
4805:−
4781:ξ
4778:−
4749:ξ
4746:−
4721:λ
4709:γ
4700:λ
4598:λ
4586:γ
4582:−
4556:λ
4544:γ
4491:λ
4448:λ
4436:γ
4421:…
4379:λ
4367:γ
4343:λ
4331:μ
4327:≤
4321:λ
4309:γ
4283:≤
4277:λ
4265:μ
4261:≤
4255:λ
4243:γ
4239:≤
4200:λ
4197:−
4188:
4182:−
4170:λ
4158:γ
4090:λ
4078:γ
3728:λ
3725:−
3523:λ
3509:μ
3488:∑
3471:μ
3457:≤
3445:λ
3432:μ
3428:≤
3359:λ
3346:μ
3337:λ
3334:−
3325:λ
3318:⋯
3304:λ
3291:μ
3282:λ
3279:−
3270:λ
3252:λ
3239:μ
3230:λ
3227:−
3218:λ
3202:λ
3199:−
2954:λ
2929:−
2900:λ
2850:λ
2847:−
2791:λ
2784:λ
2778:−
2761:λ
2758:−
2738:λ
2735:−
2712:λ
2709:−
2539:λ
2536:−
2527:λ
2520:⋯
2514:λ
2511:−
2502:λ
2492:λ
2489:−
2480:λ
2464:λ
2461:−
2431:, can be
2294:λ
2291:−
2147:λ
2144:−
2106:) is the
2042:λ
1945:∑
1912:⋯
1805:⋮
1741:⋮
1679:⋯
1642:⋮
1637:⋱
1632:⋮
1627:⋮
1605:⋯
1559:⋯
1442:−
1436:λ
1406:λ
1384:collinear
1346:−
1329:−
1283:−
1244:Consider
1198:in 1961.
951:, or the
889:λ
826:λ
818:λ
807:λ
723:Mona Lisa
652:λ
566:with the
485:λ
286:×
240:stretches
212:λ
175:λ
144:λ
25405:Category
25161:Topology
25108:Discrete
25093:Analytic
25080:Geometry
25052:Discrete
25007:Calculus
24999:Analysis
24954:Abstract
24893:Glossary
24876:Timeline
24816:Category
24755:Subspace
24750:Quotient
24701:Bivector
24615:Bilinear
24557:Matrices
24432:Glossary
24154:for the
23998:4 August
23980:45359403
23972:17700768
23755:76091646
23552:: 1–29,
23488:(1996),
23389:22275430
23335:Archived
23226:33124845
22676:Archived
22007:See also
21957:of each
20320:electron
20292:diagonal
19706:PageRank
19171:variance
16674:Rotation
15935:that is
15473:accuracy
14110: ∈
14082: ∈
14074: ∈
13803:, while
5180:Suppose
3060:spectrum
3029:are all
2433:factored
2352:and the
1380:parallel
1148:studied
1146:Poincaré
966:forms a
703:In this
542:Overview
529:matrices
428:, where
260:feedback
25417:Commons
25199:Applied
25169:General
24946:Algebra
24871:History
24427:Outline
24242:Please
24234:use of
24114:Bibcode
23944:Bibcode
23880:Bibcode
23825:Bibcode
23662:Bibcode
23634:Bibcode
23518:Bibcode
23381:2117040
23264:58-7924
23198:Bibcode
23096:Sources
22645:, §7.3.
22633:, §7.3.
