Eigenvalues and eigenvectors: Difference between revisions

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{{Short description|Concepts from linear algebra}}

[[File:Mona Lisa eigenvector grid.png|thumb|270px|In this [[shear mapping]] the red arrow changes direction but the blue arrow does not. The blue arrow is an eigenvector of this shear mapping, and since its length is unchanged its eigenvalue is 1.]]

{{Redirect|Characteristic root|the root of a characteristic equation|Characteristic equation (calculus)}}

{{Use dmy dates|date=July 2020}}

{{Use American English|date=January 2019}}

In [[linear algebra]], an '''eigenvector''' ({{IPAc-en|ˈ|aɪ|g|ən|-}} {{respell|EYE|gən|-}}) or '''characteristic vector''' is a [[Vector (mathematics and physics)|vector]] that has its [[direction (geometry)|direction]] unchanged (or reversed) by a given [[linear map|linear transformation]]. More precisely, an eigenvector <math>\mathbf v</math> of a linear transformation <math>T</math> is [[scalar multiplication|scaled by a constant factor]] <math>\lambda</math> when the linear transformation is applied to it: <math>T\mathbf v=\lambda \mathbf v</math>. The corresponding '''eigenvalue''', '''characteristic value''', or '''characteristic root''' is the multiplying factor <math>\lambda</math> (possibly a [[negative number|negative]] or [[complex number|complex]] number).

An '''eigenvector''' of a [[square matrix]] <math>A</math> is a non-zero [[vector (mathematics)|vector]] <math>v</math> that, when the matrix is [[matrix multiplication|multiplied]] by <math>v</math>, yields a constant multiple of <math>v</math>, the multiplier being commonly denoted by <math>\lambda</math>. That is:

[[Euclidean vector|Geometrically, vectors]] are multi-[[dimension]]al quantities with magnitude and direction, often pictured as arrows. A linear transformation [[Rotation (mathematics)|rotates]], [[Scaling (geometry)|stretches]], or [[Shear mapping|shears]] the vectors upon which it acts. A linear transformation's eigenvectors are those vectors that are only stretched or shrunk, with neither rotation nor shear. The corresponding eigenvalue is the factor by which an eigenvector is stretched or shrunk. If the eigenvalue is negative, the eigenvector's direction is reversed.{{sfn|Burden|Faires|1993|p=401}}

<math>A v = \lambda v</math>

The eigenvectors and eigenvalues of a linear transformation serve to characterize it, and so they play important roles in all areas where linear algebra is applied, from [[geology]] to [[quantum mechanics]]. 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 ([[feedback]]). 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 [[steady state]] of the system.

(Because this equation uses [[Matrix_product#All_matrices|post-multiplication]] by <math>v</math>, it describes a [[Eigenvalues_and_eigenvectors#Left_and_right_eigenvectors|right eigenvector]].)

{{Toclimit|3}}

The number <math>\lambda</math> is called the '''eigenvalue''' of <math>A</math> corresponding to <math>v</math>.<ref name=WolframEigenvector>

Wolfram Research, Inc. (2010) [http://mathworld.wolfram.com/Eigenvector.html ''Eigenvector'']. Accessed on 2010-01-29.

</ref>

== Matrices ==

In [[analytic geometry]], for example, a three-element vector may be seen as an arrow in three-dimensional space starting at the origin. In that case, an eigenvector <math>v</math> is an arrow whose direction is either preserved or exactly reversed after multiplication by <math>A</math>. The corresponding eigenvalue determines how the length of the arrow is changed by the operation, and whether its direction is reversed or not, determined by whether the eigenvalue is negative or positive.

For an <math>n{\times}n</math> matrix {{mvar|A}} and a nonzero vector <math>\mathbf{v}</math> of length <math>n</math>, if multiplying {{mvar|A}} by <math>\mathbf{v}</math> (denoted <math>A\mathbf{v}</math>) simply scales <math>\mathbf{v}</math> by a factor {{mvar|λ}}, where {{mvar|λ}} is a [[Scalar (mathematics)|scalar]], then <math>\mathbf{v}</math> is called an eigenvector of {{mvar|A}}, and {{mvar|λ}} is the corresponding eigenvalue. This relationship can be expressed as: <math>A\mathbf{v} = \lambda \mathbf{v}</math>.<ref>{{Cite book |last=Gilbert Strang |url=https://math.mit.edu/~gs/linearalgebra/ila5/linearalgebra5_6-1.pdf |title=Introduction to Linear Algebra |publisher=Wellesley-Cambridge Press |edition=5 |chapter=6: Eigenvalues and Eigenvectors}}</ref>

Given an [[Dimension (vector space)|''n''-dimensional vector space]] and a choice of [[Basis (linear algebra)|basis]], there is a direct correspondence between linear transformations from the vector space into itself and ''n''-by-''n'' [[Square matrix|square matrices]]. Hence, in a finite-dimensional vector space, it is equivalent to define eigenvalues and eigenvectors using either the language of linear transformations, or the language of [[Matrix (mathematics)|matrices]].{{sfn|Herstein|1964|pp=228, 229}}{{sfn|Nering|1970|p=38}}

In abstract [[linear algebra]], these concepts are naturally extended to more general situations, where the set of real scalar factors is replaced by any [[field (mathematics)|field]] of [[scalar (mathematics)|scalars]] (such as [[algebraic numbers|algebraic]] or complex numbers); the set of [[Cartesian coordinates|Cartesian]] vectors <math>\mathbb{R}^n</math> is replaced by any [[vector space]] (such as the [[continuous function]]s, the [[polynomial]]s or the [[trigonometric series]]), and matrix multiplication is replaced by any [[linear operator]] that maps vectors to vectors (such as the [[derivative (calculus)|derivative]] from [[calculus]]). In such cases, the "vector" in "eigenvector" may be replaced by a more specific term, such as "[[eigenfunction]]", "[[eigenmode]]", "[[eigenface]]", or "eigenstate". Thus, for example, the exponential function <math>f(x) = a^x</math> is an eigenfunction of the derivative operator " <math>{}'</math> ", with eigenvalue <math>\lambda = \ln a</math>, since its derivative is <math>f'(x) = (\ln a)a^x = \lambda f(x)</math>.

== Overview ==

The set of all eigenvectors of a matrix (or linear operator), each paired with its corresponding eigenvalue, is called the '''eigensystem''' of that matrix.<ref>William H. Press, Saul A. Teukolsky, William T. Vetterling, Brian P. Flannery (2007), [http://www.nr.com/ ''Numerical Recipes: The Art of Scientific Computing'', Chapter 11: ''Eigensystems.'', pages=563–597. Third edition, Cambridge University Press. ISBN 9780521880688 ]</ref> Any multiple of an eigenvector is also an eigenvector, with the same eigenvalue. An '''eigenspace''' of a matrix <math>A</math> is the set of all eigenvectors with the same eigenvalue, together with the [[zero vector]].<ref name=WolframEigenvector/> An '''eigenbasis''' for <math>A</math> is any [[basis (linear algebra)|basis]] for the set of all vectors that consists of linearly independent eigenvectors of <math>A</math>. Not every matrix has an eigenbasis, but every [[symmetric matrix]] does.

Eigenvalues and eigenvectors feature prominently in the analysis of linear transformations. The prefix ''[[wikt:eigen-|eigen-]]'' is adopted from the [[German language|German]] word ''[[wikt:eigen#German|eigen]]'' ([[cognate]] with the [[English language|English]] word ''[[wikt:own#English|own]]'') for 'proper', 'characteristic', 'own'.{{sfn|Betteridge|1965}}<ref name=":0">{{Cite web |title=Eigenvector and Eigenvalue |url=https://mathsisfun.com/algebra/eigenvalue.html |access-date=2020-08-19 |website=www.mathsisfun.com}}</ref> Originally used to study [[principal axis (mechanics)|principal axes]] of the rotational motion of [[rigid body|rigid bodies]], eigenvalues and eigenvectors have a wide range of applications, for example in [[stability theory|stability analysis]], [[vibration analysis#eigenvalue problem|vibration analysis]], [[atomic orbital]]s, [[eigenface|facial recognition]], and [[Eigendecomposition of a matrix|matrix diagonalization]].

In essence, an eigenvector '''v''' of a linear transformation ''T'' is a nonzero vector that, when ''T'' is applied to it, does not change direction. Applying ''T'' to the eigenvector only scales the eigenvector by the scalar value ''λ'', called an eigenvalue. This condition can be written as the equation

The terms '''characteristic vector''', '''characteristic value''', and '''characteristic space''' are also used for these concepts. The prefix '''[[wikt:eigen|eigen-]]''' is adopted from the [[German language|German]] word ''eigen'' for "self-" or "unique to", "peculiar to", or "belonging to."

<math display=block>T(\mathbf{v}) = \lambda \mathbf{v},</math>

referred to as the '''eigenvalue equation''' or '''eigenequation'''. In general, ''λ'' may be any [[scalar (mathematics)|scalar]]. For example, ''λ'' may be negative, in which case the eigenvector reverses direction as part of the scaling, or it may be zero or [[complex number|complex]].

[[File:Mona Lisa eigenvector grid.png|thumb|320px|In this [[shear mapping]] 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 1.]]

Eigenvalues and eigenvectors have many applications in both pure and applied mathematics. They are used in [[matrix factorization]], in [[quantum mechanics]], and in many other areas.

[[File:Eigenvectors of a linear operator.gif|thumb|200px|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 them.]]

The example here, based on the [[Mona Lisa]], 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 [[shear mapping]]. 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 ''along'' 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 eigenvalue equal to one, because the mapping does not change their length either.

==Definition==

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 [[differential operator]] like <math>\tfrac{d}{dx}</math>, in which case the eigenvectors are functions called [[eigenfunction]]s that are scaled by that differential operator, such as

===Eigenvectors and eigenvalues of a real matrix===

<math display=block>\frac{d}{dx}e^{\lambda x} = \lambda e^{\lambda x}.</math>

[[File:Eigenvalue equation.svg|thumb|right|250px|Matrix <math>A</math> acts by stretching the vector <math>x</math>, not changing its direction, so <math>x</math> is an eigenvector of <math>A</math>.]]

Alternatively, the linear transformation could take the form of an ''n'' by ''n'' matrix, in which case the eigenvectors are ''n'' by 1 matrices. If the linear transformation is expressed in the form of an ''n'' by ''n'' matrix ''A'', then the eigenvalue equation for a linear transformation above can be rewritten as the matrix multiplication

{{see also|Euclidean vector|Matrix (mathematics)}}

<math display=block>A\mathbf v = \lambda \mathbf v,</math>

<!--No need to mention "linear operators" or "linear algebra" here. There is a section for that below.-->

where the eigenvector ''v'' is an ''n'' by 1 matrix. For a matrix, eigenvalues and eigenvectors can be used to [[matrix decomposition|decompose the matrix]]—for example by [[diagonalizable matrix|diagonalizing]] it.

In many contexts, a vector can be assumed to be a list of real numbers (called ''elements''), written vertically with brackets around the entire list, such as the vectors ''u'' and ''v'' below. Two vectors are said to be [[scalar multiplication|scalar multiples]] of each other (also called [[Parallel (geometry)|parallel]] or [[collinearity|collinear]]) if they have the same number of elements, and if every element of one vector is obtained by multiplying each corresponding element in the other vector by the same number (known as a ''scaling factor'', or a ''scalar''). For example, the vectors

:<math>u = \begin{bmatrix}1\\3\\4\end{bmatrix}\quad\quad\quad</math> and <math>\quad\quad\quad v = \begin{bmatrix}-20\\-60\\-80\end{bmatrix}</math>

are scalar multiples of each other, because each element of <math>v</math> is −20 times the corresponding element of <math>u</math>.

Eigenvalues and eigenvectors give rise to many closely related mathematical concepts, and the prefix ''eigen-'' is applied liberally when naming them:

A vector with three elements, like <math>u</math> or <math>v</math> above, may represent a point in three-dimensional space, relative to some [[Cartesian coordinates|Cartesian coordinate system]]. It helps to think of such a vector as the tip of an arrow whose tail is at the origin of the coordinate system. In this case, the condition "<math>u</math> is parallel to <math>v</math>" means that the two arrows lie on the same straight line, and may differ only in length and direction along that line.

* The set of all eigenvectors of a linear transformation, each paired with its corresponding eigenvalue, is called the '''eigensystem''' of that transformation.{{sfn|Press|Teukolsky|Vetterling|Flannery|2007|p=536}}{{sfn|Wolfram.com: Eigenvector}}

* The set of all eigenvectors of ''T'' corresponding to the same eigenvalue, together with the zero vector, is called an '''eigenspace''', or the '''characteristic space''' of ''T'' associated with that eigenvalue.{{sfn|Nering|1970|p=107}}

* If a set of eigenvectors of ''T'' forms a [[basis (linear algebra)|basis]] of the ___domain of ''T'', then this basis is called an '''eigenbasis'''.

== History ==

If we [[matrix multiplication|multiply]] any square matrix <math>A</math> with <math>n</math> rows and <math>n</math> columns by such a vector <math>v</math>, the result will be another vector <math>w = A v </math>, also with <math>n</math> rows and one column. That is,

Eigenvalues are often introduced in the context of [[linear algebra]] or [[matrix (mathematics)|matrix theory]]. Historically, however, they arose in the study of [[quadratic form]]s and [[differential equation]]s.

:<math>\begin{bmatrix} v_1 \\ v_2 \\ \vdots \\ v_n \end{bmatrix} \quad\quad</math> is mapped to <math>

\begin{bmatrix} w_1 \\ w_2 \\ \vdots \\ w_n \end{bmatrix} \;=\;

In the 18th century, [[Leonhard Euler]] studied the rotational motion of a [[rigid body]], and discovered the importance of the [[Moment of inertia#Principal axes|principal axes]].{{efn|Note:

\begin{bmatrix} A_{1,1} & A_{1,2} & \ldots & A_{1,n} \\

* In 1751, Leonhard Euler proved that any body has a principal axis of rotation: Leonhard Euler (presented: October 1751; published: 1760) [https://archive.org/stream/histoiredelacad07unkngoog#page/n196/mode/2up "Du mouvement d'un corps solide quelconque lorsqu'il tourne autour d'un axe mobile"] (On the movement of any solid body while it rotates around a moving axis), ''Histoire de l'Académie royale des sciences et des belles lettres de Berlin'', pp. 176–227. [https://archive.org/stream/histoiredelacad07unkngoog#page/n232/mode/2up On p. 212], Euler proves that any body contains a principal axis of rotation: ''"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."'' (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.)

A_{2,1} & A_{2,2} & \ldots & A_{2,n} \\

* In 1755, [[Johann Andreas Segner]] proved that any body has three principal axes of rotation: Johann Andreas Segner, ''Specimen theoriae turbinum'' [Essay on the theory of tops (i.e., rotating bodies)] ( Halle ("Halae"), (Germany): Gebauer, 1755). ({{google books|plainurl=y|id=89NMAAAAcAAJ|29}} p. xxviiii [29]), Segner derives a third-degree equation in ''t'', which proves that a body has three principal axes of rotation. He then states (on the same page): ''"Non autem repugnat tres esse eiusmodi positiones plani HM, quia in aequatione cubica radices tres esse possunt, et tres tangentis t valores."'' (However, it is not inconsistent [that there] be three such positions of the plane HM, because in cubic equations, [there] can be three roots, and three values of the tangent t.)

\vdots & \vdots & \ddots & \vdots \\

* The relevant passage of Segner's work was discussed briefly by [[Arthur Cayley]]. See: A. Cayley (1862) "Report on the progress of the solution of certain special problems of dynamics," ''Report of the Thirty-second meeting of the British Association for the Advancement of Science; held at Cambridge in October 1862'', '''32''': 184–252; see especially [{{google books|plainurl=y|id=S_RJAAAAcAAJ|page=225}} pp. 225–226.]}} [[Joseph-Louis Lagrange]] realized that the principal axes are the eigenvectors of the inertia matrix.{{sfn|Hawkins|1975|loc=§2}}

A_{n,1} & A_{n,2} & \ldots & A_{n,n} \\

\end{bmatrix}

In the early 19th century, [[Augustin-Louis Cauchy]] saw how their work could be used to classify the [[Quadric|quadric surfaces]], and generalized it to arbitrary dimensions.{{sfn|Hawkins|1975|loc=§3}} Cauchy also coined the term ''racine caractéristique'' (characteristic root), for what is now called ''eigenvalue''; his term survives in ''[[Characteristic polynomial|characteristic equation]]''.{{efn|{{harvnb|Kline|1972|loc=pp. 807–808}} Augustin Cauchy (1839) "Mémoire sur l'intégration des équations linéaires" (Memoir on the integration of linear equations), ''Comptes rendus'', '''8''': 827–830, 845–865, 889–907, 931–937. [https://gallica.bnf.fr/ark:/12148/bpt6k2967c/f833.item.r{{=}}.zoom From p. 827:] ''"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'''équation caractéristique'', le degré de cette équation étant précisément l'order de l'équation différentielle qu'il s'agit d'intégrer."'' (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.)}}

\begin{bmatrix} v_1 \\ v_2 \\ \vdots \\ v_n \end{bmatrix}

Later, [[Joseph Fourier]] used the work of Lagrange and [[Pierre-Simon Laplace]] to solve the [[heat equation]] by [[separation of variables]] in his 1822 treatise ''[[Joseph Fourier#The Analytic Theory of Heat|The Analytic Theory of Heat (Théorie analytique de la chaleur)]]''.{{sfn|Kline|1972|loc=p. 673}} [[Charles-François Sturm]] elaborated on 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 [[symmetric matrix|symmetric matrices]] have real eigenvalues.{{sfn|Hawkins|1975|loc=§3}} This was extended by [[Charles Hermite]] in 1855 to what are now called [[Hermitian matrix|Hermitian matrices]].{{sfn|Kline|1972|loc=pp. 807–808}}

Around the same time, [[Francesco Brioschi]] proved that the eigenvalues of [[orthogonal matrix|orthogonal matrices]] lie on the [[unit circle]],{{sfn|Hawkins|1975|loc=§3}} and [[Alfred Clebsch]] found the corresponding result for [[skew-symmetric matrix|skew-symmetric matrices]].{{sfn|Kline|1972|loc=pp. 807–808}} Finally, [[Karl Weierstrass]] clarified an important aspect in the [[stability theory]] started by Laplace, by realizing that [[defective matrix|defective matrices]] can cause instability.{{sfn|Hawkins|1975|loc=§3}}

In the meantime, [[Joseph Liouville]] studied eigenvalue problems similar to those of Sturm; the discipline that grew out of their work is now called ''[[Sturm–Liouville theory]]''.{{sfn|Kline|1972|loc=pp. 715–716}} [[Hermann Schwarz|Schwarz]] studied the first eigenvalue of [[Laplace's equation]] on general domains towards the end of the 19th century, while [[Henri Poincaré|Poincaré]] studied [[Poisson's equation]] a few years later.{{sfn|Kline|1972|loc=pp. 706–707}}

At the start of the 20th century, [[David Hilbert]] studied the eigenvalues of [[integral operator]]s by viewing the operators as infinite matrices.{{sfn|Kline|1972|p=1063|loc=p.}} He was the first to use the [[German language|German]] word ''eigen'', which means "own",<ref name=":0" /> to denote eigenvalues and eigenvectors in 1904,{{efn|See:

* David Hilbert (1904) [https://digizeitschriften.de/dms/img/?PPN{{=}}PPN252457811_1904&DMDID{{=}}dmdlog11&LOGID{{=}}log11&PHYSID{{=}}phys57#navi "Grundzüge einer allgemeinen Theorie der linearen Integralgleichungen. (Erste Mitteilung)"] (Fundamentals of a general theory of linear integral equations. (First report)), ''Nachrichten von der Gesellschaft der Wissenschaften zu Göttingen, Mathematisch-Physikalische Klasse'' (News of the Philosophical Society at Göttingen, mathematical-physical section), pp. 49–91. [https://digizeitschriften.de/dms/img/?PPN{{=}}PPN252457811_1904&DMDID{{=}}dmdlog11&LOGID{{=}}log11&PHYSID{{=}}phys57#navi From p. 51:] {{lang|de|"Insbesondere in dieser ersten Mitteilung gelange ich zu Formeln, die die Entwickelung einer willkürlichen Funktion nach gewissen ausgezeichneten Funktionen, die ich 'Eigenfunktionen' nenne, liefern: ..."}} (In particular, in this first report I arrive at formulas that provide the [series] development of an arbitrary function in terms of some distinctive functions, which I call ''eigenfunctions'': ... ) Later on the same page: {{lang|de|"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, ... "}} (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...)

