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{{Short description|Concepts from linear algebra}}
{{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).
[[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}}
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.
{{Toclimit|3}}
== Matrices ==
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}}
== Overview ==
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
<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.]]
[[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.
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
<math display=block>\frac{d}{dx}e^{\lambda x} = \lambda e^{\lambda x}.</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
<math display=block>A\mathbf v = \lambda \mathbf v,</math>
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.
Eigenvalues and eigenvectors give rise to many closely related mathematical concepts, and the prefix ''eigen-'' is applied liberally when naming them:
* 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'''.
==
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.
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:
* 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.)
* 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.)
* 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}}
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.)}}
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>
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 row,
<math display=block>w_i = A_{i1} v_1 + A_{i2} v_2 + \cdots + A_{in} v_n = \sum_{j = 1}^n A_{ij} v_j.</math>
If it occurs that {{mvar|v}} and {{mvar|w}} are scalar multiples, that is if
{{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
{{NumBlk|:
|<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 ===
{{main|Characteristic polynomial}}
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
{{NumBlk|:
|<math>\det(A - \lambda I) = 0</math>
|{{EquationRef|3}}
}}
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)<sup>''n''</sup>''λ''<sup>''n''</sup>. This polynomial is called the ''[[characteristic polynomial]]'' of ''A''. Equation ({{EquationNote|3}}) is called the ''characteristic equation'' or the ''secular equation'' of ''A''.
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,
{{NumBlk|:
|<math>\det(A - \lambda I) = (\lambda_1 - \lambda )(\lambda_2 - \lambda) \cdots (\lambda_n - \lambda),</math>
|{{EquationRef|4}}
}}
where each ''λ''<sub>''i''</sub> may be real but in general is a complex number. The numbers ''λ''<sub>1</sub>, ''λ''<sub>2</sub>, ..., ''λ''<sub>''n''</sub>, 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
<math display=block>A = \begin{bmatrix}
2 & 1\\
1 & 2
\end{bmatrix}.</math>
Taking the determinant of {{nowrap|(''A'' − ''λI'')}}, the characteristic polynomial of ''A'' is
<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
<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.
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.
=== Spectrum of a matrix ===
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.
=== Algebraic multiplicity ===
<!-- Algebraic multiplicity, Simple eigenvalue and Semisimple eigenvalue link here. Please do not change. -->
Let ''λ''<sub>''i''</sub> be an eigenvalue of an ''n'' by ''n'' matrix ''A''. The '''algebraic multiplicity''' ''μ''<sub>''A''</sub>(''λ''<sub>''i''</sub>) 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''<sub>''i''</sub>)<sup>''k''</sup> [[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,
<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
<math display=block>\begin{align}
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''<sub>''A''</sub>(''λ''<sub>''i''</sub>) = 1, then ''λ''<sub>''i''</sub> is said to be a ''simple eigenvalue''.{{sfn|Golub|Van Loan|1996|p=316}} If ''μ''<sub>''A''</sub>(''λ''<sub>''i''</sub>) equals the geometric multiplicity of ''λ''<sub>''i''</sub>, ''γ''<sub>''A''</sub>(''λ''<sub>''i''</sub>), defined in the next section, then ''λ''<sub>''i''</sub> is said to be a ''semisimple eigenvalue''.
=== Eigenspaces, geometric multiplicity, and the eigenbasis for matrices ===
<!-- 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>.
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''.
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
<math display=block>\gamma_A(\lambda) = n - \operatorname{rank}(A - \lambda I).</math>
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''.
<math display=block> 1 \le \gamma_A(\lambda) \le \mu_A(\lambda) \le n</math>
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>.
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}},
<math display=block>\begin{align}
\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 ===
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>.
=== Left and right eigenvectors ===
{{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}}),
<math display=block>A \mathbf v = \lambda \mathbf v.</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
<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,
<math display=block>A^\textsf{T} \mathbf u^\textsf{T} = \kappa \mathbf u^\textsf{T}.</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.