21315:geology
21277:Please
21246:called
20268:inertia
19716:of the
19235:or the
16732:Matrix
16666:Scaling
14506:bounded
14416:. When
14114:, then
14106:,
14070:,
13775:is the
13626:scalars
13331:is the
13297:Hilbert
13265:is the
8317:In the
8188:similar
8016:. Then
7739:, ...,
7716:, ...,
7218:matrix
6840:(where
6668:unitary
5564:, then
4633:. Then
4131:nullity
3921:, then
3795:is the
3669:matrix
3661:of the
3095:matrix
2596:, ...,
2423:matrix
2400:or the
2344:) is a
2203:is the
2096:is the
1479:matrix
1224:Matrix
1138:Schwarz
1050:Later,
983:History
866:matrix
694:complex
568:English
564:cognate
436:, then
302:matrix
252:geology
236:rotates
25118:Finite
24974:Linear
24881:Future
24857:Major
24711:Tensor
24523:Kernel
24453:Vector
24448:Scalar
24361:Theory
24203:
24185:
23978:
23970:
23962:
23910:, SIAM
23898:
23860:
23843:
23791:
23773:
23753:
23731:
23712:
23672:
23614:
23596:
23536:
23496:
23471:
23407:
23387:
23379:
23329:
23282:
23262:
23232:
23224:
23216:
23153:
23127:
22091:Note:
21589:, and
21323:clasts
21205:atomic
21193:atomic
20990:, and
20875:is an
20811:is an
20776:where
20567:energy
20541:, the
20506:, the
20486:where
20399:proton
20381:) and
20318:of an
20288:stress
20286:, the
20264:tensor
20262:. The
20009:where
19702:Google
19633:. The
19562:, the
19302:Graphs
19286:; cf.
19218:sample
19206:matrix
19204:(PSD)
17247:
17084:
16628:, the
16550:where
14912:
14903:
14532:module
14376:is an
14282:, and
13685:is an
13515:) and
13311:. Let
13238:where
13058:For a
12903:= 4 =
12314:, and
12096:lower
11874:, and
10721:is an
10701:where
8295:has a
7354:where
7186:column
6335:, are
4924:, and
4826:since
3893:closed
3807:, and
3773:kernel
3099:. The
2364:. Its
2354:degree
2199:where
2084:is an
1165:German
1103:, and
930:eigen-
913:is an
686:scalar
600:, and
554:German
549:eigen-
462:, and
434:scalar
244:shears
84:vector
25345:lists
24888:Lists
24861:areas
24580:Minor
24565:Block
24503:Basis
24170:(PDF)
24102:(PDF)
24054:(PDF)
23976:S2CID
23917:arXiv
23896:S2CID
23841:S2CID
23815:arXiv
23747:Wiley
23534:S2CID
23385:S2CID
23338:(PDF)
23327:S2CID
23309:arXiv
23297:(PDF)
23230:S2CID
23188:arXiv
23121:Wiley
22522:, §3.
22503:, §2.
22375:(PDF)
22272:over
22193:See:
22079:Notes
21959:pixel
21078:, of
20984:Light
20322:in a
20296:shear
20258:of a
19375:) or
19312:graph
19270:. In
19197:of a
19146:0.478
19140:0.878
18975:with
18200:mult.
17864:mult.
14526:– an
14251:and α
13814:with
13636:into
13619:field
11021:Then
10725:with
7394:is a
7190:right
6736:is a
5937:trace
5531:. If
4149:) as
3851:, so
3797:union
3591:. If
2078:then
1169:eigen
1167:word
968:basis
741:like
731:along
717:them.