* For the origin and evolution of the terms eigenvalue, characteristic value, etc., see: [https://jeff560.tripod.com/e.html Earliest Known Uses of Some of the Words of Mathematics (E)]}} though he may have been following a related usage by [[Hermann von Helmholtz]]. For some time, the standard term in English was "proper value", but the more distinctive term "eigenvalue" is the standard today.{{sfn|Aldrich|2006}}

The first numerical algorithm for computing eigenvalues and eigenvectors appeared in 1929, when [[Richard von Mises]] published the [[power method]]. One of the most popular methods today, the [[QR algorithm]], was proposed independently by [[John G. F. Francis]]{{sfn|Francis|1961|pp=265–271}} and [[Vera Kublanovskaya]]{{sfn|Kublanovskaya|1962}} in 1961.{{sfn|Golub|Van Loan|1996|loc=§7.3}}{{sfn|Meyer|2000|loc=§7.3}}

== Eigenvalues and eigenvectors of matrices ==

{{See also|Euclidean vector|Matrix (mathematics)}}

Eigenvalues and eigenvectors are often introduced to students in the context of linear algebra courses focused on matrices.<ref name="CornellMathCourses">Cornell University Department of Mathematics (2016) [https://math.cornell.edu/m/Courses/Catalog/lowerlevel ''Lower-Level Courses for Freshmen and Sophomores''] {{Webarchive|url=https://web.archive.org/web/20180407035031/http://www.math.cornell.edu/m/Courses/Catalog/lowerlevel |date=7 April 2018 }}. Accessed on 2016-03-27.</ref><ref name="UMichMathCourses">University of Michigan Mathematics (2016) [https://lsa.umich.edu/UMICH/math/Home/Undergrad/Ugrad_Courses.pdf ''Math Course Catalogue''] {{webarchive|url=https://web.archive.org/web/20151101101339/https://lsa.umich.edu/UMICH/math/Home/Undergrad/Ugrad_Courses.pdf |date=2015-11-01}}. Accessed on 2016-03-27.</ref>

Furthermore, linear transformations over a finite-dimensional vector space can be represented using matrices,{{sfn|Herstein|1964|pp=228,229}}{{sfn|Nering|1970|p=38}} which is especially common in numerical and computational applications.{{sfn|Press|Teukolsky|Vetterling|Flannery|2007|p=38}}

[[File:Eigenvalue equation.svg|thumb|right|250px|Matrix ''A'' acts by stretching the vector '''x''', not changing its direction, so '''x''' is an eigenvector of ''A''.]]

Consider {{mvar|n}}-dimensional vectors that are formed as a list of {{mvar|n}} scalars, such as the three-dimensional vectors

<math display=block>\mathbf x = \begin{bmatrix}1\\-3\\4\end{bmatrix}\quad\mbox{and}\quad \mathbf y = \begin{bmatrix}-20\\60\\-80\end{bmatrix}.</math>

These vectors are said to be [[scalar multiplication|scalar multiples]] of each other, or [[Parallel (geometry)|parallel]] or [[collinearity|collinear]], if there is a scalar {{mvar|λ}} such that

<math display=block>\mathbf x = \lambda \mathbf y.</math>

In this case, <math>\lambda = -\frac{1}{20} </math>.

Now consider the linear transformation of {{mvar|n}}-dimensional vectors defined by an {{mvar|n}} by {{mvar|n}} matrix {{mvar|A}},

<math display=block>A \mathbf v = \mathbf w,</math>

or

<math display=block>\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}

</math>

where, for each index <math>i</math>row,

:<math> display=block>w_i = A_{i,1i1} v_1 + A_{i,2i2} v_2 + \cdots + A_{i,nin} v_n = \sum_{j = 1}^{n} A_{i,jij} v_j.</math>

In general, if <math>v</math> is not all zeros, the vectors <math>v</math> and <math>A v</math> will not be parallel. When they ''are'' parallel (that is, when there is some real number <math>\lambda</math> such that <math>A v = \lambda v</math>) we say that <math>v</math> is an '''eigenvector''' of <math>A</math>. In that case, the scale factor <math>\lambda</math> is said to be the '''eigenvalue''' corresponding to that eigenvector.

If it occurs that {{mvar|v}} and {{mvar|w}} are scalar multiples, that is if

In particular, multiplication by a 3×3 matrix <math>A</math> may change both the direction and the magnitude of an arrow <math>v</math> in three-dimensional space. However, if <math>v</math> is an eigenvector of <math>A</math> with eigenvalue <math>\lambda</math>, the operation may only change its length, and either keep its direction or [[point reflection|flip]] it (make the arrow point in the exact opposite direction). Specifically, the length of the arrow will increase if <math>|\lambda| > 1</math>, remain the same if <math>|\lambda| = 1</math>, and decrease it if <math>|\lambda|< 1</math>. Moreover, the direction will be precisely the same if <math>\lambda > 0</math>, and flipped if <math>\lambda < 0</math>. If <math>\lambda = 0</math>, then the length of the arrow becomes zero.

{{NumBlk|:|<math>A \mathbf v = \mathbf w = \lambda \mathbf v,</math>|{{EquationRef|1}}}}

then {{math|'''v'''}} is an '''eigenvector''' of the linear transformation {{mvar|A}} and the scale factor {{mvar|λ}} is the '''eigenvalue''' corresponding to that eigenvector. Equation ({{EquationNote|1}}) is the '''eigenvalue equation''' for the matrix {{mvar|A}}.

Equation ({{EquationNote|1}}) can be stated equivalently as

====An example====

{{NumBlk|:

[[File:Eigenvectors.gif|right|frame|The transformation matrix <math>\bigl[ \begin{smallmatrix} 2 & 1\\ 1 & 2 \end{smallmatrix} \bigr]</math> preserves the direction of vectors parallel to <math>\bigl[ \begin{smallmatrix} 1 \\ 1 \end{smallmatrix} \bigr]</math> (in blue) and <math>\bigl[ \begin{smallmatrix} 1 \\ -1 \end{smallmatrix} \bigr]</math> (in violet). The points that lie on the line through the origin, parallel to an eigenvector, remain on the line after the transformation. The vectors in red are not eigenvectors, therefore their direction is altered by the transformation. See also: [[:File:Eigenvectors-extended.gif|An extended version, showing all four quadrants]].]]

|<math>\left(A - \lambda I \right) \mathbf v = \mathbf 0,</math>

|{{EquationRef|2}}

}}

where {{mvar|I}} is the {{mvar|n}} by {{mvar|n}} [[identity matrix]] and '''0''' is the zero vector.

=== Eigenvalues and the characteristic polynomial ===

For the transformation matrix

{{main|Characteristic polynomial}}

:<math>A = \begin{bmatrix} 3 & 1\\1 & 3 \end{bmatrix},</math>

the vector

:<math>v = \begin{bmatrix} 4 \\ -4 \end{bmatrix}</math>

is an eigenvector with eigenvalue 2. Indeed,

:<math>A v = \begin{bmatrix} 3 & 1\\1 & 3 \end{bmatrix} \begin{bmatrix} 4 \\ -4 \end{bmatrix} = \begin{bmatrix} 3 \cdot 4 + 1 \cdot (-4) \\ 1 \cdot 4 + 3 \cdot (-4) \end{bmatrix}</math>

::<math> = \begin{bmatrix} 8 \\ -8 \end{bmatrix} = 2 \cdot \begin{bmatrix} 4 \\ -4 \end{bmatrix}.</math>

On the other hand the vector

:<math>v = \begin{bmatrix} 0 \\ 1 \end{bmatrix}</math>

is ''not'' an eigenvector, since

:<math>\begin{bmatrix} 3 & 1\\1 & 3 \end{bmatrix} \begin{bmatrix} 0 \\ 1 \end{bmatrix} = \begin{bmatrix} 3 \cdot 0 + 1 \cdot 1 \\ 1 \cdot 0 + 3 \cdot 1 \end{bmatrix} = \begin{bmatrix} 1 \\ 3 \end{bmatrix},</math>

and this vector is not a multiple of the original vector <math>v</math>.

Equation ({{EquationNote|2}}) has a nonzero solution ''v'' [[if and only if]] the [[determinant]] of the matrix {{nowrap|(''A'' − ''λI'')}} is zero. Therefore, the eigenvalues of ''A'' are values of ''λ'' that satisfy the equation

====Another example====

{{NumBlk|:

For the matrix

: |<math>\det(A= \begin{bmatrix}- 1 & 2 & 0\\0lambda &I) 2 &= 0\\ 0 & 0 & 3\end{bmatrix},</math>

|{{EquationRef|3}}

we have

}}

:<math>A \begin{bmatrix} 1\\0\\0 \end{bmatrix} = \begin{bmatrix} 1\\0\\0 \end{bmatrix} = 1 \cdot \begin{bmatrix} 1\\0\\0 \end{bmatrix},\quad\quad</math>

:<math>A \begin{bmatrix} 0\\0\\1 \end{bmatrix} = \begin{bmatrix} 0\\0\\3 \end{bmatrix} = 3 \cdot \begin{bmatrix} 0\\0\\1 \end{bmatrix}.\quad\quad</math>

Therefore, the vectors <math>[1,0,0]^\mathsf{T}</math> and <math>[0,0,1]^\mathsf{T}</math> are eigenvectors of <math>A</math> corresponding to the eigenvalues 1 and 3 respectively. (Here the symbol <math>{}^\mathsf{T}</math> indicates [[transpose of a matrix|matrix transposition]], in this case turning the row vectors into column vectors.)

Using the [[Leibniz formula for determinants]], the left-hand side of equation ({{EquationNote|3}}) is a [[polynomial]] function of the variable ''λ'' and the [[degree of a polynomial|degree]] of this polynomial is ''n'', the order of the matrix ''A''. Its [[coefficient]]s depend on the entries of ''A'', except that its term of degree ''n'' is always (−1)^''n''''λ''^''n''. This polynomial is called the ''[[characteristic polynomial]]'' of ''A''. Equation ({{EquationNote|3}}) is called the ''characteristic equation'' or the ''secular equation'' of ''A''.

====Trivial cases====

The [[identity matrix]] <math>I</math> (whose general element <math>I_{i j}</math> is 1 if <math>i = j</math>, and 0 otherwise) maps every vector to itself. Therefore, every vector is an eigenvector of <math>I</math>, with eigenvalue 1.

The [[fundamental theorem of algebra]] implies that the characteristic polynomial of an ''n''-by-''n'' matrix ''A'', being a polynomial of degree ''n'', can be [[factorization|factored]] into the product of ''n'' linear terms,

More generally, if <math>A</math> is a [[diagonal matrix]] (with <math>A_{i j} = 0</math> whenever <math>i \neq j</math>), and <math>v</math> is a vector parallel to axis <math>i</math> (that is, <math>v_i \neq 0</math>, and <math>v_j = 0</math> if <math>j \neq i</math>), then <math>A v = \lambda v</math> where <math>\lambda = A_{i i}</math>. That is, the eigenvalues of a diagonal matrix are the elements of its main diagonal. This is trivially the case of ''any'' 1 ×1 matrix.

{{NumBlk|:

|<math>\det(A - \lambda I) = (\lambda_1 - \lambda )(\lambda_2 - \lambda) \cdots (\lambda_n - \lambda),</math>

|{{EquationRef|4}}

}}

where each ''λ''_''i'' may be real but in general is a complex number. The numbers ''λ''₁, ''λ''₂, ..., ''λ''_''n'', which may not all have distinct values, are roots of the polynomial and are the eigenvalues of ''A''.

As a brief example, which is described in more detail in the examples section later, consider the matrix

===General definition===

<math display=block>A = \begin{bmatrix}

The concept of eigenvectors and eigenvalues extends naturally to abstract [[linear transformation]]s on abstract [[vector space]]s. Namely, let <math>V</math> be any vector space over some [[field (algebra)|field]] <math>K</math> of [[scalar (mathematics)|scalars]], and let <math>T</math> be a linear transformation mapping <math>V</math> into <math>V</math>. We say that a non-zero vector <math>v</math> of <math>V</math> is an '''eigenvector''' of <math>T</math> if (and only if) there is a scalar <math>\lambda</math> in <math>K</math> such that

2 & 1\\

:<math>T(v)=\lambda v</math>.

1 & 2

This equation is called the [[eigenvalue equation]] for <math>T</math>, and the scalar <math>\lambda</math> is the '''eigenvalue''' of <math>T</math> corresponding to the eigenvector <math>v</math>. Note that <math>T(v)</math> means the result of applying the operator <math>T</math> to the vector <math>v</math>, while <math>\lambda v </math> means the product of the scalar <math>\lambda</math> by <math>v</math>.<ref>See {{Harvnb|Korn|Korn|2000|loc=Section 14.3.5a}}; {{Harvnb|Friedberg|Insel|Spence|1989|loc=p. 217}}</ref>

\end{bmatrix}.</math>

Taking the determinant of {{nowrap|(''A'' − ''λI'')}}, the characteristic polynomial of ''A'' is

The matrix-specific definition is a special case of this abstract definition. Namely, the vector space <math>V</math> is the set of all column vectors of a certain size <math>n</math>×1, and <math>T</math> is the linear transformation that consists in multiplying a vector by the given <math>n\times n</math> matrix <math>A</math>.

<math display=block>\det(A - \lambda I) = \begin{vmatrix}

2 - \lambda & 1 \\

1 & 2 - \lambda

\end{vmatrix} =

3 - 4\lambda + \lambda^2.

</math>

Setting the characteristic polynomial equal to zero, it has roots at {{nowrap|1=λ=1}} and {{nowrap|1=λ=3}}, which are the two eigenvalues of ''A''. The eigenvectors corresponding to each eigenvalue can be found by solving for the components of the eigenvectors '''v''' in the equation {{nowrap|<math>\left(A - \lambda I\right) \mathbf v = \mathbf 0</math>}} at each eigenvalue {{nowrap|1=λ}}. In this example, the eigenvectors are any nonzero scalar multiples of

Some authors allow <math>v</math> to be the [[zero vector]] in the definition of eigenvector.<ref>{{Citation|last=Axler|first= Sheldon |title=Linear Algebra Done Right|edition=2nd |chapter=Ch. 5|page= 77}}</ref> This is reasonable as long as we define eigenvalues and eigenvectors carefully: If we would like the zero vector to be an eigenvector, then we must first define an eigenvalue of <math> T </math> as a scalar <math> \lambda </math> in <math>K</math> such that there is a ''nonzero'' vector <math> v </math> in <math>V</math> with <math> T(v) = \lambda v </math>. We then define an eigenvector to be a vector <math> v </math> in <math>V</math> such that there is an eigenvalue <math> \lambda </math> in <math>K</math> with <math> T(v) = \lambda v </math>. This way, we ensure that it is not the case that every scalar is an eigenvalue corresponding to the zero vector.

<math display="block">\mathbf v_{\lambda=1} = \begin{bmatrix} 1 \\ -1 \end{bmatrix}, \quad \mathbf v_{\lambda=3} = \begin{bmatrix} 1 \\ 1 \end{bmatrix}.</math>

If the entries of the matrix ''A'' 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 [[irrational number]]s even if all the entries of ''A'' are [[rational number]]s or even if they are all integers. However, if the entries of ''A'' are all [[algebraic number]]s, which include the rationals, the eigenvalues must also be algebraic numbers.

===Eigenspace and spectrum=== <!-- Geometric multiplicity links here -->

The non-real roots of a real polynomial with real coefficients can be grouped into pairs of [[complex conjugate]]s, 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 [[intermediate value theorem]] at least one of the roots is real. Therefore, any [[real matrix]] 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.

If <math>v</math> is an eigenvector of <math>T</math>, with eigenvalue <math>\lambda</math>, then any [[scalar multiplication|scalar multiple]] <math>\alpha v </math> of <math>v</math> with nonzero <math>\alpha</math> is also an eigenvector with eigenvalue <math>\lambda</math>, since <math>T(\alpha v) = \alpha T(v) = \alpha(\lambda v) = \lambda(\alpha v)</math>. Moreover, if <math>u</math> and <math>v</math> are eigenvectors with the same eigenvalue <math>\lambda</math>, then <math>u+v</math> is also an eigenvector with the same eigenvalue <math>\lambda</math>. Therefore, the set of all eigenvectors with the same eigenvalue <math>\lambda</math>, together with the zero vector, is a [[linear subspace]] of <math>V</math>, called the '''eigenspace''' of <math>T</math> associated to <math>\lambda</math>.<ref>{{Harvnb|Shilov|1977|loc=p. 109}}</ref><ref>[[b:The Book of Mathematical Proofs/Algebra/Linear Transformations#Lemma for the eigenspace|Lemma for the eigenspace]]</ref> If that subspace has dimension 1, it is sometimes called an '''eigenline'''.<ref>''[http://books.google.com/books?id=pkESXAcIiCQC&pg=PA111 Schaum's Easy Outline of Linear Algebra]'', p. 111</ref>

=== Spectrum of a matrix ===

The '''geometric multiplicity''' <math>\gamma_T(\lambda)</math> of an eigenvalue <math>\lambda</math> is the dimension of the eigenspace associated to <math>\lambda</math>, i.e. number of [[linear independence|linearly independent]] eigenvectors with that eigenvalue.