=== Diagonalization and the eigendecomposition ===
{{main|Eigendecomposition of a matrix}}
Suppose the eigenvectors of ''A'' form a basis, or equivalently ''A'' has ''n'' linearly independent eigenvectors '''v'''<sub>1</sub>, '''v'''<sub>2</sub>, ..., '''v'''<sub>''n''</sub> with associated eigenvalues ''λ''<sub>1</sub>, ''λ''<sub>2</sub>, ..., ''λ''<sub>''n''</sub>. The eigenvalues need not be distinct. Define a [[square matrix]] ''Q'' whose columns are the ''n'' linearly independent eigenvectors of ''A'',
: <math>Q = \begin{bmatrix} \mathbf v_1 & \mathbf v_2 & \cdots & \mathbf v_n \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,
: <math>AQ = \begin{bmatrix} \lambda_1 \mathbf v_1 & \lambda_2 \mathbf v_2 & \cdots & \lambda_n \mathbf v_n \end{bmatrix}.</math>
With this in mind, define a diagonal matrix Λ where each diagonal element Λ<sub>''ii''</sub> is the eigenvalue associated with the ''i''th column of ''Q''. Then
: <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''<sup>−1</sup>,
: <math>A = Q\Lambda Q^{-1},</math>
or by instead left multiplying both sides by ''Q''<sup>−1</sup>,
: <math>Q^{-1}AQ = \Lambda.</math>
''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 Λ.
Conversely, suppose a matrix ''A'' is diagonalizable. Let ''P'' be a non-singular square matrix such that ''P''<sup>−1</sup>''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'''<sub>''λ''=1</sub> = [1 −1]<sup>T</sup> and blue vectors parallel to '''v'''<sub>''λ''=3</sub> = [1 1]<sup>T</sup>. 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'' − ''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''<sub>1</sub> = −''v''<sub>2</sub> 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''<sub>1</sub> = ''v''<sub>2</sub> 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'''<sub>''λ''=1</sub> and '''v'''<sub>''λ''=3</sub> 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 − ''λ''<sup>3</sup>, 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''<sub>1</sub> = 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>
and
<math display=block>
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 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''<sub>2</sub> and ''λ''<sub>3</sub>, respectively. The two complex eigenvectors also appear in a complex conjugate pair,
==== Diagonal matrix example ====
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
which has the roots {{nowrap|1=''λ''<sub>1</sub> = 1}}, {{nowrap|1=''λ''<sub>2</sub> = 2}}, and {{nowrap|1=''λ''<sub>3</sub> = 3}}. These roots are the diagonal elements as well as the eigenvalues of ''A''.
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,
<math display=block>
\mathbf v_{\lambda_1} = \begin{bmatrix} 1\\ 0\\ 0 \end{bmatrix},\quad
\mathbf v_{\lambda_2} = \begin{bmatrix} 0\\ 1\\ 0 \end{bmatrix},\quad
\mathbf v_{\lambda_3} = \begin{bmatrix} 0\\ 0\\ 1 \end{bmatrix},
</math>
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''<sub>1</sub> = 1}}, {{nowrap|1=''λ''<sub>2</sub> = 2}}, and {{nowrap|1=''λ''<sub>3</sub> = 3}}. These roots are the diagonal elements as well as the eigenvalues of ''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>
has a characteristic polynomial that is the product of its diagonal elements,
<math display=block>\det(A - \lambda I) = \begin{vmatrix}
2 - \lambda & 0 & 0 & 0 \\
1 & 2- \lambda & 0 & 0 \\
0 & 1 & 3- \lambda & 0 \\
0 & 0 & 1 & 3- \lambda
\end{vmatrix} =
(2 - \lambda)^2(3 - \lambda)^2.
</math>
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''<sub>''A''</sub> = 4 = ''n'', the order of the characteristic polynomial and the dimension of ''A''.