570:word
559:eigen
556:word
525:basis
432:is a
355:with
242:, or
200:, or
132:, is
82:is a
78:) or
75:-gən-
43:, an
24735:Dual
24590:Rank
24201:ISBN
24183:ISBN
24040:2019
24020:2020
24000:2019
23968:PMID
23960:ISSN
23858:ISBN
23789:ISBN
23771:ISBN
23751:LCCN
23729:ISBN
23710:ISBN
23670:ISBN
23612:ISBN
23594:ISBN
23494:ISBN
23469:ISBN
23405:ISBN
23377:PMID
23280:ISBN
23260:LCCN
23222:PMID
23214:ISSN
23151:ISBN
23125:ISBN
22445:2020
21753:>
21740:>
21700:>
21533:360°
21219:via
21207:and
21195:and
21146:and
20835:and
20669:The
20646:and
20514:and
20397:, a
20310:The
19940:and
19776:(or
19193:The
19187:and
17836:sinh
17824:cosh
17771:sinh
17759:cosh
17306:cosh
17042:cosh
17031:sinh
17018:sinh
17007:cosh
14549:and
14541:The
14305:The
14090:and
14078:and
14066:for
11582:and
11484:and
11204:and
10370:and
9732:and
9711:and
9281:For
8919:For
8905:and
7698:has
7464:and
7291:left
7094:are
6981:are
6471:are
6116:The
5935:The
5753:Let
5200:has
4185:rank
4133:of (
4036:and
4011:) =
3994:) ∈
3981:and
3949:) =
3931:) ∈
3899:and
3156:and
3078:Let
3057:The
3021:are
2818:and
2419:-by-
2411:The
2245:the
2214:and
2006:and
1210:and
1194:and
999:and
925:it.
509:-by-
24250:or
24132:hdl
24122:doi
24110:123
24075:doi
24071:692
23952:doi
23888:doi
23833:doi
23692:doi
23642:doi
23554:doi
23526:doi
23451:doi
23429:doi
23367:hdl
23359:doi
23319:doi
23206:doi
23184:125
21949:In
21531:of
21313:In
21281:to
21187:In
21159:min
21126:max
20940:to
20815:of
20429:in
20282:In
20246:In
19896:In
19827:or
19704:'s
19600:deg
19306:In
19294:in
19282:of
19056:sin
19041:cos
19001:sin
17640:sin
17625:cos
17569:sin
17554:cos
17205:cos
16910:cos
16899:sin
16886:sin
16872:cos
16583:of
16434:If
14438:If
14380:of
13917:of
13913:or
13779:of
13689:of
13624:of
13307:on
13299:or
12681:det
12322:= 3
12312:= 2
12302:= 1
12211:det
12203:is
11882:= 3
11872:= 2
11862:= 1
11771:det
11763:is
10230:22.
9854:det
9842:is
9150:= −
8638:det
7868:by
7444:of
7287:row
7244:),
6716:If
6666:is
6646:If
6619:If
6608:is
6538:If
6421:If
6414:is
6150:det
6120:of
5939:of
5489:sum
5299:is
5014:det
4956:det
4932:det
3831:of
3827:or
3819:).
3690:),
3675:set
3665:by
3560:If
3409:as
3387:If
3190:det
3091:by
2820:λ=3
2816:λ=1
2700:det
2692:is
2452:det
2404:of
2386:of
2282:det
2207:by
1519:or
1475:by
1382:or
1307:and
1062:by
991:or
955:of
862:by
850:by
676:or
573:own
254:to
73:EYE
39:In
25447::
24150:.
24146:.
24130:,
24120:,
24108:,
24104:,
24069:.
24065:.
24031:.
24011:.
23991:.
23974:.
23966:.
23958:.
23950:.
23940:32
23938:.
23934:.
23894:,
23886:,
23876:66
23874:,
23839:.
23831:.
23823:.
23811:89
23809:.
23805:.
23749:,
23686:,
23668:,
23640:,
23630:24
23628:,
23548:,
23532:,
23524:,
23514:25
23512:,
23484:;
23445:,
23423:,
23403:,
23383:,
23375:,
23365:,
23355:28
23353:,
23349:,
23333:.
23325:.
23317:.
23305:59
23303:.
23299:.
23258:,
23228:.
23220:.
23212:.
23204:.
23196:.
23182:.
23178:.
23149:,
23123:,
23036:.
22964:^
22933:^
22815:^
22740:^
22711:^
22539:^
22508:^
22477:^
22436:.
22425:^
22398:^
22383:^
22169:,
22146:32
22144:,
21562:,
21440:;
21250:.
20986:,
20975:.
20569:.
20433::
20274:.
20209:0.
20157:,
19960:a
19733:A
19298:.