The '''[[Spectrum of a matrix|spectrum]]''' of a matrix is the list of eigenvalues, repeated according to multiplicity; in an alternative notation the set of eigenvalues with their multiplicities.

An important quantity associated with the spectrum is the maximum absolute value of any eigenvalue. This is known as the [[Spectral radius#Matrices|spectral radius]] of the matrix.

The eigenspaces of ''T'' always form a direct sum (and as a consequence any family of eigenvectors for different eigenvalues is always linearly independent). Therefore the sum of the dimensions of the eigenspaces cannot exceed the dimension ''n'' of the space on which ''T'' operates, and in particular there cannot be more than ''n'' distinct eigenvalues.<ref name="Shilov_lemma">For a proof of this lemma, see {{Harvnb|Roman|2008|loc=Theorem 8.2 on p. 186}}; {{Harvnb|Shilov|1977|loc=p. 109}}; {{Harvnb|Hefferon|2001|loc=p. 364}}; {{Harvnb|Beezer|2006|loc=Theorem EDELI on p. 469}}; and [[b:Famous Theorems of Mathematics/Algebra/Linear Transformations#Lemma for linear independence of eigenvectors|Lemma for linear independence of eigenvectors]]</ref>

=== Algebraic multiplicity ===

Any subspace spanned by eigenvectors of <math>T</math> is an [[invariant subspace]] of <math>T</math>, and the restriction of ''T'' to such a subspace is diagonalizable.

<!-- Algebraic multiplicity, Simple eigenvalue and Semisimple eigenvalue link here. Please do not change. -->

Let ''λ''_''i'' be an eigenvalue of an ''n'' by ''n'' matrix ''A''. The '''algebraic multiplicity''' ''μ''_''A''(''λ''_''i'') of the eigenvalue is its [[Multiple roots of a polynomial|multiplicity as a root]] of the characteristic polynomial, that is, the largest integer ''k'' such that (''λ'' − ''λ''_''i'')^''k'' [[polynomial division|divides evenly]] that polynomial.{{sfn|Nering|1970|p=107}}{{sfn|Fraleigh|1976|p=358}}{{sfn|Golub|Van Loan|1996|p=316}}

Suppose a matrix ''A'' has dimension ''n'' and ''d'' ≤ ''n'' distinct eigenvalues. Whereas equation ({{EquationNote|4}}) factors the characteristic polynomial of ''A'' into the product of ''n'' linear terms with some terms potentially repeating, the characteristic polynomial can also be written as the product of ''d'' terms each corresponding to a distinct eigenvalue and raised to the power of the algebraic multiplicity,

The set of eigenvalues of <math>T</math> is sometimes called the [[Spectrum of a matrix|spectrum]] of <math>T</math>.

<math display=block>\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)}.</math>

If ''d'' = ''n'' then the right-hand side is the product of ''n'' linear terms and this is the same as equation ({{EquationNote|4}}). The size of each eigenvalue's algebraic multiplicity is related to the dimension ''n'' as

===Eigenbasis===

<math display=block>\begin{align}

An '''eigenbasis''' for a linear operator <math>T</math> that operates on a vector space <math>V</math> is a basis for <math>V</math> that consists entirely of eigenvectors of <math>T</math> (possibly with different eigenvalues). Such a basis may not exist.

1 &\leq \mu_A(\lambda_i) \leq n, \\

\mu_A &= \sum_{i=1}^d \mu_A\left(\lambda_i\right) = n.

\end{align} </math>

If ''μ''_''A''(''λ''_''i'') = 1, then ''λ''_''i'' is said to be a ''simple eigenvalue''.{{sfn|Golub|Van Loan|1996|p=316}} If ''μ''_''A''(''λ''_''i'') equals the geometric multiplicity of ''λ''_''i'', ''γ''_''A''(''λ''_''i''), defined in the next section, then ''λ''_''i'' is said to be a ''semisimple eigenvalue''.

Suppose <math>V</math> has finite dimension <math>n</math>, and let <math>\boldsymbol{\gamma}_T</math> be the sum of the geometric multiplicities <math>\gamma_T(\lambda_i)</math> over all distinct eigenvalues <math>\lambda_i</math> of <math>T</math>. This integer is the maximum number of linearly independent eigenvectors of <math>T</math>, and therefore cannot exceed <math>n</math>. If <math>\boldsymbol{\gamma}_T</math> is exactly <math>n</math>, then <math>T</math> admits an eigenbasis; that is, there exists a basis for <math>V</math> that consists of <math>n</math> eigenvectors. The matrix <math>A</math> that represents <math>T</math> relative to this basis is a diagonal matrix, whose diagonal elements are the eigenvalues associated to each basis vector.

=== Eigenspaces, geometric multiplicity, and the eigenbasis for matrices ===

Conversely, if the sum <math>\boldsymbol{\gamma}_T</math> is less than <math>n</math>, then <math>T</math> admits no eigenbasis, and there is no choice of coordinates that will allow <math>T</math> to be represented by a diagonal matrix.

<!-- Geometric multiplicity redirects here -->

Given a particular eigenvalue ''λ'' of the ''n'' by ''n'' matrix ''A'', define the [[Set (mathematics)|set]] ''E'' to be all vectors '''v''' that satisfy equation ({{EquationNote|2}}),

<math display=block>E = \left\{\mathbf{v} : \left(A - \lambda I\right) \mathbf{v} = \mathbf{0}\right\}.</math>

On one hand, this set is precisely the [[kernel (linear algebra)|kernel]] or nullspace of the matrix {{math|(''A'' − ''λI'')}}. On the other hand, by definition, any nonzero vector that satisfies this condition is an eigenvector of ''A'' associated with ''λ''. So, the set ''E'' is the [[Union (set theory)|union]] of the zero vector with the set of all eigenvectors of ''A'' associated with ''λ'', and ''E'' equals the nullspace of {{math|(''A'' − ''λI'').}} ''E'' is called the '''eigenspace''' or '''characteristic space''' of ''A'' associated with ''λ''.{{sfn|Anton|1987|pp=305,307}}{{sfn|Nering|1970|p=107}} In general ''λ'' is a complex number and the eigenvectors are complex ''n'' by 1 matrices. A property of the nullspace is that it is a [[linear subspace]], so ''E'' is a linear subspace of <math>\mathbb{C}^n</math>.

Note that <math>\boldsymbol{\gamma}_T</math> is at least equal to the number of ''distinct'' eigenvalues of <math>T</math>, but may be larger than that.<ref>{{Citation|first=Gilbert|last=Strang|title=Linear Algebra and Its Applications|edition=3rd|publisher=Harcourt|___location= San Diego| year=1988}}</ref> For example, the identity operator <math>I</math> on <math>V</math> has <math>\boldsymbol{\gamma}_I = n</math>, and any basis of <math>V</math> is an eigenbasis of <math>I</math>; but its only eigenvalue is 1, with <math>\gamma_T(1) = n</math>.

Because the eigenspace ''E'' is a linear subspace, it is [[closure (mathematics)|closed]] under addition. That is, if two vectors '''u''' and '''v''' belong to the set ''E'', written {{math|'''u''', '''v''' ∈ ''E''}}, then {{math|('''u''' + '''v''') ∈ ''E''}} or equivalently {{math|1=''A''('''u''' + '''v''') = ''λ''('''u''' + '''v''')}}. This can be checked using the [[distributive property]] of matrix multiplication. Similarly, because ''E'' is a linear subspace, it is closed under scalar multiplication. That is, if {{math|'''v''' ∈ ''E''}} and ''α'' is a complex number, {{math|(''α'''''v''') ∈ ''E''}} or equivalently {{math|1=''A''(''α'''''v''') = ''λ''(''α'''''v''')}}. This can be checked by noting that multiplication of complex matrices by complex numbers is [[commutative property|commutative]]. As long as '''u''' + '''v''' and ''α'''''v''' are not zero, they are also eigenvectors of ''A'' associated with ''λ''.

==Generalizations to infinite-dimensional spaces==

{{details|Spectral theorem}}

The definition of eigenvalue of a linear transformation <math>T</math> remains valid even if the underlying space <math>V</math> is an infinite dimensional [[Hilbert space|Hilbert]] or [[Banach space]]. Namely, a scalar <math>\lambda</math> is an eigenvalue if and only if there is some nonzero vector <math>v</math> such that <math>T(v) = \lambda v</math>.

The dimension of the eigenspace ''E'' associated with ''λ'', or equivalently the maximum number of linearly independent eigenvectors associated with ''λ'', is referred to as the eigenvalue's '''geometric multiplicity''' <math>\gamma_A(\lambda)</math>. Because ''E'' is also the nullspace of {{math|(''A'' − ''λI'')}}, the geometric multiplicity of ''λ'' is the dimension of the nullspace of {{math|(''A'' − ''λI''),}} also called the ''nullity'' of {{math|(''A'' − ''λI''),}} which relates to the dimension and rank of {{math|(''A'' − ''λI'')}} as

===Eigenfunctions===

<math display=block>\gamma_A(\lambda) = n - \operatorname{rank}(A - \lambda I).</math>

A widely used class of linear operators acting on infinite dimensional spaces are the [[differential operator]]s on [[function space]]s. Let <math>D</math> be a linear differential operator in on the space <math>\mathbf{C^\infty}</math> of infinitely [[derivative|differentiable]] real functions of a real argument <math>t</math>. The eigenvalue equation for <math>D</math> is the [[differential equation]]

:<math>D f = \lambda f</math>

The functions that satisfy this equation are commonly called '''[[eigenfunctions]]'''. For the derivative operator <math>d/dt</math>, an eigenfunction is a function that, when differentiated, yields a constant times the original function. If <math>\lambda</math> is zero, the generic solution is a constant function. If <math>\lambda</math> is non-zero, the solution is an [[exponential function]]

: <math>f(t) = Ae^{\lambda t}.\ </math>

Eigenfunctions are an essential tool in the solution of differential equations and many other applied and theoretical fields. For instance, the exponential functions are eigenfunctions of any [[shift invariant operator|shift invariant linear operator]]. This fact is the basis of powerful [[Fourier transform]] methods for solving all sorts of problems.

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 that an eigenvalue's algebraic multiplicity cannot exceed ''n''.

===Spectral theory===

<math display=block> 1 \le \gamma_A(\lambda) \le \mu_A(\lambda) \le n</math>

If <math>\lambda</math> is an eigenvalue of <math>T</math>, then the operator <math>T-\lambda I</math> is not one-to-one, and therefore its inverse <math>(T-\lambda I)^{-1}</math> is not defined. The converse is true for finite-dimensional vector spaces, but not for infinite-dimensional ones. In general, the operator <math>T - \lambda I</math> may not have an inverse, even if <math>\lambda</math> is not an eigenvalue.

To prove the inequality <math>\gamma_A(\lambda)\le\mu_A(\lambda)</math>, consider how the definition of geometric multiplicity implies the existence of <math>\gamma_A(\lambda)</math> [[Orthonormality|orthonormal]] eigenvectors <math>\boldsymbol{v}_1,\, \ldots,\, \boldsymbol{v}_{\gamma_A(\lambda)}</math>, such that <math>A \boldsymbol{v}_k = \lambda \boldsymbol{v}_k</math>. We can therefore find a (unitary) matrix {{mvar|V}} whose first <math>\gamma_A(\lambda)</math> columns are these eigenvectors, and whose remaining columns can be any orthonormal set of <math>n - \gamma_A(\lambda)</math> vectors orthogonal to these eigenvectors of {{mvar|A}}. Then {{mvar|V}} has full rank and is therefore invertible. Evaluating <math>D:=V^TAV</math>, we get a matrix whose top left block is the diagonal matrix <math>\lambda I_{\gamma_A(\lambda)}</math>. This can be seen by evaluating what the left-hand side does to the first column basis vectors. By reorganizing and adding <math>-\xi V</math> on both sides, we get <math>(A - \xi I)V = V(D - \xi I)</math> since {{mvar|I}} commutes with {{mvar|V}}. In other words, <math>A - \xi I</math> is similar to <math>D - \xi I</math>, and <math>\det(A - \xi I) = \det(D - \xi I)</math>. But from the definition of {{mvar|D}}, we know that <math>\det(D - \xi I)</math> contains a factor <math>(\xi - \lambda)^{\gamma_A(\lambda)}</math>, which means that the algebraic multiplicity of <math>\lambda</math> must satisfy <math>\mu_A(\lambda) \ge \gamma_A(\lambda)</math>.

For this reason, in [[functional analysis]] one defines the [[spectrum (functional analysis)|spectrum of a linear operator]] <math>T</math> as the set of all scalars <math>\lambda</math> for which the operator <math>T-\lambda I</math> has no [[bounded operator|bounded]] inverse. Thus the spectrum of an operator always contains all its eigenvalues, but is not limited to them.

Suppose {{mvar|A}} has <math>d \leq n</math> distinct eigenvalues <math>\lambda_1, \ldots, \lambda_d</math>, where the geometric multiplicity of <math>\lambda_i</math> is <math>\gamma_A (\lambda_i)</math>. The total geometric multiplicity of {{mvar|A}},

===Associative algebras and representation theory===

<math display=block>\begin{align}

More algebraically, rather than generalizing the vector space to an infinite dimensional space, one can generalize the algebraic object that is acting on the space, replacing a single operator acting on a vector space with an [[algebra representation]] – an [[associative algebra]] acting on a [[module (mathematics)|module]]. The study of such actions is the field of [[representation theory]].

\gamma_A &= \sum_{i=1}^d \gamma_A(\lambda_i), \\

d &\le \gamma_A \le n,

\end{align}</math>

is the dimension of the [[Linear subspace#Sum|sum]] of all the eigenspaces of {{mvar|A}}'s eigenvalues, or equivalently the maximum number of linearly independent eigenvectors of {{mvar|A}}. If <math>\gamma_A=n</math>, then

* The direct sum of the eigenspaces of all of {{mvar|A}}'s eigenvalues is the entire vector space <math>\mathbb{C}^n</math>.

* A basis of <math>\mathbb{C}^n</math> can be formed from {{mvar|n}} linearly independent eigenvectors of {{mvar|A}}; such a basis is called an '''eigenbasis'''

* Any vector in <math>\mathbb{C}^n</math> can be written as a linear combination of eigenvectors of {{mvar|A}}.

=== Additional properties ===

A closer analog of eigenvalues is given by the [[weight (representation theory)|representation-theoretical concept of weight]], with the analogs of eigenvectors and eigenspaces being ''weight vectors'' and ''weight spaces.''

Let <math>A</math> be an arbitrary <math>n \times n</math> matrix of complex numbers with eigenvalues <math>\lambda_1, \ldots, \lambda_n</math>. Each eigenvalue appears <math>\mu_A(\lambda_i)</math> times in this list, where <math>\mu_A(\lambda_i)</math> is the eigenvalue's algebraic multiplicity. The following are properties of this matrix and its eigenvalues:

* The [[trace (linear algebra)|trace]] of <math>A</math>, defined as the sum of its diagonal elements, is also the sum of all eigenvalues,{{sfn|Beauregard|Fraleigh|1973|p=307}}{{sfn|Herstein|1964|p=272}}{{sfn|Nering|1970|pp=115–116}}

*: <math>\operatorname{tr}(A) = \sum_{i=1}^n a_{ii} = \sum_{i=1}^n \lambda_i = \lambda_1 + \lambda_2 + \cdots + \lambda_n.</math>

* The [[determinant]] of <math>A</math> is the product of all its eigenvalues,{{sfn|Beauregard|Fraleigh|1973|p=307}}{{sfn|Herstein|1964|p=290}}{{sfn|Nering|1970|p=116}}

*: <math>\det(A) = \prod_{i=1}^n \lambda_i = \lambda_1\lambda_2 \cdots \lambda_n.</math>

* The eigenvalues of the <math>k</math>th power of <math>A</math>; i.e., the eigenvalues of <math>A^k</math>, for any positive integer <math>k</math>, are <math>\lambda_1^k, \ldots, \lambda_n^k</math>.

* The matrix <math>A</math> is [[invertible matrix|invertible]] if and only if every eigenvalue is nonzero.

* If <math>A</math> is invertible, then the eigenvalues of <math>A^{-1}</math> are <math display="inline">\frac{1}{\lambda_1}, \ldots, \frac{1}{\lambda_n}</math> and each eigenvalue's geometric multiplicity coincides. Moreover, since the characteristic polynomial of the inverse is the [[reciprocal polynomial]] of the original, the eigenvalues share the same algebraic multiplicity.

* If <math>A</math> is equal to its [[conjugate transpose]] <math>A^*</math>, or equivalently if <math>A</math> is [[Hermitian matrix|Hermitian]], then every eigenvalue is real. The same is true of any [[symmetric matrix|symmetric]] real matrix.

* If <math>A</math> is not only Hermitian but also [[positive-definite matrix|positive-definite]], positive-semidefinite, negative-definite, or negative-semidefinite, then every eigenvalue is positive, non-negative, negative, or non-positive, respectively.

* If <math>A</math> is [[unitary matrix|unitary]], every eigenvalue has absolute value <math>|\lambda_i|=1</math>.

* If <math>A</math> is a <math>n\times n</math> matrix and <math>\{\lambda_1,\ldots,\lambda_k\}</math> are its eigenvalues, then the eigenvalues of matrix <math>I+A</math> (where <math>I</math> is the identity matrix) are <math>\{\lambda_1+1,\ldots,\lambda_k+1\}</math>. Moreover, if <math>\alpha\in\mathbb C</math>, the eigenvalues of <math>\alpha I+A</math> are <math>\{\lambda_1+\alpha,\ldots,\lambda_k+\alpha\}</math>. More generally, for a polynomial <math>P</math> the eigenvalues of matrix <math>P(A)</math> are <math>\{P(\lambda_1), \ldots, P(\lambda_k)\}</math>.