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''<sub>''A''</sub> is 2, which is the smallest it could be for a matrix with two distinct eigenvalues. Geometric multiplicities are defined in a later section.
=== Eigenvector-eigenvalue identity ===
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 ==
{{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'''<sup>∞</sup> of infinitely [[derivative|differentiable]] real functions of a real argument ''t''. The eigenvalue equation for ''D'' is the [[differential equation]]
<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'''.
=== Derivative operator example ===
Consider the derivative operator <math>\tfrac{d}{dt}</math> with eigenvalue equation
<math display=block>\frac{d}{dt}f(t) = \lambda f(t).</math>
This differential equation can be solved by multiplying both sides by ''dt''/''f''(''t'') and [[Integration (calculus)|integrating]]. Its solution, the [[exponential function]]
<math display=block>f(t) = f(0)e^{\lambda t},</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.
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 ''λ''. ''E'' is called the '''eigenspace''' or '''characteristic space''' of ''T'' associated with ''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''', '''y''' ∈ ''V'' and ''α'' ∈ ''K''. Therefore, if '''u''' and '''v''' are eigenvectors of ''T'' associated with eigenvalue ''λ'', namely '''u''', '''v''' ∈ ''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''<sub>''T''</sub>(''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''<sub>''T''</sub>(''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'')<sup>−1</sup> 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'' – 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
: <math>x_t = c_1\lambda_1^t + \cdots + c_k\lambda_k^t.</math>
A similar procedure is used for solving a [[differential equation]] of the form
: <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 ===
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]].
==== Eigenvalues ====
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]].{{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 <math> n \times n </math> matrix is a sum of <math> n! </math> different products.{{efn|By doing [[Gaussian elimination]] over [[formal power series]] truncated to <math>n</math> 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.
==== 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>
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{aligned} 4x + y &= 6x \\ 6x + 3y &= 6y\end{aligned} \right.
</math> {{spaces|4}} that is {{spaces|4}} <math>
\left\{ \begin{aligned} -2x + y &= 0 \\ 6x - 3y &= 0\end{aligned} \right.
</math>
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>.
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>.
=== Simple iterative methods ===
{{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
: <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>.
===
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.
==
=== 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)|Scaling]]
! scope="col" | Unequal scaling
! scope="col" | [[Rotation (geometry)|Rotation]]
! scope="col" | [[Shear mapping|Horizontal shear]]
! scope="col" | [[Hyperbolic rotation]]
|-
! scope="row" | Illustration
| [[File:Homothety in two dim.svg|100px|alt=Equal scaling ([[Homothetic transformation|homothety]])]]
| [[File:Unequal scaling.svg|100px|alt=Vertical shrink and horizontal stretch 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]]
|- 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
| <math>\ (\lambda - k)^2</math>
| <math>(\lambda - k_1)(\lambda - k_2)</math>
| <math>\lambda^2 - 2\cos(\theta)\lambda + 1</math>
| <math>\ (\lambda - 1)^2</math>
| <math>\lambda^2 - 2\cosh(\varphi)\lambda + 1</math>
|-
! scope="row" | Eigenvalues, <math>\lambda_i</math>
| <math>\lambda_1 = \lambda_2 = k</math>
| <math>\begin{align}\lambda_1 &= k_1 \\ \lambda_2 &= k_2\end{align}</math>
| <math>\begin{align}\lambda_1 &= e^{i\theta} \\ &= \cos\theta + i\sin\theta \\ \lambda_2 &= e^{-i\theta} \\ &= \cos\theta - i\sin\theta \end{align}</math>
| <math>\lambda_1 = \lambda_2 = 1</math>
| <math>\begin{align}\lambda_1 &= e^\varphi \\ &= \cosh\varphi + \sinh\varphi \\ \lambda_2 &= e^{-\varphi} \\ &= \cosh\varphi - \sinh\varphi \end{align}</math>
|-
! scope="row" | Algebraic {{abbr|mult.|multiplicity}},<br /><math>\mu_i = \mu(\lambda_i)</math>
| <math>\mu_1 = 2</math>
| <math>\begin{align}\mu_1 &= 1 \\ \mu_2 &= 1 \end{align}</math>
| <math>\begin{align}\mu_1 &= 1 \\ \mu_2 &= 1 \end{align}</math>
| <math>\mu_1 = 2</math>
| <math>\begin{align}\mu_1 &= 1 \\ \mu_2 &= 1 \end{align}</math>