16605:.
16334:.
16174:.
15413:.
15375:0.
14563:.
14538:.
14502:λI
14500:−
14472:λI
14470:−
14464:λI
14462:−
14452:λI
14450:−
14302:.
14278:∈
14271:,
14267:+
14247:+
13925:.
13905:.
13818:.
13787:.
13697:∈
13681:∈
13640:,
13505:dt
13394:.
13070:,
12911:.
12328:.
12304:,
11888:.
11864:,
10221:35
10205:14
10176:16
9737:=3
9730:=1
9719:=3
9709:=1
9574:=
9286:=3
8924:=1
8916:.
8910:=3
8903:=1
8626:,
8617:λI
8523:=3
8513:=1
8450:=
8303:.
8237:PD
8235:=
8233:AP
8230:,
8220:AP
8120:,
8065:,
8005:ii
7763:,
7732:,
7709:,
5969:tr
5362:,
5177:.
4664::=
4228:.
4147:λI
4145:−
4139:λI
4137:−
4127:λI
4125:−
4115:λI
4113:−
4051:.
4032:+
3976:∈
3957:+
3945:+
3927:+
3916:∈
3912:,
3884:.
3817:λI
3815:−
3781:λI
3779:−
3649:.
3618:,
3391:=
3160:≤
3141:)
3132:−
2684:λI
2682:−
2609:.
2589:,
2408:.
2257:λI
2255:−
2114:.
1598:22
1586:21
1552:12
1540:11
1483:,
1464:.
1450:20
1349:80
1339:60
1332:20
1136:.
1088:.
1072:.
1047:.
1018:.
1003:.
708:1.
696:.
604:.
596:,
592:,
588:,
538:.
502:.
238:,
224:.
196:,
136:,
66:-/
63:ən
57:aɪ
24850:e
24843:t
24836:v
24408:e
24401:t
24394:v
24331:n
24325:n
24277:)
24271:(
24266:)
24262:(
24258:.
24240:.
24158:.
24134::
24124::
24116::
24083:.
24077::
24042:.
24022:.
24002:.
23982:.
23954::
23946::
23925:.
23919::
23890::
23882::
23847:.
23835::
23827::
23817::
23718:.
23694::
23688:1
23664::
23644::
23636::
23556::
23550:2
23528::
23520::
23453::
23447:4
23431::
23425:4
23369::
23361::
23321::
23311::
23236:.
23208::
23200::
23190::
23089:.
23077:.
23040:.
22947:.
22849:.
22837:.
22825:.
22810:.
22621:.
22597:.
22472:.
22447:.
22420:.
22320:)
22315:4
22311:n
22307:(
22304:O
22284:n
22171:8
22127:t
21910:0
21906:R
21883:G
21879:t
21856:G
21852:t
21829:0
21825:R
21802:0
21798:R
21761:3
21757:E
21748:2
21744:E
21735:1
21731:E
21708:3
21704:E
21695:2
21691:E
21687:=
21682:1
21678:E
21655:3
21651:E
21647:=
21642:2
21638:E
21634:=
21629:1
21625:E
21602:3
21598:E
21575:2
21571:E
21548:1
21544:E
21513:3
21508:v
21484:2
21479:v
21455:1
21450:v
21426:3
21422:E
21413:2
21409:E
21400:1
21396:E
21373:3
21368:v
21363:,
21358:2
21353:v
21348:,
21343:1
21338:v
21304:)
21298:(
21293:)
21289:(
21275:.
21167:0
21164:=
21134:1
21131:=
21099:t
21088:t
21045:t
21034:t
21011:t
20958:E
20949:|
20928:H
20903:E
20894:|
20890:H
20863:H
20843:E
20823:H
20794:E
20785:|
20756:E
20747:|
20743:E
20740:=
20732:E
20723:|
20719:H
20691:E
20682:|
20654:H
20632:E
20589:E
20553:E
20527:E
20494:H
20468:E
20460:E
20457:=
20452:E
20444:H
20413:T
20401:.