==Eigenvalues= Left and eigenvectorsright ofeigenvectors matrices===

{{See also|left and right (algebra)}}

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 '''right eigenvector''', namely a ''column'' vector that ''right'' multiplies the <math>n \times n</math> matrix <math>A</math> in the defining equation, equation ({{EquationNote|1}}),

===Characteristic polynomial===

<math display=block>A \mathbf v = \lambda \mathbf v.</math>

The eigenvalue equation for a matrix <math>A</math> is

: <math>A v - \lambda v = 0,</math>

which is equivalent to

: <math>(A-\lambda I)v = 0,</math>

where <math>I</math> is the <math>n\times n</math> [[identity matrix]]. It is a fundamental result of linear algebra that an equation <math>M v = 0</math> has a non-zero solution <math>v</math> if, and only if, the [[determinant]] <math>\det(M)</math> of the matrix <math>M</math> is zero. It follows that the eigenvalues of <math>A</math> are precisely the real numbers <math>\lambda</math> that satisfy the equation

: <math>\det(A-\lambda I) = 0</math>

The eigenvalue and eigenvector problem can also be defined for ''row'' vectors that ''left'' multiply matrix <math>A</math>. In this formulation, the defining equation is

The left-hand side of this equation can be seen (using [[Leibniz formula for determinants|Leibniz' rule]] for the determinant) to be a [[polynomial]] function of the variable <math>\lambda</math>. The [[degree of a polynomial|degree]] of this polynomial is <math>n</math>, the order of the matrix. Its [[coefficient]]s depend on the entries of <math>A</math>, except that its term of degree <math>n</math> is always <math>(-1)^n\lambda^n</math>. This polynomial is called the ''[[characteristic polynomial]]'' of <math>A</math>; and the above equation is called the ''characteristic equation'' (or, less often, the ''secular equation'') of <math>A</math>.

<math display=block>\mathbf u A = \kappa \mathbf u,</math>

where <math>\kappa</math> is a scalar and <math>u</math> is a <math>1 \times n</math> matrix. Any row vector <math>u</math> satisfying this equation is called a '''left eigenvector''' of <math>A</math> and <math>\kappa</math> is its associated eigenvalue. Taking the transpose of this equation,

For example, let <math>A</math> be the matrix

<math display=block>A^\textsf{T} \mathbf u^\textsf{T} = \kappa \mathbf u^\textsf{T}.</math>

:<math>A =

\begin{bmatrix}

2 & 0 & 0 \\

0 & 3 & 4 \\

0 & 4 & 9

\end{bmatrix}</math>

Comparing this equation to equation ({{EquationNote|1}}), it follows immediately that a left eigenvector of <math>A</math> is the same as the transpose of a right eigenvector of <math>A^\textsf{T}</math>, with the same eigenvalue. Furthermore, since the characteristic polynomial of <math>A^\textsf{T}</math> is the same as the characteristic polynomial of <math>A</math>, the left and right eigenvectors of <math>A</math> are associated with the same eigenvalues.

The characteristic polynomial of <math>A</math> is

:<math>\det (A-\lambda I) \;=\; \det \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) \;=\;

\det \begin{bmatrix}

2 - \lambda & 0 & 0 \\

0 & 3 - \lambda & 4 \\

0 & 4 & 9 - \lambda

\end{bmatrix}</math>

which is

:<math> (2 - \lambda) \bigl[ (3 - \lambda) (9 - \lambda) - 16 \bigr] = -\lambda^3 + 14\lambda^2 - 35\lambda + 22</math>

The roots of this polynomial are 2, 1, and 11. Indeed these are the only three eigenvalues of <math>A</math>, corresponding to the eigenvectors <math>[1,0,0]',</math> <math>[0,2,-1]',</math> and <math>[0,1,2]'</math> (or any non-zero multiples thereof).

====In Diagonalization and the realeigendecomposition ___domain====

{{main|Eigendecomposition of a matrix}}

Since the eigenvalues are roots of the characteristic polynomial, an <math>n\times n</math> matrix has at most <math>n</math> eigenvalues. If the matrix has real entries, the coefficients of the characteristic polynomial are all real; but it may have fewer than <math>n</math> real roots, or no real roots at all.

Suppose the eigenvectors of ''A'' form a basis, or equivalently ''A'' has ''n'' linearly independent eigenvectors '''v'''₁, '''v'''₂, ..., '''v'''_''n'' with associated eigenvalues ''λ''₁, ''λ''₂, ..., ''λ''_''n''. The eigenvalues need not be distinct. Define a [[square matrix]] ''Q'' whose columns are the ''n'' linearly independent eigenvectors of ''A'',

For example, consider the [[permutation matrix|cyclic permutation matrix]]

: <math>AQ = \begin{bmatrix} 0\mathbf & 1v_1 & 0\\0 &mathbf 0v_2 & 1\\ 1cdots & 0\mathbf &v_n 0\end{bmatrix}.</math>

This matrix shifts the coordinates of the vector up by one position, and moves the first coordinate to the bottom. Its characteristic polynomial is <math>1 - \lambda^3</math> which has one real root <math>\lambda_1 = 1</math>. Any vector with three equal non-zero elements is an eigenvector for this eigenvalue. For example,

:<math>

A \begin{bmatrix} 5\\5\\5 \end{bmatrix} =

\begin{bmatrix} 5\\5\\5 \end{bmatrix} =

1 \cdot \begin{bmatrix} 5\\5\\5 \end{bmatrix}

</math>

Since each column of ''Q'' is an eigenvector of ''A'', right multiplying ''A'' by ''Q'' scales each column of ''Q'' by its associated eigenvalue,

====In the complex ___domain====

: <math>AQ = \begin{bmatrix} \lambda_1 \mathbf v_1 & \lambda_2 \mathbf v_2 & \cdots & \lambda_n \mathbf v_n \end{bmatrix}.</math>

The [[fundamental theorem of algebra]] implies that the characteristic polynomial of an <math>n\times n</math> matrix <math>A</math>, being a polynomial of degree <math>n</math>, has exactly <math>n</math> complex [[root]]s. More precisely, it can be [[factorization|factored]] into the product of <math>n</math> linear terms,

:<math> \det(A-\lambda I) = (\lambda_1 - \lambda )(\lambda_2 - \lambda)\cdots(\lambda_n - \lambda)</math>

where each <math>\lambda_i</math> is a complex number. The numbers <math>\lambda_1</math>, <math>\lambda_2</math>, ... <math>\lambda_n</math>, (which may not be all distinct) are roots of the polynomial, and are precisely the eigenvalues of <math>A</math>.

With this in mind, define a diagonal matrix Λ where each diagonal element Λ_''ii'' is the eigenvalue associated with the ''i''th column of ''Q''. Then

Even if the entries of <math>A</math> are all real numbers, the eigenvalues may still have non-zero imaginary parts (and the elements of the corresponding eigenvectors will therefore also have non-zero imaginary parts). Also, the eigenvalues may be [[irrational number]]s even if all the entries of <math>A</math> are [[rational number]]s, or all are integers. However, if the entries of <math>A</math> are [[algebraic number]]s (which include the rationals), the eigenvalues will be (complex) algebraic numbers too.

: <math>AQ = Q\Lambda.</math>

Because the columns of ''Q'' are linearly independent, Q is invertible. Right multiplying both sides of the equation by ''Q''⁻¹,

The non-real roots of a real polynomial with real coefficients can be grouped into pairs of [[complex conjugate]] values, namely with the two members of each pair having the same real part and imaginary parts that differ only in sign. If the degree is odd, then by the [[intermediate value theorem]] at least one of the roots will be real. Therefore, any real matrix with odd order will have at least one real eigenvalue; whereas a real matrix with even order may have no real eigenvalues.

: <math>A = Q\Lambda Q^{-1},</math>

or by instead left multiplying both sides by ''Q''⁻¹,

In the example of the 3×3 cyclic permutation matrix <math>A</math>, above, the characteristic polynomial <math>1 - \lambda^3</math> has two additional non-real roots, namely

: <math>Q^{-1}AQ = \Lambda.</math>

:<math>\lambda_2 = -1/2 + \mathbf{i}\sqrt{3}/2\quad\quad</math> and <math>\quad\quad\lambda_3 = \lambda_2^* = -1/2 - \mathbf{i}\sqrt{3}/2</math>,

where <math>\mathbf{i}= \sqrt{-1}</math> is the imaginary unit. Note that <math>\lambda_2\lambda_3 = 1</math>, <math>\lambda_2^2 = \lambda_3</math>, and <math>\lambda_3^2 = \lambda_2</math>. Then

''A'' 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 [[eigendecomposition of a matrix|eigendecomposition]] and it is a [[matrix similarity|similarity transformation]]. Such a matrix ''A'' is said to be ''similar'' to the diagonal matrix Λ or ''[[diagonalizable matrix|diagonalizable]]''. The matrix ''Q'' is the change of basis matrix of the similarity transformation. Essentially, the matrices ''A'' 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&nbsp;Λ.

:<math>

Conversely, suppose a matrix ''A'' is diagonalizable. Let ''P'' be a non-singular square matrix such that ''P''⁻¹''AP'' is some diagonal matrix ''D''. Left multiplying both by ''P'', {{nowrap|1=''AP'' = ''PD''}}. Each column of ''P'' must therefore be an eigenvector of ''A'' whose eigenvalue is the corresponding diagonal element of ''D''. Since the columns of ''P'' must be linearly independent for ''P'' to be invertible, there exist ''n'' linearly independent eigenvectors of ''A''. It then follows that the eigenvectors of ''A'' form a basis if and only if ''A'' is diagonalizable.

A matrix that is not diagonalizable is said to be [[defective matrix|defective]]. For defective matrices, the notion of eigenvectors generalizes to [[generalized eigenvector]]s and the diagonal matrix of eigenvalues generalizes to the [[Jordan normal form]]. Over an algebraically closed field, any matrix ''A'' has a [[Jordan normal form]] and therefore admits a basis of generalized eigenvectors and a decomposition into [[generalized eigenspace]]s.

=== Variational characterization ===

{{main|Min-max theorem}}

In the [[Hermitian matrix|Hermitian]] case, eigenvalues can be given a variational characterization. The largest eigenvalue of <math>H</math> is the maximum value of the [[quadratic form]] <math>\mathbf x^\textsf{T} H \mathbf x / \mathbf x^\textsf{T} \mathbf x</math>. A value of <math>\mathbf x</math> that realizes that maximum is an eigenvector.

=== Matrix examples ===

==== Two-dimensional matrix example ====

[[File:Eigenvectors.gif|right|frame|The transformation matrix ''A'' = <math>\left[\begin{smallmatrix} 2 & 1\\ 1 & 2 \end{smallmatrix}\right]</math> preserves the direction of magenta vectors parallel to '''v'''_''λ''=1 = [1 −1]^T and blue vectors parallel to '''v'''_''λ''=3 = [1 1]^T. The red vectors are not parallel to either eigenvector, so, their directions are changed by the transformation. The lengths of the magenta 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: [[:File:Eigenvectors-extended.gif|An extended version, showing all four quadrants]].]]

Consider the matrix

<math display=block>A = \begin{bmatrix}

2 & 1\\

1 & 2

\end{bmatrix}.</math>

The figure on the right shows the effect of this transformation on point coordinates in the plane. The eigenvectors ''v'' of this transformation satisfy equation ({{EquationNote|1}}), and the values of ''λ'' for which the determinant of the matrix (''A''&nbsp;−&nbsp;''λI'') equals zero are the eigenvalues.

Taking the determinant to find characteristic polynomial of ''A'',

<math display=block>\begin{align}

\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} \\[6pt]

&= 3 - 4\lambda + \lambda^2 \\[6pt]

&= (\lambda - 3)(\lambda - 1).

\end{align}</math>

Setting the characteristic polynomial equal to zero, it has roots at {{nowrap|1=''λ''=1}} and {{nowrap|1=''λ''=3}}, which are the two eigenvalues of ''A''.

For {{nowrap|1=''λ''=1}}, equation ({{EquationNote|2}}) becomes,

<math display=block>(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}</math>

<math display=block>1v_1 + 1v_2 = 0</math>

Any nonzero vector with ''v''₁ = −''v''₂ solves this equation. Therefore,

<math display=block>\mathbf{v}_{\lambda=1} = \begin{bmatrix} v_1 \\ -v_1 \end{bmatrix} = \begin{bmatrix} 1 \\ -1 \end{bmatrix}</math>

is an eigenvector of ''A'' corresponding to ''λ'' = 1, as is any scalar multiple of this vector.

For {{nowrap|1=''λ''=3}}, equation ({{EquationNote|2}}) becomes

<math display=block>\begin{align}

(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{align}</math>

Any nonzero vector with ''v''₁ = ''v''₂ solves this equation. Therefore,

<math display=block>\mathbf v_{\lambda=3} = \begin{bmatrix} v_1 \\ v_1 \end{bmatrix} = \begin{bmatrix} 1 \\ 1 \end{bmatrix}</math>

is an eigenvector of ''A'' corresponding to ''λ'' = 3, as is any scalar multiple of this vector.

Thus, the vectors '''v'''_''λ''=1 and '''v'''_''λ''=3 are eigenvectors of ''A'' associated with the eigenvalues {{nowrap|1=''λ''=1}} and {{nowrap|1=''λ''=3}}, respectively.

==== Three-dimensional matrix example ====

Consider the matrix

<math display=block>A = \begin{bmatrix}

2 & 0 & 0 \\

0 & 3 & 4 \\

0 & 4 & 9

\end{bmatrix}.</math>

The characteristic polynomial of ''A'' is

<math display=block>\begin{align}

\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}, \\[6pt]

&= (2 - \lambda)\bigl[(3 - \lambda)(9 - \lambda) - 16\bigr]

= -\lambda^3 + 14\lambda^2 - 35\lambda + 22.

\end{align}</math>

The roots of the characteristic polynomial are 2, 1, and 11, which are the only three eigenvalues of ''A''. These eigenvalues correspond to the eigenvectors {{nowrap|<math>\begin{bmatrix} 1 & 0 & 0 \end{bmatrix}^\textsf{T}</math>,}} {{nowrap|<math>\begin{bmatrix} 0 & -2 & 1 \end{bmatrix}^\textsf{T}</math>,}} and {{nowrap|<math>\begin{bmatrix} 0 & 1 & 2 \end{bmatrix}^\textsf{T}</math>,}} or any nonzero multiple thereof.

==== Three-dimensional matrix example with complex eigenvalues ====

Consider the [[permutation matrix|cyclic permutation matrix]]

<math display=block>A = \begin{bmatrix}

0 & 1 & 0\\

0 & 0 & 1\\

1 & 0 & 0

\end{bmatrix}.</math>

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&nbsp;−&nbsp;''λ''³, whose roots are

<math display=block>\begin{align}

\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{align}</math>

where <math>i</math> is an [[imaginary unit]] with {{nowrap|<math>i^2 = -1</math>.}}

For the real eigenvalue ''λ''₁ = 1, any vector with three equal nonzero entries is an eigenvector. For example,

<math display=block>

A \begin{bmatrix} 5\\ 5\\ 5 \end{bmatrix} =

\begin{bmatrix} 5\\ 5\\ 5 \end{bmatrix} =

1 \cdot \begin{bmatrix} 5\\ 5\\ 5 \end{bmatrix}.

</math>

For the complex conjugate pair of imaginary eigenvalues,

<math display=block>\lambda_2\lambda_3 = 1, \quad \lambda_2^2 = \lambda_3, \quad \lambda_3^2 = \lambda_2.</math>

Then

<math display=block>

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},

</math>

\quad\quad

and

</math> and <math>

<math display=block>

\quad\quad

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}.

</math>

Therefore, the vectors <math>[1,\lambda_2,\lambda_3]'</math> and <math>[1,\lambda_3,\lambda_2]'</math> are eigenvectors of <math>A</math>, with eigenvalues <math>\lambda_2</math>, and <math>\lambda_3</math>, respectively.

Therefore, the other two eigenvectors of ''A'' are complex and are <math>\mathbf v_{\lambda_2} = \begin{bmatrix} 1 & \lambda_2 & \lambda_3\end{bmatrix}^\textsf{T}</math> and <math>\mathbf v_{\lambda_3} = \begin{bmatrix} 1 & \lambda_3 & \lambda_2\end{bmatrix}^\textsf{T}</math> with eigenvalues ''λ''₂ and ''λ''₃, respectively. The two complex eigenvectors also appear in a complex conjugate pair,

===Algebraic multiplicities===

<math display=block>\mathbf v_{\lambda_2} = \mathbf v_{\lambda_3}^*.</math>

Let <math>\lambda_i</math> be an eigenvalue of an <math>n\times n</math> matrix <math>A</math>. The ''algebraic multiplicity'' <math>\mu_A(\lambda_i)</math> of <math>\lambda_i</math> is its [[multiplicity (mathematics)#Multiplicity of a root of a polynomial|multiplicity as a root]] of the characteristic polynomial, that is, the largest integer <math>k</math> such that <math>(\lambda - \lambda_i)^k</math> [[polynomial division|divides evenly]] that polynomial.

==== Diagonal matrix example ====

Like the geometric multiplicity <math>\gamma_A(\lambda_i)</math>, the algebraic multiplicity is an integer between 1 and <math>n</math>; and the sum <math>\boldsymbol{\mu}_A</math> of <math>\mu_A(\lambda_i)</math> over all ''distinct'' eigenvalues also cannot exceed <math>n</math>. If complex eigenvalues are considered, <math>\boldsymbol{\mu}_A</math> is exactly <math>n</math>.

Matrices with entries only along the main diagonal are called ''[[diagonal matrices]]''. The eigenvalues of a diagonal matrix are the diagonal elements themselves. Consider the matrix

<math display=block>A = \begin{bmatrix} 1 & 0 & 0\\ 0 & 2 & 0\\ 0 & 0 & 3\end{bmatrix}.</math>

The characteristic polynomial of ''A'' is

It can be proved that the geometric multiplicity <math>\gamma_A(\lambda_i)</math> of an eigenvalue never exceeds its algebraic multiplicity <math>\mu_A(\lambda_i)</math>. Therefore, <math>\boldsymbol{\gamma}_A</math> is at most <math>\boldsymbol{\mu}_A</math>.

<math display=block>\det(A - \lambda I) = (1 - \lambda)(2 - \lambda)(3 - \lambda),</math>

which has the roots {{nowrap|1=''λ''₁ = 1}}, {{nowrap|1=''λ''₂ = 2}}, and {{nowrap|1=''λ''₃ = 3}}. These roots are the diagonal elements as well as the eigenvalues of&nbsp;''A''.

If <math>\gamma_A(\lambda_i) = \mu_A(\lambda_i)</math>, then <math>\lambda_i</math> is said to be a ''semisimple eigenvalue''.

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,

====Example====

<math display=block>

For the matrix:

<math>A \mathbf v_{\lambda_1} = \begin{bmatrix} 1\\ 0\\ 0 \end{bmatrix},\quad

\mathbf v_{\lambda_2} = \begin{bmatrix} 0\\ 1\\ 0 \end{bmatrix},\quad

2 & 0 & 0 & 0 \\

\mathbf v_{\lambda_3} = \begin{bmatrix} 0\\ 0\\ 1 \end{bmatrix},

1 & 2 & 0 & 0 \\

</math>

0 & 1 & 3 & 0 \\

0 & 0 & 1 & 3

respectively, as well as scalar multiples of these vectors.