|-
! scope="row" | Geometric {{abbr|mult.|multiplicity}},<br /><math>\gamma_i = \gamma(\lambda_i)</math>
| <math>\gamma_1 = 2</math>
| <math>\begin{align}\gamma_1 &= 1 \\ \gamma_2 &= 1 \end{align}</math>
| <math>\begin{align}\gamma_1 &= 1 \\ \gamma_2 &= 1 \end{align}</math>
| <math>\gamma_1 = 1</math>
| <math>\begin{align}\gamma_1 &= 1 \\ \gamma_2 &= 1 \end{align}</math>
|-
! scope="row" | Eigenvectors
| All nonzero vectors
| <math>\begin{align}
\mathbf u_1 &= \begin{bmatrix} 1\\ 0\end{bmatrix} \\
\mathbf u_2 &= \begin{bmatrix} 0\\ 1\end{bmatrix}
\end{align}</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>
|}
The characteristic equation for a rotation is a [[quadratic equation]] with [[discriminant]] <math>D = -4(\sin\theta)^2</math>, which is a negative number whenever {{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 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.
=== Principal component 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 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.
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.
=== Graphs ===
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.
=== Markov chains ===
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 ===
[[File:Mode Shape of a Tuning Fork at Eigenfrequency 440.09 Hz.gif|thumb|Mode shape of a tuning fork at eigenfrequency 440.09 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
<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).
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 display=block>kx = \omega^2 mx</math>
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 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.
=== Tensor of moment of inertia ===
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 ===
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 ===
<!-- This section is linked from [[Eigenstate]] -->
[[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]].]]
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]]:
: <math>H\psi_E = E\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]].
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.
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:
: <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>.
=== 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}}
=== 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. 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]].
=== Geology and glaciology ===
{{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 [[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>
<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}}
=== Basic reproduction number ===
{{main|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 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 ===
[[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.
== See also ==
* [[Antieigenvalue theory]]
* [[Eigenoperator]]
* [[Eigenplane]]
* [[Eigenmoments]]
* [[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]]
== Notes ==
{{Notelist}}
=== Citations ===
{{Reflist|30em}}
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{{Refend}}
== Further reading ==
{{Refbegin}}
* {{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}}
* {{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}}
* {{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}}
* {{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 ==
{{external links|date=December 2019}}
{{Wikibooks|Linear Algebra|Eigenvalues and Eigenvectors}}
* [https://physlink.com/education/AskExperts/ae520.cfm What are Eigen Values?] – non-technical introduction from PhysLink.com's "Ask the Experts"
* [https://people.revoledu.com/kardi/tutorial/LinearAlgebra/EigenValueEigenVector.html Eigen Values and Eigen Vectors Numerical Examples] – Tutorial and Interactive Program from Revoledu.
* [https://web.archive.org/web/20100325112901/https://khanexercises.appspot.com/video?v=PhfbEr2btGQ Introduction to Eigen Vectors and Eigen Values] – lecture from Khan Academy
* [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]]
* [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.)
{{sister-inline|project=v|links= Wikiversity uses introductory physics to introduce [[v:Physics/A/Eigenvalues for beginners|'''Eigenvalues and eigenvectors''']]|short=yes
}}
=== Theory ===
* [https://sosmath.com/matrix/eigen1/eigen1.html Computation of Eigenvalues]
* [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]]
{{Linear algebra}}
{{Areas of mathematics |collapsed}}
{{Authority control}}
{{DEFAULTSORT:Eigenvalues And Eigenvectors}}
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