20365:,
20362:3
20358:,
20355:2
20351:,
20348:1
20345:=
20342:n
20206:=
20203:x
20199:)
20195:k
20192:+
20189:c
20183:+
20180:m
20175:2
20166:(
20141:0
20138:=
20135:x
20132:k
20129:+
20120:x
20114:c
20111:+
20102:x
20096:m
20072:k
20022:2
19997:x
19994:m
19989:2
19981:=
19978:x
19975:k
19948:k
19924:m
19904:n
19881:x
19859:x
19856:k
19850:=
19841:x
19835:m
19815:0
19812:=
19809:x
19806:k
19803:+
19794:x
19788:m
19681:k
19661:k
19641:k
19619:)
19614:i
19610:v
19606:(
19594:/
19590:1
19570:i
19548:2
19544:/
19540:1
19533:D
19510:i
19506:v
19483:i
19480:i
19476:D
19455:D
19429:2
19425:/
19421:1
19414:D
19410:A
19405:2
19401:/
19397:1
19390:D
19383:I
19359:A
19353:D
19325:A
19173:.
19149:)
19143:,
19137:(
19117:)
19114:3
19111:,
19108:1
19105:(
19053:i
19031:θ
19015:2
19011:)
18998:(
18995:4
18989:=
18986:D
18948:]
18942:1
18932:1
18926:[
18921:=
18912:2
18907:u
18896:]
18890:1
18883:1
18877:[
18872:=
18863:1
18858:u
18829:]
18823:0
18816:1
18810:[
18805:=
18800:1
18795:u
18766:]
18760:i
18757:+
18750:1
18744:[
18739:=
18730:2
18725:u
18714:]
18708:i
18698:1
18692:[
18687:=
18678:1
18673:u
18640:]
18634:1
18627:0
18621:[
18616:=
18607:2
18602:u
18591:]
18585:0
18578:1
18572:[
18567:=
18558:1
18553:u
18514:1
18511:=
18502:2
18490:1
18487:=
18478:1
18448:1
18445:=
18440:1
18410:1
18407:=
18398:2
18386:1
18383:=
18374:1
18340:1
18337:=
18328:2
18316:1
18313:=
18304:1
18274:2
18271:=
18266:1
18240:)
18235:i
18227:(
18221:=
18216:i
18202:,
18178:1
18175:=
18166:2
18154:1
18151:=
18142:1
18112:2
18109:=
18104:1
18074:1
18071:=
18062:2
18050:1
18047:=
18038:1
18004:1
18001:=
17992:2
17980:1
17977:=
17968:1
17938:2
17935:=
17930:1
17904:)
17899:i
17891:(
17885:=
17880:i
17866:,
17821:=
17802:e
17798:=
17789:2
17768:+
17756:=
17740:e
17736:=
17727:1
17697:1
17694:=
17689:2
17681:=
17676:1
17637:i
17622:=
17607:i
17600:e
17596:=
17587:2
17566:i
17563:+
17551:=
17536:i
17532:e
17528:=
17519:1
17483:2
17479:k
17475:=
17466:2
17452:1
17448:k
17444:=
17435:1
17405:k
17402:=
17397:2
17389:=
17384:1
17356:i
17327:1
17324:+
17318:)
17312:(
17303:2
17295:2
17267:2
17263:)
17259:1
17250:(
17226:1
17223:+
17217:)
17211:(
17202:2
17194:2
17168:)
17163:2
17159:k
17149:(
17146:)
17141:1
17137:k
17127:(
17104:2
17100:)
17096:k
17087:(
17054:]
17001:[
16976:]
16970:1
16965:0
16958:k
16953:1
16947:[
16922:]
16866:[
16841:]
16833:2
16829:k
16823:0
16816:0
16809:1
16805:k
16798:[
16773:]
16767:k
16762:0
16755:0
16750:k
16744:[
16592:v
16560:v
16531:v
16520:v
16512:v
16508:A
16498:v
16490:=
16464:A
16443:v
16430:.