==== Triangular matrix example ====

A matrix whose elements above the main diagonal are all zero is called a ''lower [[triangular matrix]]'', while a matrix whose elements below the main diagonal are all zero is called an ''upper triangular matrix''. As with diagonal matrices, the eigenvalues of triangular matrices are the elements of the main diagonal.

Consider the lower triangular matrix,

<math display=block>A = \begin{bmatrix}

1 & 0 & 0\\

1 & 2 & 0\\

2 & 3 & 3

\end{bmatrix}.</math>

The characteristic polynomial of ''A'' is

<math display=block>\det(A - \lambda I) = (1 - \lambda)(2 - \lambda)(3 - \lambda),</math>

which has the roots {{nowrap|1=''λ''₁ = 1}}, {{nowrap|1=''λ''₂ = 2}}, and {{nowrap|1=''λ''₃ = 3}}. These roots are the diagonal elements as well as the eigenvalues of&nbsp;''A''.

These eigenvalues correspond to the eigenvectors,

<math display=block>

\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},

</math>

respectively, as well as scalar multiples of these vectors.

==== Matrix with repeated eigenvalues example ====

As in the previous example, the lower triangular matrix

<math display=block>A = \begin{bmatrix}

2 & 0 & 0 & 0 \\

1 & 2 & 0 & 0 \\

0 & 1 & 3 & 0 \\

0 & 0 & 1 & 3

\end{bmatrix},</math>

:thehas a characteristic polynomial of <math>A</math>that is <math>\detthe product of (A-\lambdaits I)diagonal \;=\;elements,

<math display=block>\det(A - \lambda I) = \begin{bmatrixvmatrix}

2 - \lambda & 0 & 0 & 0 \\

1 & 2- \lambda & 0 & 0 \\

0 & 1 & 3- \lambda & 0 \\

0 & 0 & 1 & 3- \lambda

\end{bmatrixvmatrix} =
(2 - \lambda)^2 (3 - \lambda)^2 .
</math>,

:being the product of the diagonal with a lower triangular matrix.

The roots of this polynomial, and hence the eigenvalues, are 2 and 3. The ''algebraic multiplicity'' of each eigenvalue is 2; in other words they are both double roots. The sum of the algebraic multiplicities of all distinct eigenvalues is ''μ''_''A'' = 4 = ''n'', the order of the characteristic polynomial and the dimension of ''A''.

The roots of this polynomial, and hence the eigenvalues, are 2 and 3.

The ''algebraic multiplicity'' of each eigenvalue is 2; in other words they are both double roots.

On the other hand, the ''geometric multiplicity'' of the eigenvalue 2 is only 1, because its eigenspace is spanned by the vector <math>[0,1,-1,1]</math>, and is therefore 1 dimensional.

Similarly, the geometric multiplicity of the eigenvalue 3 is 1 because its eigenspace is spanned by <math>[0,0,0,1]</math>. Hence, the total algebraic multiplicity of A, denoted <math>\mu_A</math>, is 4, which is the most it could be for a 4 by 4 matrix. The geometric multiplicity <math>\gamma_A</math> is 2, which is the smallest it could be for a matrix which has two distinct eigenvalues.

On the other hand, the ''geometric multiplicity'' of the eigenvalue 2 is only 1, because its eigenspace is spanned by just one vector <math>\begin{bmatrix} 0 & 1 & -1 & 1 \end{bmatrix}^\textsf{T}</math> and is therefore 1-dimensional. Similarly, the geometric multiplicity of the eigenvalue 3 is 1 because its eigenspace is spanned by just one vector <math>\begin{bmatrix} 0 & 0 & 0 & 1 \end{bmatrix}^\textsf{T}</math>. The total geometric multiplicity ''γ''_''A'' is 2, which is the smallest it could be for a matrix with two distinct eigenvalues. Geometric multiplicities are defined in a later section.

===Diagonalization and eigendecomposition===

If the sum <math>\boldsymbol{\gamma}_A</math> of the geometric multiplicities of all eigenvalues is exactly <math>n</math>, then <math>A</math> has a set of <math>n</math> linearly independent eigenvectors. Let <math>Q</math> be a square matrix whose columns are those eigenvectors, in any order. Then we will have <math>A Q = Q\Lambda </math>, where <math>\Lambda</math> is the diagonal matrix such that <math>\Lambda_{i i}</math> is the eigenvalue associated to column <math>i</math> of <math>Q</math>. Since the columns of <math>Q</math> are linearly independent, the matrix <math>Q</math> is invertible. Premultiplying both sides by <math>Q^{-1}</math> we get <math>Q^{-1}A Q = \Lambda</math>. By definition, therefore, the matrix <math>A</math> is [[diagonalizable matrix|diagonalizable]].

=== Eigenvector-eigenvalue identity ===

Conversely, if <math>A</math> is diagonalizable, let <math>Q</math> be a non-singular square matrix such that <math>Q^{-1} A Q</math> is some diagonal matrix <math>D</math>. Multiplying both sides on the left by <math>Q</math> we get <math>A Q = Q D </math>. Therefore each column of <math>Q</math> must be an eigenvector of <math>A</math>, whose eigenvalue is the corresponding element on the diagonal of <math>D</math>. Since the columns of <math>Q</math> must be linearly independent, it follows that <math>\boldsymbol{\gamma}_A = n</math>. Thus <math>\boldsymbol{\gamma}_A</math> is equal to <math>n</math> if and only if <math>A</math> is diagonalizable.

For a [[Hermitian matrix]] ''A'', the norm squared of the ''α''-th component of a normalized eigenvector can be calculated using only the matrix eigenvalues and the eigenvalues of the corresponding [[Minor (linear algebra)|minor matrix]],

<math display=block>|v_{i\alpha}|^2 = \frac{\prod_{k}{(\lambda_i(A)-\lambda_k(A_\alpha))}}{\prod_{k \neq i}{(\lambda_i(A)-\lambda_k(A))}},</math>

where <math display="inline">A_\alpha</math> is the [[submatrix]] formed by removing the ''α''-th row and column from the original matrix.{{sfn|Wolchover|2019}}{{sfn|Denton|Parke|Tao|Zhang|2022}}{{sfn|Van Mieghem|2014}} This identity also extends to [[Diagonalizable matrix|diagonalizable matrices]], and has been rediscovered many times in the literature.{{sfn|Denton|Parke|Tao|Zhang|2022}}{{sfn|Van Mieghem|2024}}

== Eigenvalues and eigenfunctions of differential operators ==

If <math>A</math> is diagonalizable, the space of all <math>n</math>-element vectors can be decomposed into the direct sum of the eigenspaces of <math>A</math>. This decomposition is called the [[eigendecomposition of a matrix|eigendecomposition]] of <math>A</math>, and it is the preserved under change of coordinates.

{{main|Eigenfunction}}

The definitions of eigenvalue and eigenvectors of a linear transformation ''T'' remains valid even if the underlying vector space is an infinite-dimensional [[Hilbert space|Hilbert]] or [[Banach space]]. A widely used class of linear transformations acting on infinite-dimensional spaces are the [[differential operator]]s on [[function space]]s. Let ''D'' be a linear differential operator on the space '''C'''^∞ of infinitely [[derivative|differentiable]] real functions of a real argument ''t''. The eigenvalue equation for ''D'' is the [[differential equation]]

A matrix that is not diagonalizable is said to be [[defective matrix|defective]]. For defective matrices, the notion of eigenvector can be generalized to [[generalized eigenvector]]s, and that of diagonal matrix to a [[Jordan form]] matrix. Over an algebraically closed field, any matrix <math>A</math> has a [[Jordan form]] and therefore admits a basis of generalized eigenvectors, and a decomposition into [[generalized eigenspace]]s

<math display=block>D f(t) = \lambda f(t)</math>

The functions that satisfy this equation are eigenvectors of ''D'' and are commonly called '''eigenfunctions'''.

===Further properties===

Let <math>A</math> be an arbitrary <math>n\times n</math> matrix of complex numbers with eigenvalues <math>\lambda_1</math>, <math>\lambda_2</math>, ... <math>\lambda_n</math>. (Here it is understood that an eigenvalue with algebraic multiplicity <math>\mu</math> occurs <math>\mu</math> times in this list.) Then

* The [[trace (linear algebra)|trace]] of <math>A</math>, defined as the sum of its diagonal elements, is also the sum of all eigenvalues:

:<math>\operatorname{tr}(A) = \sum_{i=1}^n A_{i i} = \sum_{i=1}^n \lambda_i = \lambda_1+ \lambda_2 +\cdots+ \lambda_n</math>.

* The [[determinant]] of <math>A</math> is the product of all eigenvalues:

:<math>\operatorname{det}(A) = \prod_{i=1}^n \lambda_i=\lambda_1\lambda_2\cdots\lambda_n</math>.

* The eigenvalues of the <math>k</math>th power of <math>A</math>, i.e. the eigenvalues of <math>A^k</math>, for any positive integer <math>k</math>, are <math>\lambda_1^k,\lambda_2^k,\dots,\lambda_n^k</math>

* The matrix <math>A</math> is invertible if and only if all the eigenvalues <math>\lambda_i</math> are nonzero.

* If <math>A</math> is invertible, then the eigenvalues of <math>A^{-1}</math> are <math>1/\lambda_1,1/\lambda_2,\dots,1/\lambda_n</math>

* If <math>A</math> is equal to its [[conjugate transpose]] <math>A^*</math> (in other words, if <math>A</math> is [[Hermitian matrix|Hermitian]]), then every eigenvalue is real. The same is true of any a [[symmetric matrix|symmetric]] real matrix. If <math>A</math> is also [[Positive-definite matrix|positive-definite]], positive-semidefinite, negative-definite, or negative-semidefinite every eigenvalue is positive, non-negative, negative, or non-positive respectively.

* Every eigenvalue of a [[unitary matrix]] has absolute value <math>|\lambda|=1</math>.

=== LeftDerivative andoperator right eigenvectorsexample ===

Consider the derivative operator <math>\tfrac{d}{dt}</math> with eigenvalue equation

{{see also|left and right (algebra)}}

<math display=block>\frac{d}{dt}f(t) = \lambda f(t).</math>

The use of matrices with a single column (rather than a single row) to represent vectors is traditional in many disciplines. For that reason, the word "eigenvector" almost always means a '''right eigenvector''', namely a ''column'' vector that must be placed to the ''right'' of the matrix <math>A</math> in the defining equation

:<math>A v = \lambda v</math>.

This differential equation can be solved by multiplying both sides by ''dt''/''f''(''t'') and [[Integration (calculus)|integrating]]. Its solution, the [[exponential function]]

There may be also single-''row'' vectors that are unchanged when they occur on the ''left'' side of a product with a square matrix <math>A</math>; that is, which satisfy the equation

:<math>u Adisplay=block>f(t) = f(0)e^{\lambda ut},</math>

is the eigenfunction of the derivative operator. In this case the eigenfunction is itself a function of its associated eigenvalue. In particular, for ''λ'' = 0 the eigenfunction ''f''(''t'') is a constant.

Any such row vector <math>u</math> is called a '''left eigenvector''' of <math>A</math>.

The main [[eigenfunction]] article gives other examples.

== General definition ==

The concept of eigenvalues and eigenvectors extends naturally to arbitrary [[linear map|linear transformations]] on arbitrary vector spaces. Let ''V'' be any vector space over some [[field (algebra)|field]] ''K'' of [[scalar (mathematics)|scalars]], and let ''T'' be a linear transformation mapping ''V'' into ''V'',

<math display=block>T:V \to V.</math>

We say that a nonzero vector '''v''' ∈ ''V'' is an '''eigenvector''' of ''T'' if and only if there exists a scalar ''λ'' ∈ ''K'' such that

{{NumBlk|:

|<math>T(\mathbf{v}) = \lambda \mathbf{v}.</math>

| {{EquationRef|5}}

}}

This equation is called the eigenvalue equation for ''T'', and the scalar ''λ'' is the '''eigenvalue''' of ''T'' corresponding to the eigenvector '''v'''. ''T''('''v''') is the result of applying the transformation ''T'' to the vector '''v''', while ''λ'''''v''' is the product of the scalar ''λ'' with '''v'''.{{sfn|Korn|Korn|2000|loc=Section 14.3.5a}}{{sfn|Friedberg|Insel|Spence|1989|loc=p. 217}}

=== Eigenspaces, geometric multiplicity, and the eigenbasis ===

Given an eigenvalue ''λ'', consider the set

<math display=block>E = \left\{\mathbf{v} : T(\mathbf{v}) = \lambda \mathbf{v}\right\},</math>

which is the union of the zero vector with the set of all eigenvectors associated with&nbsp;''λ''. ''E'' is called the '''eigenspace''' or '''characteristic space''' of ''T'' associated with&nbsp;''λ''.<ref>{{harvnb|Roman|2008|loc=p. 186 §8}}</ref>

By definition of a linear transformation,

<math display=block>\begin{align}

T(\mathbf{x} + \mathbf{y}) &= T(\mathbf{x}) + T(\mathbf{y}),\\

T(\alpha \mathbf{x}) &= \alpha T(\mathbf{x}),

\end{align}</math>

for '''x''',&nbsp;'''y'''&nbsp;∈ ''V'' and ''α''&nbsp;∈ ''K''. Therefore, if '''u''' and '''v''' are eigenvectors of ''T'' associated with eigenvalue ''λ'', namely '''u''',&nbsp;'''v'''&nbsp;∈ ''E'', then

<math display=block>\begin{align}

T(\mathbf{u} + \mathbf{v}) &= \lambda (\mathbf{u} + \mathbf{v}),\\

T(\alpha \mathbf{v}) &= \lambda (\alpha \mathbf{v}).

\end{align}</math>

So, both '''u''' + '''v''' and α'''v''' are either zero or eigenvectors of ''T'' associated with ''λ'', namely '''u''' + '''v''', ''α'''''v''' ∈ ''E'', and ''E'' is closed under addition and scalar multiplication. The eigenspace ''E'' associated with ''λ'' is therefore a linear subspace of ''V''.<ref>{{harvnb|Nering|1970|p=107}}; {{harvnb|Shilov|1977|p=109}} [[b:The Book of Mathematical Proofs/Algebra/Linear Transformations#Lemma for the eigenspace|Lemma for the eigenspace]]</ref>

If that subspace has dimension 1, it is sometimes called an '''eigenline'''.{{sfn|Lipschutz|Lipson|2002|p=111}}

The '''geometric multiplicity''' ''γ''_''T''(''λ'') of an eigenvalue ''λ'' is the dimension of the eigenspace associated with ''λ'', i.e., the maximum number of linearly independent eigenvectors associated with that eigenvalue.{{sfn|Nering|1970|p=107}}{{sfn|Golub|Van Loan|1996|p=316}}{{sfn|Roman|2008|loc=p. 189 §8}} By the definition of eigenvalues and eigenvectors, ''γ''_''T''(''λ'') ≥ 1 because every eigenvalue has at least one eigenvector.

The eigenspaces of ''T'' always form a [[direct sum]]. As a consequence, eigenvectors of ''different'' eigenvalues are always linearly independent. Therefore, the sum of the dimensions of the eigenspaces cannot exceed the dimension ''n'' of the vector space on which ''T'' operates, and there cannot be more than ''n'' distinct eigenvalues.{{efn|For a proof of this lemma, see {{Harvnb|Roman|2008|loc=Theorem 8.2 on p. 186}}; {{Harvnb|Shilov|1977|loc=p. 109}}; {{Harvnb|Hefferon|2001|loc=p. 364}}; {{Harvnb|Beezer|2006|loc=Theorem EDELI on p. 469}}; and [[b:Famous Theorems of Mathematics/Algebra/Linear Transformations#Lemma for linear independence of eigenvectors|Lemma for linear independence of eigenvectors]]}}

Any subspace spanned by eigenvectors of ''T'' is an [[invariant subspace]] of ''T'', and the restriction of ''T'' to such a subspace is diagonalizable. Moreover, if the entire vector space ''V'' can be spanned by the eigenvectors of ''T'', or equivalently if the direct sum of the eigenspaces associated with all the eigenvalues of ''T'' is the entire vector space ''V'', then a basis of ''V'' called an '''eigenbasis''' can be formed from linearly independent eigenvectors of ''T''. When ''T'' admits an eigenbasis, ''T'' is diagonalizable.

=== Spectral theory ===

{{main|Spectral theory}}

If ''λ'' is an eigenvalue of ''T'', then the operator (''T'' − ''λI'') is not [[One to one correspondence|one-to-one]], and therefore its inverse (''T'' − ''λI'')⁻¹ does not exist. The converse is true for finite-dimensional vector spaces, but not for infinite-dimensional vector spaces. In general, the operator (''T'' − ''λI'') may not have an inverse even if ''λ'' is not an eigenvalue.

For this reason, in [[functional analysis]] eigenvalues can be generalized to the [[spectrum (functional analysis)|spectrum of a linear operator]] ''T'' as the set of all scalars ''λ'' for which the operator (''T'' − ''λI'') has no [[bounded operator|bounded]] inverse. The spectrum of an operator always contains all its eigenvalues but is not limited to them.

=== Associative algebras and representation theory ===

{{main|Weight (representation theory)}}

One can generalize the algebraic object that is acting on the vector space, replacing a single operator acting on a vector space with an [[algebra representation]] – an [[associative algebra]] acting on a [[module (mathematics)|module]]. The study of such actions is the field of [[representation theory]].

The [[weight (representation theory)|representation-theoretical concept of weight]] is an analog of eigenvalues, while ''weight vectors'' and ''weight spaces'' are the analogs of eigenvectors and eigenspaces, respectively.

[[Hecke eigensheaf]] is a tensor-multiple of itself and is considered in [[Langlands correspondence]].

== Dynamic equations ==

The simplest [[difference equation]]s have the form

: <math>x_t = a_1 x_{t-1} + a_2 x_{t-2} + \cdots + a_k x_{t-k}.</math>

The solution of this equation for ''x'' in terms of ''t'' is found by using its characteristic equation

: <math>\lambda^k - a_1\lambda^{k-1} - a_2\lambda^{k-2} - \cdots - a_{k-1}\lambda-a_k = 0,</math>

which can be found by stacking into matrix form a set of equations consisting of the above difference equation and the ''k''&nbsp;–&nbsp;1 equations <math>x_{t-1} = x_{t-1},\ \dots,\ x_{t-k+1} = x_{t-k+1},</math> giving a ''k''-dimensional system of the first order in the stacked variable vector <math>\begin{bmatrix} x_t & \cdots & x_{t-k+1} \end{bmatrix}</math> in terms of its once-lagged value, and taking the characteristic equation of this system's matrix. This equation gives ''k'' characteristic roots <math>\lambda_1,\, \ldots,\, \lambda_k,</math> for use in the solution equation

The left eigenvectors of <math>A</math> are transposes of the right eigenvectors of the transposed matrix <math>A^\mathsf{T}</math>, since their defining equation is equivalent to

: <math>A^\mathsf{T}x_t u^= c_1\mathsf{T}lambda_1^t =+ \lambdacdots u^+ c_k\mathsf{T}lambda_k^t.</math>

A similar procedure is used for solving a [[differential equation]] of the form

It follows that, if <math>A</math> is [[Hermitian matrix|Hermitian]], its left and right eigenvectors are [[complex conjugate vector space|complex conjugates]]. In particular if <math>A</math> is a real symmetric matrix, they are the same except for transposition.