16417:C
16399:;
16385:1
16378:)
16374:I
16365:A
16362:(
16322:b
16299:T
16292:]
16286:b
16283:3
16275:b
16269:[
16246:0
16243:=
16240:y
16237:+
16234:x
16231:3
16211:1
16208:=
16185:A
16162:6
16159:=
16136:A
16116:a
16093:T
16086:]
16080:a
16077:2
16072:a
16066:[
16043:x
16040:2
16037:=
16034:y
16005:0
16002:=
15995:y
15992:3
15986:x
15983:6
15976:0
15973:=
15966:y
15963:+
15960:x
15957:2
15947:{
15913:y
15910:6
15907:=
15900:y
15897:3
15894:+
15891:x
15888:6
15881:x
15878:6
15875:=
15868:y
15865:+
15862:x
15859:4
15852:{
15824:]
15818:y
15811:x
15805:[
15797:6
15794:=
15789:]
15783:y
15776:x
15770:[
15763:]
15757:3
15752:6
15745:1
15740:4
15734:[
15712:v
15709:6
15706:=
15703:v
15700:A
15676:]
15670:3
15665:6
15658:1
15653:4
15647:[
15642:=
15639:A
15599:n
15575:n
15551:n
15524:!
15521:n
15501:n
15495:n
15455:2
15449:2
15429:A
15372:=
15369:x
15364:0
15360:a
15356:+
15350:t
15347:d
15342:x
15339:d
15331:1
15327:a
15323:+
15317:+
15309:1
15303:k
15299:t
15295:d
15290:x
15285:1
15279:k
15275:d
15266:1
15260:k
15256:a
15252:+
15244:k
15240:t
15236:d
15231:x
15226:k
15222:d
15188:.
15183:t
15178:k
15168:k
15164:c
15160:+
15154:+
15149:t
15144:1
15134:1
15130:c
15126:=
15121:t
15117:x
15093:,
15088:k
15079:,
15072:,
15067:1
15052:k
15036:]
15028:1
15025:+
15022:k
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15012:x
14999:t
14995:x
14988:[
14976:k
14962:,
14957:1
14954:+
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14945:t
14941:x
14937:=
14932:1
14929:+
14926:k
14920:t
14916:x
14909:,
14900:,
14895:1
14889:t
14885:x
14881:=
14876:1
14870:t
14866:x
14855:k
14838:,
14835:0
14832:=
14827:k
14823:a
14811:1
14805:k
14801:a
14786:2
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14757:1
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14728:k
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14689:.
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14660:+
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14633:2
14629:a
14625:+
14620:1
14614:t
14610:x
14604:1
14600:a
14596:=
14591:t
14587:x
14498:T
14494:λ
14490:T
14476:λ
14468:T
14460:T
14448:T
14444:T
14440:λ
14422:T
14418:T
14414:T
14406:V
14402:V
14398:T
14394:T
14390:V
14386:T
14382:T
14374:T
14367:n
14363:T
14359:n
14347:T
14340:λ
14338:(
14335:T
14331:γ
14327:λ
14323:λ
14319:λ
14317:(
14314:T
14310:γ
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14292:λ
14288:E
14284:E
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14269:v
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14257:T
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14225:.
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14218:v
14211:(
14205:=
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14194:v
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14184:T
14177:,
14174:)
14170:v
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3154:n
3150:A
3138:i
3134:λ
3130:λ
3126:k
3117:i
3113:λ
3111:(
3108:A
3104:μ
3097:A
3093:n
3089:n
3084:i
3080:λ
3027:A
3019:A
3010:A
2994:.
2989:]
2983:1
2976:1
2970:[
2965:=
2960:3
2957:=
2949:v
2943:,
2938:]
2932:1
2922:1
2916:[
2911:=
2906:1
2903:=
2895:v
2883:.
2870:0
2866:=
2862:v
2857:)
2853:I
2844:A
2840:(
2828:v
2824:A
2800:.
2795:2
2787:+
2781:4
2775:3
2772:=
2767:|
2755:2
2750:1
2743:1
2732:2
2726:|
2721:=
2718:)
2715:I
2706:A
2703:(
2690:A
2686:)
2680:A
2678:(
2662:.
2657:]
2651:2
2646:1
2639:1
2634:2
2628:[
2623:=
2620:A
2607:A
2602:n
2598:λ
2594:2
2591:λ
2587:1
2584:λ
2579:i
2575:λ
2566:)
2564:4
2562:(
2545:,
2542:)
2531:n
2523:(
2517:)
2506:2
2498:(
2495:)
2484:1
2476:(
2473:=
2470:)
2467:I
2458:A
2455:(
2437:n
2429:n
2425:A
2421:n
2417:n
2406:A
2393:3
2388:A
2378:λ
2374:n
2370:A
2362:A
2358:n
2350:λ
2341:3
2327:)
2325:3
2323:(
2306:0
2303:=
2300:)
2297:I
2288:A
2285:(
2267:λ
2263:A
2259:)
2253:A
2251:(
2240:v
2235:2
2216:0
2209:n
2205:n
2201:I
2192:)
2190:2
2188:(
2171:,
2167:0
2163:=
2159:v
2154:)
2150:I
2141:A
2137:(
2120:1
2112:A
2103:1
2094:λ
2090:A
2081:v
2071:)
2069:1
2067:(
2050:,
2046:v
2039:=
2035:w
2031:=
2027:v
2023:A
2008:w
2004:v
1988:.
1983:j
1979:v
1973:j
1970:i
1966:A
1960:n
1955:1
1952:=
1949:j
1941:=
1936:n
1932:v
1926:n
1923:i
1919:A
1915:+
1909:+
1904:2
1900:v
1894:2
1891:i
1887:A
1883:+
1878:1
1874:v
1868:1
1865:i
1861:A
1857:=
1852:i
1848:w
1825:]
1817:n
1813:w
1796:2
1792:w
1782:1
1778:w
1771:[
1766:=
1761:]
1753:n
1749:v
1732:2
1728:v
1718:1
1714:v
1707:[
1700:]
1692:n
1689:n
1685:A
1672:2
1669:n
1665:A
1657:1
1654:n
1650:A
1618:n
1615:2
1611:A
1594:A
1582:A
1572:n
1569:1
1565:A
1548:A
1536:A
1529:[
1507:,
1503:w
1499:=
1495:v
1491:A
1481:A
1477:n
1473:n
1469:n
1447:1
1439:=
1414:.
1410:y
1403:=
1399:x
1388:λ
1360:.
1355:]
1323:[
1318:=
1314:y
1299:]
1293:4
1286:3
1276:1
1270:[
1265:=
1261:x
1250:n
1246:n
1240:.
1238:A
1234:x
1230:x
1226:A
978:.
972:T
964:T
957:T
945:T
915:n
911:v
897:,
893:v
886:=
882:v
878:A
868:A
864:n
860:n
856:n
852:n
848:n
834:.
829:x
822:e
815:=
810:x
803:e
796:x
793:d
789:d
759:x
756:d
752:d
690:λ
682:λ
660:,
656:v
649:=
646:)
642:v
638:(
635:T
625:λ
621:T
617:T
613:T
609:v
562:(
519:n
511:n
507:n
489:v
482:=
478:v
474:A
464:λ
460:A
445:v
430:λ
426:λ
411:v
389:v
385:A
364:v
353:A
339:.
336:n
315:v
304:A
290:n
282:n
179:v
172:=
168:v
164:T
120:T
99:v
60:ɡ
54:ˈ
51:/
47:(
34:.
20:)
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