: <math>\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.</math>

== Calculation ==

{{main|Eigenvalue algorithm}}

The calculation of eigenvalues and eigenvectors is a topic where theory, as presented in elementary linear algebra textbooks, is often very far from practice.

=== Classical method ===

===Computing the eigenvalues===

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 [[floating-point]].

The eigenvalues of a matrix <math>A</math> can be determined by finding the roots of the characteristic polynomial. Explicit [[algebraic solution|algebraic formulas]] for the roots of a polynomial exist only if the degree <math>n</math> is 4 or less. According to the [[Abel–Ruffini theorem]] there is no general, explicit and exact algebraic formula for the roots of a polynomial with degree 5 or more.

==== Eigenvalues ====

It turns out that any polynomial with degree <math>n</math> is the characteristic polynomial of some [[companion matrix]] of order <math>n</math>. 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 [[numerical method]]s.

The eigenvalues of a matrix <math>A</math> can be determined by finding the roots of the characteristic polynomial. This is easy for <math> 2 \times 2 </math> matrices, but the difficulty increases rapidly with the size of the matrix.

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 [[accuracy]].<ref name=TrefethenBau/> {{sfn|Trefethen|Bau|1997}} However, this approach is not viable in practice because the coefficients would be contaminated by unavoidable [[round-off error]]s, and the roots of a polynomial can be an extremely sensitive function of the coefficients (as exemplified by [[Wilkinson's polynomial]]).{{sfn|Trefethen|Bau|1997}} Even for matrices whose elements are integers the calculation becomes nontrivial, because the sums are very long; the constant term is the [[determinant]], which for an <refmath> name=TrefethenBaun \times n </math> matrix is a sum of <math> n! </math> different products.{{Citationefn|first1=LloydBy N.doing |last1=Trefethen[[Gaussian |first2=elimination]] David|last2=over Bau|title=Numerical[[formal Linearpower Algebra|publisher=SIAM|year=1997}}series]] truncated to <math>n</refmath> terms it is possible to get away with <math>O(n^4)</math> operations, but that does not take [[combinatorial explosion]] into account.}}

Explicit [[algebraic solution|algebraic formulas]] for the roots of a polynomial exist only if the degree <math>n</math> is 4 or less. According to the [[Abel–Ruffini theorem]] 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 <math>n</math> is the characteristic polynomial of some [[companion matrix]] of order <math>n</math>.) 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 [[numerical method]]s. Even the [[Cubic function#General solution to the cubic equation with real coefficients|exact formula]] for the roots of a degree 3 polynomial is numerically impractical.

Efficient, accurate methods to compute eigenvalues and eigenvectors of arbitrary matrices were not known until the advent of the [[QR algorithm]] in 1961.

<ref name=TrefethenBau/> Combining the [[Householder transformation]] with the LU decomposition results in an algorithm with better convergence than the QR algorithm.{{citation needed|date=March 2013}} For large [[Hermitian matrix|Hermitian]] [[sparse matrix|sparse matrices]], the [[Lanczos algorithm]] is one example of an efficient [[iterative method]] to compute eigenvalues and eigenvectors, among several other possibilities.<ref name=TrefethenBau/>

==== Eigenvectors ====

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 [[linear system|system of linear equations]] with known coefficients. For example, once it is known that 6 is an eigenvalue of the matrix

<math display=block>A = \begin{bmatrix} 4 & 1\\ 6 & 3\end{bmatrix}</math>

===Computing the eigenvectors===

Once the (exact) value of an eigenvalue is known, the corresponding eigenvectors can be found by finding non-zero solutions of the eigenvalue equation, that becomes a [[linear system|system of linear equations]] with known coefficients. For example, once it is known that 6 is an eigenvalue of the matrix

:<math>A = \begin{bmatrix} 4 & 1\\6 & 3 \end{bmatrix}</math>

we can find its eigenvectors by solving the equation <math>A v = 6 v</math>, that is

:<math display=block>\begin{bmatrix} 4 & 1\\ 6 & 3 \end{bmatrix}\begin{bmatrix}x \\y\end{bmatrix} = 6 \cdot \begin{bmatrix}x \\y\end{bmatrix}</math>

This matrix equation is equivalent to two [[linear equation]]s

:<math display=block>

\left\{ \begin{matrixaligned} 4x + {\ }y &{}= 6x \\ 6x + 3y &{}=6 y6y\end{matrixaligned} \right.

\quad\quad\quad</math> {{spaces|4}} that is {{spaces|4}} <math>

\left\{ \begin{matrixaligned} -2x + {\ }y &{}= 0 \\+ 6x - 3y &{}= 0\end{matrixaligned} \right.

</math>

Both equations reduce to the single linear equation <math>y=2x</math>. Therefore, any vector of the form <math>[a,2a]'</math>, for any non-zero real number <math>a</math>, is an eigenvector of <math>A</math> with eigenvalue <math>\lambda = 6</math>.

The matrix <math>A</math> above has another eigenvalue <math>\lambda=1</math>. A similar calculation shows that the corresponding eigenvectors are the non-zero solutions of <math>3x+y=0</math>, that is, any vector of the form <math>[b,-3b]'</math>, for any non-zero real number <math>b</math>.

Some numeric methods that compute the eigenvalues of a matrix also determine a set of corresponding eigenvectors as a by-product of the computation.

==History==

Eigenvalues are often introduced in the context of [[linear algebra]] or [[matrix (mathematics)|matrix theory]]. Historically, however, they arose in the study of [[quadratic form]]s and [[differential equation]]s.

Both equations reduce to the single linear equation <math>y=2x</math>. Therefore, any vector of the form <math>\begin{bmatrix} a & 2a \end{bmatrix}^\textsf{T}</math>, for any nonzero real number <math>a</math>, is an eigenvector of <math>A</math> with eigenvalue <math>\lambda = 6</math>.

In the 18th century [[Leonhard Euler|Euler]] studied the rotational motion of a [[rigid body]] and discovered the importance of the [[Principal axis (mechanics)|principal axes]]. [[Lagrange]] realized that the principal axes are the eigenvectors of the inertia matrix.<ref>See {{Harvnb|Hawkins|1975|loc=§2}}</ref> In the early 19th century, [[Augustin Louis Cauchy|Cauchy]] saw how their work could be used to classify the [[quadric surface]]s, and generalized it to arbitrary dimensions.<ref name="hawkins3">See {{Harvnb|Hawkins|1975|loc=§3}}</ref> Cauchy also coined the term ''racine caractéristique'' (characteristic root) for what is now called ''eigenvalue''; his term survives in ''[[Characteristic polynomial#Characteristic equation|characteristic equation]]''.<ref name="kline807">See {{Harvnb|Kline|1972|loc=pp. 807–808}}</ref>

The matrix <math>A</math> above has another eigenvalue <math>\lambda=1</math>. A similar calculation shows that the corresponding eigenvectors are the nonzero solutions of <math>3x+y=0</math>, that is, any vector of the form <math>\begin{bmatrix} b & -3b \end{bmatrix}^\textsf{T}</math>, for any nonzero real number <math>b</math>.

[[Joseph Fourier|Fourier]] used the work of Laplace and Lagrange to solve the [[heat equation]] by [[separation of variables]] in his famous 1822 book ''[[Théorie analytique de la chaleur]]''.<ref>See {{Harvnb|Kline|1972|loc=p. 673}}</ref> [[Jacques Charles François Sturm|Sturm]] 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 symmetric matrices have real eigenvalues.<ref name="hawkins3" /> This was extended by [[Charles Hermite|Hermite]] in 1855 to what are now called [[Hermitian matrix|Hermitian matrices]].<ref name="kline807" /> Around the same time, [[Francesco Brioschi|Brioschi]] proved that the eigenvalues of [[orthogonal matrix|orthogonal matrices]] lie on the [[unit circle]],<ref name="hawkins3" /> and [[Alfred Clebsch|Clebsch]] found the corresponding result for [[skew-symmetric matrix|skew-symmetric matrices]].<ref name="kline807" /> Finally, [[Karl Weierstrass|Weierstrass]] clarified an important aspect in the [[stability theory]] started by Laplace by realizing that [[defective matrix|defective matrices]] can cause instability.<ref name="hawkins3" />

=== Simple iterative methods ===

In the meantime, [[Joseph Liouville|Liouville]] studied eigenvalue problems similar to those of Sturm; the discipline that grew out of their work is now called ''[[Sturm–Liouville theory]]''.<ref>See {{Harvnb|Kline|1972|loc=pp. 715–716}}</ref> [[Hermann Schwarz|Schwarz]] studied the first eigenvalue of [[Laplace's equation]] on general domains towards the end of the 19th century, while [[Henri Poincaré|Poincaré]] studied [[Poisson's equation]] a few years later.<ref>See {{Harvnb|Kline|1972|loc=pp. 706–707}}</ref>

{{main|Power iteration}}

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 (optionally normalizing the vector to keep its elements of reasonable size); this makes the vector converge towards an eigenvector. [[inverse iteration|A variation]] is to instead multiply the vector by {{nowrap|<math>(A - \mu I)^{-1}</math>;}} this causes it to converge to an eigenvector of the eigenvalue closest to {{nowrap|<math>\mu \in \mathbb{C}</math>.}}

If <math>\mathbf{v}</math> is (a good approximation of) an eigenvector of <math>A</math>, then the corresponding eigenvalue can be computed as

At the start of the 20th century, [[David Hilbert|Hilbert]] studied the eigenvalues of [[integral operator]]s by viewing the operators as infinite matrices.<ref>See {{Harvnb|Kline|1972|loc=p. 1063}}</ref> He was the first to use the [[German language|German]] word ''eigen'' to denote eigenvalues and eigenvectors in 1904, though he may have been following a related usage by [[Helmholtz]]. For some time, the standard term in English was "proper value", but the more distinctive term "eigenvalue" is standard today.<ref>See {{Harvnb|Aldrich|2006}}</ref>

: <math> \lambda = \frac{\mathbf{v}^* A\mathbf{v}}{\mathbf{v}^* \mathbf{v}}</math>

where <math>\mathbf{v}^*</math> denotes the [[conjugate transpose]] of <math>\mathbf{v}</math>.

=== Modern methods ===

The first numerical algorithm for computing eigenvalues and eigenvectors appeared in 1929, when [[Richard Edler von Mises|Von Mises]] published the [[power method]]. One of the most popular methods today, the [[QR algorithm]], was proposed independently by [[John G.F. Francis]]<ref>{{Citation|first=J. G. F. |last=Francis|title=The QR Transformation, I (part 1)|journal=The Computer Journal|volume= 4|issue= 3|pages =265–271 |year=1961|doi=10.1093/comjnl/4.3.265}} and {{Citation|doi=10.1093/comjnl/4.4.332|first=J. G. F. |last=Francis|title=The QR Transformation, II (part 2)|journal=The Computer Journal|volume=4|issue= 4| pages= 332–345|year=1962}}</ref> and [[Vera Kublanovskaya]]<ref>{{Citation|first=Vera N. |last=Kublanovskaya|title=On some algorithms for the solution of the complete eigenvalue problem|journal=USSR Computational Mathematics and Mathematical Physics|volume= 3| pages= 637–657 |year=1961}}. Also published in: {{Citation|journal=Zhurnal Vychislitel'noi Matematiki i Matematicheskoi Fiziki|volume=1|issue=4| pages =555–570 |year=1961}}</ref> in 1961.<ref>See {{Harvnb|Golub|van Loan|1996|loc=§7.3}}; {{Harvnb|Meyer|2000|loc=§7.3}}</ref>

Efficient, accurate methods to compute eigenvalues and eigenvectors of arbitrary matrices were not known until the [[QR algorithm]] was designed in 1961.{{sfn|Trefethen|Bau|1997}} Combining the [[Householder transformation]] with the LU decomposition results in an algorithm with better convergence than the QR algorithm.{{citation needed|date=March 2013}} For large [[Hermitian matrix|Hermitian]] [[sparse matrix|sparse matrices]], the [[Lanczos algorithm]] is one example of an efficient [[iterative method]] to compute eigenvalues and eigenvectors, among several other possibilities.{{sfn|Trefethen|Bau|1997}}

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.

==Applications==

== Applications ==

===Eigenvalues of geometric transformations===

=== Geometric transformations ===

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.

{| class="wikitable" style="text-align:center; margin:1em auto 1em auto;"

|+ Eigenvalues of geometric transformations

|-

!

|

! scope="col" | [[Scaling (geometry)|scalingScaling]]

! scope="col" | unequalUnequal scaling

! scope="col" | [[Rotation (geometry)|rotationRotation]]

! scope="col" | [[Shear mapping|horizontalHorizontal shear]]

! scope="col" | [[hyperbolicHyperbolic rotation]]

|-

! scope="row" | Illustration

|illustration

|| [[File:Homothety in two dim.svg|100px|alt=Equal scaling ([[Homothetic transformation|homothety]])]]

|| [[File:Unequal scaling.svg|100px|alt=Vertical shrink (<math>k_2 < 1</math>) and horizontal stretch (<math>k_1 > 1</math>) of a unit square.]]

|| [[File:Rotation.png|100px|alt=Rotation by 50 degrees]]

|| [[File:Shear.svg|100px|center|alt=Horizontal shear mapping]]

|| [[File:Squeeze r=1.5.svg|100px|<math>e^\mathsf{T} = \frac 3 2</math>]]

|- style="vertical-align:top"

! scope="row" | Matrix

| <math>\begin{bmatrix}k & 0\\ 0 & k\end{bmatrix}</math>

| <math>\begin{bmatrix}k_1 & 0\\ 0 & k_2\end{bmatrix}</math>

| <math>\begin{bmatrix}\cos\theta & -\sin\theta\\ \sin\theta & \cos\theta\end{bmatrix}</math>

| <math> \begin{bmatrix}1 & k\\ 0 & 1\end{bmatrix}</math>

| <math>\begin{bmatrix}\cosh\varphi & \sinh\varphi\\ \sinh\varphi & \cosh\varphi\end{bmatrix}</math>

|-

! scope="row" | Characteristic<br />polynomial

|matrix

| <math> \begin{bmatrix}k & 0\\0 & k\end{bmatrix}</math><br />&nbsp;<br />&nbsp;

| <math> \begin{bmatrix}k_1 & 0\\0 & k_2\end{bmatrix}</math><br />&nbsp;<br />&nbsp;

| <math> \begin{bmatrix}c & -s \\ s & c\end{bmatrix} </math><br /><math>c=\cos\theta</math><br /><math>s=\sin\theta</math>

| <math> \begin{bmatrix}1 & k\\ 0 & 1\end{bmatrix} </math><br />&nbsp;<br />&nbsp;

| <math>\begin{bmatrix} c & s \\ s & c \end{bmatrix}</math><br /><math>c=\cosh \varphi</math><br /><math>s=\sinh \varphi</math>

|-

|characteristic<br />polynomial

| <math>\ (\lambda - k)^2</math>

| <math>(\lambda - k_1)(\lambda - k_2)</math>

| <math>\lambda^2 - 2c2\cos(\theta)\lambda + 1</math>

| <math>\ (\lambda - 1)^2</math>

| <math>\lambda^2 - 2c2\cosh(\varphi)\lambda + 1</math>

|-

! scope="row" |eigenvalues Eigenvalues, <math>\lambda_i</math>

| <math>\lambda_1 = \lambda_2 = k</math>

| <math>\begin{align}\lambda_1 &= k_1</math><br /><math>\\ \lambda_2 &= k_2\end{align}</math>

| <math>\begin{align}\lambda_1 &= e^{\mathbf{i}\theta} \\ &=c+s \mathbf{cos\theta + i}</math><br\sin\theta \\ /><math>\lambda_2 &= e^{-\mathbf{i}\theta} \\ &=c \cos\theta -s i\mathbfsin\theta \end{ialign}</math>

| <math>\lambda_1 = \lambda_2 = 1</math>

| <math>\begin{align}\lambda_1 &= e^\varphi</math><br /><math>\\ &= \cosh\varphi + \sinh\varphi \\ \lambda_2 &= e^{-\varphi} \\ &= \cosh\varphi - \sinh\varphi \end{align}</math>,

|-

! scope="row" |algebraic multiplAlgebraic {{abbr|mult.|multiplicity}},<br /><math>\mu_i = \mu(\lambda_i)</math>

| <math>\mu_1 = 2</math>

| <math>\begin{align}\mu_1 &= 1</math><br /><math>\\ \mu_2 &= 1 \end{align}</math>

| <math>\begin{align}\mu_1 &= 1</math><br /><math>\\ \mu_2 &= 1 \end{align}</math>

| <math>\mu_1 = 2</math>

| <math>\begin{align}\mu_1 &= 1</math><br /><math>\\ \mu_2 &= 1 \end{align}</math>

|-

! scope="row" |geometric multiplGeometric {{abbr|mult.|multiplicity}},<br /><math>\gamma_i = \gamma(\lambda_i)</math>

| <math>\gamma_1 = 2</math>

| <math>\begin{align}\gamma_1 &= 1</math><br /><math>\\ \gamma_2 &= 1 \end{align}</math>

| <math>\begin{align}\gamma_1 &= 1</math><br /><math>\\ \gamma_2 &= 1 \end{align}</math>

| <math>\gamma_1 = 1</math>

| <math>\begin{align}\gamma_1 &= 1</math><br /><math>\\ \gamma_2 &= 1 \end{align}</math>

|-

! scope="row" | Eigenvectors

|eigenvectors

| All non-zerononzero vectors

| <math>\begin{align}

|<math>u_1 = \begin{bmatrix}1\\0\end{bmatrix}</math><br /><math>u_2 = \begin{bmatrix}0\\1\end{bmatrix}</math>

|<math>u_1 = \begin{bmatrix}{\ }1\\-\mathbf{i}\end{bmatrix}</math><br /><math>u_2u_1 &= \begin{bmatrix}{\ }1\\ +\mathbf{i}0\end{bmatrix}</math> \\

|<math>u_1 \mathbf u_2 &= \begin{bmatrix}1 0\\0 1\end{bmatrix}</math>

\end{align}</math>

|<math>u_1 = \begin{bmatrix}{\ }1\\{\ }1\end{bmatrix}</math><br /><math>u_2 = \begin{bmatrix}{\ }1\\-1\end{bmatrix}.</math>

| <math>\begin{align}

\mathbf u_1 &= \begin{bmatrix} 1\\ -i\end{bmatrix} \\

\mathbf u_2 &= \begin{bmatrix} 1\\ +i\end{bmatrix}

\end{align}</math>

| <math>\mathbf u_1 = \begin{bmatrix} 1\\ 0 \end{bmatrix}</math>

| <math>\begin{align}

\mathbf u_1 &= \begin{bmatrix} 1\\ 1\end{bmatrix} \\

\mathbf u_2 &= \begin{bmatrix} 1\\ -1\end{bmatrix}

\end{align}</math>

|}

Note that theThe characteristic equation for a rotation is a [[quadratic equation]] with [[discriminant]] <math>D = -4(\sin\theta)^2</math>, which is a negative number whenever <math>\theta</math>{{mvar|θ}} is not an integer multiple of 180°. Therefore, except for these special cases, the two eigenvalues are complex numbers, <math>\cos\theta \pm \mathbf{i}\sin\theta</math>; 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.

A linear transformation that takes a square to a rectangle of the same area (a [[squeeze mapping]]) has reciprocal eigenvalues.

===Schrödinger equation===<!-- This section is linked from [[Eigenstate]] -->

=== Principal component analysis ===

[[File:HAtomOrbitals.png|thumb|271px|The [[wavefunction]]s associated with the [[bound state]]s of an [[electron]] in a [[hydrogen atom]] can be seen as the eigenvectors of the [[hydrogen atom|hydrogen atom Hamiltonian]] as well as of the [[angular momentum operator]]. They are associated with eigenvalues interpreted as their energies (increasing downward: <math>n=1,2,3,\ldots</math>) and [[angular momentum]] (increasing across: <!-- do not italicize! -->s, p, d, ...). The illustration shows the square of the absolute value of the wavefunctions. Brighter areas correspond to higher [[probability density function|probability density]] for a position [[measurement in quantum mechanics|measurement]]. The center of each figure is the [[atomic nucleus]], a [[proton]].]]

[[File:GaussianScatterPCA.png|thumb|right|PCA of the [[multivariate Gaussian distribution]] centered at <math>(1, 3)</math> with a standard deviation of 3 in roughly the <math>(0.878, 0.478)</math> direction and of&nbsp;1 in the orthogonal direction. The vectors shown are unit eigenvectors of the (symmetric, positive-semidefinite) [[covariance matrix]] scaled by the square root of the corresponding eigenvalue. Just as in the one-dimensional case, the square root is taken because the [[standard deviation]] is more readily visualized than the [[variance]].]]

{{Main|Principal component analysis}}

{{See also|Positive semidefinite matrix|Factor analysis}}

The [[Eigendecomposition of a matrix#Real symmetric matrices|eigendecomposition]] of a [[symmetric matrix|symmetric]] [[positive semidefinite matrix|positive semidefinite]] (PSD) [[positive semidefinite matrix|matrix]] yields an [[orthogonal basis]] of eigenvectors, each of which has a nonnegative eigenvalue. The orthogonal decomposition of a PSD matrix is used in [[multivariate statistics|multivariate analysis]], where the [[sample variance|sample]] [[covariance matrix|covariance matrices]] are PSD. This orthogonal decomposition is called [[principal component analysis]] (PCA) in statistics. PCA studies [[linear relation]]s among variables. PCA is performed on the [[covariance matrix]] or the [[correlation matrix]] (in which each variable is scaled to have its [[sample variance]] equal to one). For the covariance or correlation matrix, the eigenvectors correspond to [[principal component analysis|principal components]] and the eigenvalues to the [[explained variance|variance explained]] by the principal components. Principal component analysis of the correlation matrix provides an [[orthogonal basis]] 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.

An example of an eigenvalue equation where the transformation <math>T</math> is represented in terms of a differential operator is the time-independent [[Schrödinger equation]] in [[quantum mechanics]]:

Principal component analysis is used as a means of [[dimensionality reduction]] in the study of large [[data set]]s, such as those encountered in [[bioinformatics]]. In [[Q methodology]], the eigenvalues of the correlation matrix determine the Q-methodologist's judgment of ''practical'' significance (which differs from the [[statistical significance]] of [[hypothesis testing]]; cf. [[Scree's test|criteria for determining the number of factors]]). More generally, principal component analysis can be used as a method of [[factor analysis]] in [[structural equation model]]ing.

: <math>H\psi_E = E\psi_E \,</math>

=== Graphs ===

where <math>H</math>, the [[Hamiltonian (quantum mechanics)|Hamiltonian]], is a second-order [[differential operator]] and <math>\psi_E</math>, the [[wavefunction]], is one of its eigenfunctions corresponding to the eigenvalue <math>E</math>, interpreted as its [[energy]].

In [[spectral graph theory]], an eigenvalue of a [[graph theory|graph]] is defined as an eigenvalue of the graph's [[adjacency matrix]] <math>A</math>, or (increasingly) of the graph's [[Laplacian matrix]] due to its [[discrete Laplace operator]], which is either <math>D - A</math> (sometimes called the ''combinatorial Laplacian'') or <math>I - D^{-1/2}A D^{-1/2}</math> (sometimes called the ''normalized Laplacian''), where <math>D</math> is a diagonal matrix with <math>D_{ii}</math> equal to the degree of vertex <math>v_i</math>, and in <math>D^{-1/2}</math>, the <math>i</math>th diagonal entry is <math display="inline">1/\sqrt{\deg(v_i)}</math>. The <math>k</math>th principal eigenvector of a graph is defined as either the eigenvector corresponding to the <math>k</math>th largest or <math>k</math>th smallest eigenvalue of the Laplacian. The first principal eigenvector of the graph is also referred to merely as the principal eigenvector.

The principal eigenvector is used to measure the [[eigenvector centrality|centrality]] of its vertices. An example is [[Google]]'s [[PageRank]] algorithm. The principal eigenvector of a modified [[adjacency matrix]] of the World Wide Web graph gives the page ranks as its components. This vector corresponds to the [[stationary distribution]] of the [[Markov chain]] 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 [[spectral clustering]]. Other methods are also available for clustering.

However, in the case where one is interested only in the [[bound state]] solutions of the Schrödinger equation, one looks for <math>\psi_E</math> within the space of [[Square-integrable function|square integrable]] functions. Since this space is a [[Hilbert space]] with a well-defined [[scalar product]], one can introduce a [[Basis (linear algebra)|basis set]] in which <math>\psi_E</math> and <math>H</math> can be represented as a one-dimensional array and a matrix respectively. This allows one to represent the Schrödinger equation in a matrix form.

=== Markov chains ===

[[Bra-ket notation]] 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 <math>|\Psi_E\rangle</math>. In this notation, the Schrödinger equation is:

A [[Markov chain]] is represented by a matrix whose entries are the [[transition probabilities]] 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 [[Perron–Frobenius theorem]] gives sufficient conditions for a Markov chain to have a unique dominant eigenvalue, which governs the convergence of the system to a steady state.

=== Vibration analysis ===

: <math>H|\Psi_E\rangle = E|\Psi_E\rangle</math>

[[File:Mode Shape of a Tuning Fork at Eigenfrequency 440.09 Hz.gif|thumb|Mode shape of a tuning fork at eigenfrequency 440.09&nbsp;Hz]]

{{Main|Vibration}}

Eigenvalue problems occur naturally in the vibration analysis of mechanical structures with many [[Degrees of freedom (mechanics)|degrees of freedom]]. The eigenvalues are the [[Natural frequency|natural frequencies]] (or '''eigenfrequencies''') of vibration, and the eigenvectors are the shapes of these vibrational modes. In particular, undamped vibration is governed by

where <math>|\Psi_E\rangle</math> is an '''eigenstate''' of <math>H</math>. It is a [[self adjoint operator]], the infinite dimensional analog of Hermitian matrices (''see [[Observable]]''). As in the matrix case, in the equation above <math>H|\Psi_E\rangle</math> is understood to be the vector obtained by application of the transformation <math>H</math> to <math>|\Psi_E\rangle</math>.

<math display=block>m\ddot{x} + kx = 0</math>

or

<math display=block>m\ddot{x} = -kx</math>

That is, acceleration is proportional to position (i.e., we expect <math>x</math> to be sinusoidal in time).

===Molecular orbitals===

In [[quantum mechanics]], and in particular in [[atomic physics|atomic]] and [[molecular physics]], within the [[Hartree–Fock]] theory, the [[atomic orbital|atomic]] and [[molecular orbital]]s can be defined by the eigenvectors of the [[Fock operator]]. The corresponding eigenvalues are interpreted as [[ionization potential]]s via [[Koopmans' theorem]]. 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. If one wants to underline this aspect one speaks of nonlinear eigenvalue problem. Such equations are usually solved by an [[iteration]] procedure, called in this case [[self-consistent field]] method. In [[quantum chemistry]], one often represents the Hartree–Fock equation in a non-[[orthogonal]] [[basis set (chemistry)|basis set]]. This particular representation is a [[generalized eigenvalue problem]] called [[Roothaan equations]].

In <math>n</math> dimensions, <math>m</math> becomes a [[mass matrix]] and <math>k</math> a [[stiffness matrix]]. Admissible solutions are then a linear combination of solutions to the [[generalized eigenvalue problem]]

===Geology and glaciology===

<math display=block>kx = \omega^2 mx</math>

In [[geology]], especially in the study of [[glacial till]], 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 [[clasts]] in a soil sample, which can only be compared graphically such as in a Tri-Plot (Sneed and Folk) diagram,<ref>{{Citation|doi=10.1002/1096-9837(200012)25:13<1473::AID-ESP158>3.0.CO;2-C|last1=Graham|first1=D.|last2=Midgley|first2= N.|title=Graphical representation of particle shape using triangular diagrams: an Excel spreadsheet method|year= 2000|journal= Earth Surface Processes and Landforms |volume=25|pages=1473–1477|issue=13}}</ref><ref>{{Citation|doi=10.1086/626490|last1=Sneed|first1= E. D.|last2=Folk|first2= R. L.|year= 1958|title=Pebbles in the lower Colorado River, Texas, a study of particle morphogenesis|journal= Journal of Geology|volume= 66|issue=2|pages=114–150}}</ref> or as a Stereonet on a Wulff Net.<ref>{{Citation |doi=10.1016/S0098-3004(97)00122-2 |last1=Knox-Robinson |year=1998 |first1=C |pages=243 |volume=24 |journal=Computers & Geosciences|title= GIS-stereoplot: an interactive stereonet plotting module for ArcView 3.0 geographic information system |issue=3 |last2=Gardoll |first2=Stephen J}}</ref>

where <math>\omega^2</math> is the eigenvalue and <math>\omega</math> is the (imaginary) [[angular frequency]]. The principal [[vibration mode]]s are different from the principal compliance modes, which are the eigenvectors of <math>k</math> alone. Furthermore, [[damped vibration]], governed by

<math display=block>m\ddot{x} + c\dot{x} + kx = 0</math>

leads to a so-called [[quadratic eigenvalue problem]],

<math display=block>\left(\omega^2 m + \omega c + k\right)x = 0.</math>

This can be reduced to a generalized eigenvalue problem by [[quadratic eigenvalue problem#Methods of Solution|algebraic manipulation]] at the cost of solving a larger system.

The output for the orientation tensor is in the three orthogonal (perpendicular) axes of space. The three eigenvectors are ordered <math>v_1, v_2, v_3</math> by their eigenvalues <math>E_1 \geq E_2 \geq E_3</math>;<ref>[http://www.ruhr-uni-bochum.de/hardrock/downloads.htm Stereo32 software]</ref> <math>v_1</math> then is the primary orientation/dip of clast, <math>v_2</math> is the secondary and <math>v_3</math> is the tertiary, in terms of strength. The clast orientation is defined as the direction of the eigenvector, on a [[compass rose]] of [[turn (geometry)|360°]]. 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 <math>E_1</math>, <math>E_2</math>, and <math>E_3</math> are dictated by the nature of the sediment's fabric. If <math>E_1 = E_2 = E_3</math>, the fabric is said to be isotropic. If <math>E_1 = E_2 > E_3</math>, the fabric is said to be planar. If <math>E_1 > E_2 > E_3</math>, the fabric is said to be linear.<ref>{{Citation|last1=Benn|first1= D.|last2=Evans|first2=D.|year=2004|title= A Practical Guide to the study of Glacial Sediments|___location= London|publisher=Arnold|pages=103–107}}</ref>

The orthogonality properties of the eigenvectors allows decoupling of the [[differential equation]]s so that the system can be represented as linear summation of the eigenvectors. The eigenvalue problem of complex structures is often solved using [[finite element analysis]], but neatly generalize the solution to scalar-valued vibration problems.

===Principal components analysis===

[[File:GaussianScatterPCA.png|thumb|right|PCA of the [[multivariate Gaussian distribution]] centered at <math>(1,3)</math> with a standard deviation of 3 in roughly the <math>(0.878,0.478)</math> direction and of&nbsp;1 in the orthogonal direction. The vectors shown are unit eigenvectors of the (symmetric, positive-semidefinite) [[covariance matrix]] scaled by the square root of the corresponding eigenvalue. (Just as in the one-dimensional case, the square root is taken because the [[standard deviation]] is more readily visualized than the [[variance]].]]

{{Main|Principal component analysis}}

{{See also|Positive semidefinite matrix|Factor analysis}}

=== Tensor of moment of inertia ===

The [[Eigendecomposition of a matrix#Symmetric matrices|eigendecomposition]] of a [[symmetric matrix|symmetric]] [[positive semidefinite matrix|positive semidefinite]] (PSD) [[positive semidefinite matrix|matrix]] yields an [[orthogonal basis]] of eigenvectors, each of which has a nonnegative eigenvalue. The orthogonal decomposition of a PSD matrix is used in [[multivariate statistics|multivariate analysis]], where the [[sample variance|sample]] [[covariance matrix|covariance matrices]] are PSD. This orthogonal decomposition is called [[principal components analysis]] (PCA) in statistics. PCA studies [[linear relation]]s among variables. PCA is performed on the [[covariance matrix]] or the [[correlation matrix]] (in which each variable is scaled to have its [[sample variance]] equal to one). For the covariance or correlation matrix, the eigenvectors correspond to [[principal components analysis|principal components]] and the eigenvalues to the [[explained variance|variance explained]] by the principal components. Principal component analysis of the correlation matrix provides an [[orthogonal basis|orthonormal eigen-basis]] 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.

In [[mechanics]], the eigenvectors of the [[inertia tensor|moment of inertia tensor]] define the [[principal axis (mechanics)|principal axes]] of a [[rigid body]]. The [[tensor]] of moment of [[inertia]] is a key quantity required to determine the rotation of a rigid body around its [[center of mass]].

=== Stress tensor ===

Principal component analysis is used to study [[data mining|large]] [[data set]]s, such as those encountered in [[data mining]], [[chemometrics|chemical research]], [[psychometrics|psychology]], and in [[marketing]]. PCA is popular especially in psychology, in the field of [[psychometrics]]. In [[Q methodology]], the eigenvalues of the correlation matrix determine the Q-methodologist's judgment of ''practical'' significance (which differs from the [[statistical significance]] of [[hypothesis testing]]; cf. [[Factor analysis#Criteria for determining the number of factors|criteria for determining the number of factors]]). More generally, principal component analysis can be used as a method of [[factor analysis]] in [[structural equation model]]ing.

In [[solid mechanics]], the [[stress (mechanics)|stress]] tensor is symmetric and so can be decomposed into a [[diagonal]] tensor with the eigenvalues on the diagonal and eigenvectors as a basis. Because it is diagonal, in this orientation, the stress tensor has no [[Shear (mathematics)|shear]] components; the components it does have are the principal components.

=== Schrödinger equation ===

===Vibration analysis===

<!-- This section is linked from [[Eigenstate]] -->

[[File:beam mode 1.gif|thumb|225px|1st lateral bending (See [[vibration]] for more types of vibration)]]

[[File:HAtomOrbitals.png|thumb|271px|The [[wavefunction]]s associated with the [[bound state]]s of an [[electron]] in a [[hydrogen atom]] can be seen as the eigenvectors of the [[hydrogen atom|hydrogen atom Hamiltonian]] as well as of the [[angular momentum operator]]. They are associated with eigenvalues interpreted as their energies (increasing downward: <math>n = 1,\, 2,\, 3,\, \ldots</math>) and [[angular momentum]] (increasing across: <!-- do not italicize! -->s, p, d, ...). The illustration shows the square of the absolute value of the wavefunctions. Brighter areas correspond to higher [[probability density function|probability density]] for a position [[measurement in quantum mechanics|measurement]]. The center of each figure is the [[atomic nucleus]], a [[proton]].]]

{{Main|Vibration}}

An example of an eigenvalue equation where the transformation <math>T</math> is represented in terms of a differential operator is the time-independent [[Schrödinger equation]] in [[quantum mechanics]]:

Eigenvalue problems occur naturally in the vibration analysis of mechanical structures with many [[Degrees of freedom (mechanics)|degrees of freedom]]. The eigenvalues are used to determine the natural frequencies (or '''eigenfrequencies''') of vibration, and the eigenvectors determine the shapes of these vibrational modes. In particular, undamped vibration is governed by

: <math>mH\ddot x + kxpsi_E = 0E\psi_E \,</math>

where <math>H</math>, the [[Hamiltonian (quantum mechanics)|Hamiltonian]], is a second-order [[differential operator]] and <math>\psi_E</math>, the [[wavefunction]], is one of its eigenfunctions corresponding to the eigenvalue <math>E</math>, interpreted as its [[energy]].

or

:<math>m\ddot x = -k x</math>

that is, acceleration is proportional to position (i.e., we expect <math>x</math> to be sinusoidal in time).

However, in the case where one is interested only in the [[bound state]] solutions of the Schrödinger equation, one looks for <math>\psi_E</math> within the space of [[Square-integrable function|square integrable]] functions. Since this space is a [[Hilbert space]] with a well-defined [[scalar product]], one can introduce a [[Basis (linear algebra)|basis set]] in which <math>\psi_E</math> and <math>H</math> 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.

In <math>n</math> dimensions, <math>m</math> becomes a [[mass matrix]] and <math>k</math> a [[stiffness matrix]]. Admissible solutions are then a linear combination of solutions to the [[generalized eigenvalue problem]]

:<math>-k x = \omega^2 m x</math>

where <math>\omega^2</math> is the eigenvalue and <math>\omega</math> is the [[angular frequency]]. Note that the principal vibration modes are different from the principal compliance modes, which are the eigenvectors of <math>k</math> alone. Furthermore, [[damped vibration]], governed by

:<math>m\ddot x + c \dot x + kx = 0</math>

leads to what is called a so-called [[quadratic eigenvalue problem]],

:<math>(\omega^2 m + \omega c + k)x = 0.</math>

This can be reduced to a generalized eigenvalue problem by [[Quadratic eigenvalue problem#Methods of Solution|clever use of algebra]] at the cost of solving a larger system.

The [[bra–ket notation]] 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 <math>|\Psi_E\rangle</math>. In this notation, the Schrödinger equation is:

The orthogonality properties of the eigenvectors allows decoupling of the differential equations so that the system can be represented as linear summation of the eigenvectors. The eigenvalue problem of complex structures is often solved using [[finite element analysis]], but neatly generalize the solution to scalar-valued vibration problems.

: <math>H|\Psi_E\rangle = E|\Psi_E\rangle</math>

where <math>|\Psi_E\rangle</math> is an '''eigenstate''' of <math>H</math> and <math>E</math> represents the eigenvalue. <math>H</math> is an [[observable]] [[self-adjoint operator]], the infinite-dimensional analog of Hermitian matrices. As in the matrix case, in the equation above <math>H|\Psi_E\rangle</math> is understood to be the vector obtained by application of the transformation <math>H</math> to <math>|\Psi_E\rangle</math>.

===Eigenfaces Wave transport ===

[[Light]], [[acoustic wave]]s, and [[microwave]]s are randomly [[Scattering theory|scattered]] numerous times when traversing a static [[disordered system]]. 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 <math>\mathbf{t}</math>.{{sfn|Vellekoop|Mosk|2007|pp=2309-2311}}{{sfn|Rotter|Gigan|2017|p=15005}} The eigenvectors of the transmission operator <math>\mathbf{t}^\dagger\mathbf{t}</math> 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, <math>\tau</math>, of <math>\mathbf{t}^\dagger\mathbf{t}</math> 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 <math>\tau_\max = 1</math> and <math>\tau_\min = 0</math>.{{sfn|Rotter|Gigan|2017|p=15005}} Furthermore, one of the striking properties of open eigenchannels, beyond the perfect transmittance, is the statistically robust spatial profile of the eigenchannels.{{sfn|Bender|Yamilov|Yilmaz|Cao|2020|p=165901}}

[[File:Eigenfaces.png|thumb|200px|[[Eigenface]]s as examples of eigenvectors]]

{{Main|Eigenface}}

In [[image processing]], processed images of [[face]]s can be seen as vectors whose components are the [[brightness]]es of each [[pixel]].<ref>{{Citation

| last=Xirouhakis

| first=A.

| first2=G.

| last2=Votsis

| first3=A.

| last3=Delopoulus

| title=Estimation of 3D motion and structure of human faces

| publisher=Online paper in PDF format, National Technical University of Athens

| url=http://www.image.ece.ntua.gr/papers/43.pdf

|format=PDF| year=2004

}}</ref> The dimension of this vector space is the number of pixels. The eigenvectors of the [[covariance matrix]] associated with a large set of normalized pictures of faces are called '''[[eigenface]]s'''; this is an example of [[principal components analysis]]. They are very useful for expressing any face image as a [[linear combination]] of some of them. In the [[Facial recognition system|facial recognition]] branch of [[biometrics]], eigenfaces provide a means of applying [[data compression]] to faces for [[Recognition of human individuals|identification]] purposes. Research related to eigen vision systems determining hand gestures has also been made.

=== Molecular orbitals ===

Similar to this concept, '''eigenvoices''' 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 recognition systems, for speaker adaptation.

In [[quantum mechanics]], and in particular in [[atomic physics|atomic]] and [[molecular physics]], within the [[Hartree–Fock]] theory, the [[atomic orbital|atomic]] and [[molecular orbital]]s can be defined by the eigenvectors of the [[Fock operator]]. The corresponding eigenvalues are interpreted as [[ionization potential]]s via [[Koopmans' theorem]]. 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 [[iteration]] procedure, called in this case [[self-consistent field]] method. In [[quantum chemistry]], one often represents the Hartree–Fock equation in a non-[[orthogonal]] [[basis set (chemistry)|basis set]]. This particular representation is a [[generalized eigenvalue problem]] called [[Roothaan equations]].

===Tensor ofGeology momentand ofglaciology inertia===

{{technical|section|reason=This section uses a lot of geology jargon without introduction or explanation ("clast", "fabric", "dip", "Tri-Plot", "Steronet", "Wulff Net").|date=December 2023}}

In [[mechanics]], the eigenvectors of the [[moment of inertia#Inertia tensor|moment of inertia tensor]] define the [[principal axis (mechanics)|principal axes]] of a [[rigid body]]. The [[tensor]] of moment of [[inertia]] is a key quantity required to determine the rotation of a rigid body around its [[center of mass]].

In [[geology]], especially in the study of [[glacial till]], eigenvectors and eigenvalues are used as a method by which a mass of information of a [[Clastic rock|clast's]] [[Fabric (geology)|fabric]] 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 clasts in a soil sample, which can be compared graphically or as a [[stereographic projection]]. Graphically, many geologists use a Tri-Plot (Sneed and Folk) diagram,.{{sfn|Graham|Midgley|2000|pp=1473–1477}}{{sfn|Sneed|Folk|1958|pp=114–150}} A stereographic projection projects 3-dimensional spaces onto a two-dimensional plane. A type of stereographic projection is Wulff Net, which is commonly used in [[crystallography]] to create [[stereograms]].{{sfn|Knox-Robinson|Gardoll|1998|p=243}}

The output for the orientation tensor is in the three orthogonal (perpendicular) axes of space. The three eigenvectors are ordered <math>\mathbf v_1, \mathbf v_2, \mathbf v_3</math> by their eigenvalues <math>E_1 \geq E_2 \geq E_3</math>;<ref>{{Cite web |last=Busche |first=Christian |last2=Schiller |first2=Beate |title=Endogene Geologie - Ruhr-Universität Bochum |url=https://ruhr-uni-bochum.de/hardrock/downloads.html |website=www.ruhr-uni-bochum.de}}</ref>

===Stress tensor===

<math>\mathbf v_1</math> then is the primary orientation/dip of clast, <math>\mathbf v_2</math> is the secondary and <math>\mathbf v_3</math> is the tertiary, in terms of strength. The clast orientation is defined as the direction of the eigenvector, on a [[compass rose]] of [[turn (geometry)|360°]]. 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 <math>E_1</math>, <math>E_2</math>, and <math>E_3</math> are dictated by the nature of the sediment's fabric. If <math>E_1 = E_2 = E_3</math>, the fabric is said to be isotropic. If <math>E_1 = E_2 > E_3</math>, the fabric is said to be planar. If <math>E_1 > E_2 > E_3</math>, the fabric is said to be linear.{{sfn|Benn|Evans|2004|pp=103–107}}

In [[solid mechanics]], the [[stress (mechanics)|stress]] tensor is symmetric and so can be decomposed into a [[diagonal]] tensor with the eigenvalues on the diagonal and eigenvectors as a basis. Because it is diagonal, in this orientation, the stress tensor has no [[Shear (mathematics)|shear]] components; the components it does have are the principal components.

=== Basic reproduction number ===

===Eigenvalues of a graph===

{{main|Basic reproduction number}}

In [[spectral graph theory]], an eigenvalue of a [[graph theory|graph]] is defined as an eigenvalue of the graph's [[adjacency matrix]] <math>A</math>, or (increasingly) of the graph's [[Laplacian matrix]] (see also [[Discrete Laplace operator]]), which is either <math>T - A</math> (sometimes called the ''combinatorial Laplacian'') or <math>I - T^{-1/2}A T^{-1/2}</math> (sometimes called the ''normalized Laplacian''), where <math>T</math> is a diagonal matrix with <math>T_{i i}</math> equal to the degree of vertex <math>v_i</math>, and in <math>T^{-1/2}</math>, the <math>i</math>th diagonal entry is <math>\sqrt{\operatorname{deg}(v_i)}</math>. The <math>k</math>th principal eigenvector of a graph is defined as either the eigenvector corresponding to the <math>k</math>th largest or <math>k</math>th smallest eigenvalue of the Laplacian. The first principal eigenvector of the graph is also referred to merely as the principal eigenvector.

The basic reproduction number (<math>R_0</math>) 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 <math>R_0</math> is the average number of people that one typical infectious person will infect. The generation time of an infection is the time, <math>t_G</math>, 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 <math>t_G</math> has passed. The value <math>R_0</math> is then the largest eigenvalue of the next generation matrix.{{sfn|Diekmann|Heesterbeek|Metz|1990|pp=365–382}}{{sfn|Heesterbeek|Diekmann|2000}}

=== Eigenfaces ===

The principal eigenvector is used to measure the [[eigenvector centrality|centrality]] of its vertices. An example is [[Google]]'s [[PageRank]] algorithm. The principal eigenvector of a modified [[adjacency matrix]] of the World Wide Web graph gives the page ranks as its components. This vector corresponds to the [[stationary distribution]] of the [[Markov chain]] 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 [[spectral clustering]]. Other methods are also available for clustering.

[[File:Eigenfaces.png|thumb|200px|[[Eigenface]]s as examples of eigenvectors]]

{{Main|Eigenface}}

In [[image processing]], processed images of faces can be seen as vectors whose components are the [[brightness]]es of each [[pixel]].{{sfn|Xirouhakis|Votsis|Delopoulus|2004}} The dimension of this vector space is the number of pixels. The eigenvectors of the [[covariance matrix]] associated with a large set of normalized pictures of faces are called '''[[eigenface]]s'''; this is an example of [[principal component analysis]]. They are very useful for expressing any face image as a [[linear combination]] of some of them. In the [[Facial recognition system|facial recognition]] branch of [[biometrics]], eigenfaces provide a means of applying [[data compression]] to faces for [[Recognition of human individuals|identification]] purposes. Research related to eigen vision systems determining hand gestures has also been made.

Similar to this concept, '''eigenvoices''' 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 recognition systems for speaker adaptation.

===Basic reproduction number===

::''See [[Basic reproduction number]]''

The basic reproduction number (<math>R_0</math>) 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 <math>R_0</math> is the average number of people that one typical infectious person will infect. The generation time of an infection is the time, <math>t_G</math>, from one person becoming infected to the next person becoming infected. In a heterogenous population, the next generation matrix defines how many people in the population will become infected after time <math>t_G</math> has passed. <math>R_0</math> is then the largest eigenvalue of the next generation matrix.<ref>{{Citation

| author = Diekmann O, Heesterbeek JAP, Metz JAJ

| year = 1990

| title = On the definition and the computation of the basic reproduction ratio R0 in models for infectious diseases in heterogeneous populations

| journal = Journal of Mathematical Biology

| volume = 28

| issue = 4

| pages =365–382

| pmid = 2117040

| doi = 10.1007/BF00178324

}}</ref><ref>{{Citation

| author = Odo Diekmann and J. A. P. Heesterbeek

| title = Mathematical epidemiology of infectious diseases

| series = Wiley series in mathematical and computational biology

| publisher = John Wiley & Sons

| ___location = West Sussex, England

| year = 2000

}}</ref>

== See also ==

* [[Antieigenvalue theory]]

* [[Eigenoperator]]

* [[Eigenplane]]

* [[Eigenmoments]]

* [[Introduction to eigenstates]]

* [[Eigenvalue algorithm]]

* [[Quantum states]]

* [[Jordan normal form]]

* [[List of numerical -analysis software]]

* [[Nonlinear eigenproblem]]

* [[Normal eigenvalue]]

* [[Quadratic eigenvalue problem]]

* [[Singular value]]

* [[Spectrum of a matrix]]

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* {{Cite book |last=Xirouhakis |first=A. |url=http://link.springer.com/10.1007/978-94-011-4840-5_30 |title=Estimation of 3D motion and structure of human faces |last2=Votsis |first2=G. |last3=Delopoulus |first3=A. |date= |publisher=National Technical University of Athens |year=1999 |isbn=978-1-4020-0393-6 |editor-last=Tzafestas |editor-first=Spyros G. |___location=Dordrecht |pages=333–344 |language=en |doi=10.1007/978-94-011-4840-5_30}}

* {{Cite journal |last=Van Mieghem |first=P. |date=2024 |title=Eigenvector components of symmetric, graph-related matrices |journal=Linear Algebra and Its Applications |volume=692 |pages=91–134 |doi=10.1016/j.laa.2024.03.035 |doi-access=free}}

{{Refend}}

== Further reading ==

{{Wikibooks|The Book of Mathematical Proofs|Algebra/Linear Transformations}}

{{Refbegin}}

* [http://www.physlink.com/education/AskExperts/ae520.cfm What are Eigen Values?] – non-technical introduction from PhysLink.com's "Ask the Experts"

* {{Citation |last=Golub |first=Gene F. |title=Eigenvalue Computation in the 20th Century |work=Journal of Computational and Applied Mathematics |volume=123 |issue=1–2 |pages=35–65 |year=2000 |url=https://dspace.library.uu.nl/bitstream/1874/2663/1/eighistory.pdf |bibcode=2000JCoAM.123...35G |doi=10.1016/S0377-0427(00)00413-1 |hdl=1874/2663 |last2=van der Vorst |first2=Henk A. |doi-access=free}}

* [http://people.revoledu.com/kardi/tutorial/LinearAlgebra/EigenValueEigenVector.html Eigen Values and Eigen Vectors Numerical Examples] – Tutorial and Interactive Program from Revoledu.

* {{Cite web |last=Hill |first=Roger |year=2009 |title=λ – Eigenvalues |url=https://sixtysymbols.com/videos/eigenvalues.htm |website=Sixty Symbols |publisher=[[Brady Haran]] for the [[University of Nottingham]] |ref=none}}

* [http://khanexercises.appspot.com/video?v=PhfbEr2btGQ Introduction to Eigen Vectors and Eigen Values] – lecture from Khan Academy

* {{Citation |last=Kuttler |first=Kenneth |title=An introduction to linear algebra |date=2017 |url=https://math.byu.edu/~klkuttle/Linearalgebra.pdf |access-date=26 November 2023 |archive-url=https://web.archive.org/web/20231217154253/https://math.byu.edu/~klkuttle/Linearalgebra.pdf |archive-date=17 December 2023 |url-status=dead |publisher=Brigham Young University}}

* {{cite web|last=Hill|first=Roger|title=λ – Eigenvalues|url=http://www.sixtysymbols.com/videos/eigenvalues.htm|work=Sixty Symbols|publisher=[[Brady Haran]] for the [[University of Nottingham]]|year=2009}}

* {{Citation |last=Strang |first=Gilbert |title=Introduction to linear algebra |year=1993 |place=Wellesley, MA |publisher=Wellesley-Cambridge Press |isbn=978-0-9614088-5-5}}

*{{Citation |last=Strang |first=Gilbert |title=Linear algebra and its applications |year=2006 |place=Belmont, CA |publisher=Thomson, Brooks/Cole |isbn=978-0-03-010567-8}}

{{Refend}}

== External links ==

'''Theory'''

{{external links|date=December 2019}}

* {{springer|title=Eigen value|id=p/e035150}}

{{Wikibooks|Linear Algebra|Eigenvalues and Eigenvectors}}

* {{springer|title=Eigen vector|id=p/e035180}}

* [https://physlink.com/education/AskExperts/ae520.cfm What are Eigen Values?] – non-technical introduction from PhysLink.com's "Ask the Experts"

* {{planetmath reference|id=4397|title=Eigenvalue (of a matrix)}}

* [https://people.revoledu.com/kardi/tutorial/LinearAlgebra/EigenValueEigenVector.html Eigen Values and Eigen Vectors Numerical Examples] – Tutorial and Interactive Program from Revoledu.

* [http://mathworld.wolfram.com/Eigenvector.html Eigenvector] – Wolfram [[MathWorld]]

* [https://web.archive.org/web/20100325112901/https://khanexercises.appspot.com/video?v=PhfbEr2btGQ Introduction to Eigen Vectors and Eigen Values] – lecture from Khan Academy

* [http://ocw.mit.edu/ans7870/18/18.06/javademo/Eigen/ Eigen Vector Examination working applet]

* [https://youtube.com/watch?v=PFDu9oVAE-g&list=PLZHQObOWTQDPD3MizzM2xVFitgF8hE_ab&index=14 Eigenvectors and eigenvalues | Essence of linear algebra, chapter 10] – A visual explanation with [[3Blue1Brown]]

* [http://web.mit.edu/18.06/www/Demos/eigen-applet-all/eigen_sound_all.html Same Eigen Vector Examination as above in a Flash demo with sound]

* [https://symbolab.com/solver/matrix-eigenvectors-calculator Matrix Eigenvectors Calculator] from Symbolab (Click on the bottom right button of the 2×12 grid to select a matrix size. Select an <math>n \times n</math> size (for a square matrix), then fill out the entries numerically and click on the Go button. It can accept complex numbers as well.)

* [http://www.sosmath.com/matrix/eigen1/eigen1.html Computation of Eigenvalues]

{{sister-inline|project=v|links= Wikiversity uses introductory physics to introduce [[v:Physics/A/Eigenvalues for beginners|'''Eigenvalues and eigenvectors''']]|short=yes

* [http://www.cs.utk.edu/~dongarra/etemplates/index.html Numerical solution of eigenvalue problems] Edited by Zhaojun Bai, [[James Demmel]], Jack Dongarra, Axel Ruhe, and [[Henk van der Vorst]]

}}

* Eigenvalues and Eigenvectors on the Ask Dr. Math forums: [http://mathforum.org/library/drmath/view/55483.html], [http://mathforum.org/library/drmath/view/51989.html]

=== Theory ===

'''Online calculators'''

* [https://sosmath.com/matrix/eigen1/eigen1.html Computation of Eigenvalues]

* [http://www.arndt-bruenner.de/mathe/scripts/engl_eigenwert.htm arndt-bruenner.de]

* [https://cs.utk.edu/~dongarra/etemplates/index.html Numerical solution of eigenvalue problems] Edited by Zhaojun Bai, [[James Demmel]], Jack Dongarra, Axel Ruhe, and [[Henk van der Vorst]]

* [http://www.bluebit.gr/matrix-calculator/ bluebit.gr]

* [http://wims.unice.fr/wims/wims.cgi?session=6S051ABAFA.2&+lang=en&+module=tool%2Flinear%2Fmatrix.en wims.unice.fr]

'''Demonstration applets'''

* [http://scienceapplets.blogspot.com/2012/03/eigenvalues-and-eigenvectors.html Java applet about eigenvectors in the real plane]

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