Eigenvalues and eigenvectors

In linear algebra, an eigenvector (Template:IPAc-en Template:Respell) or characteristic vector is a (nonzero) vector that has its direction unchanged (or reversed) by a given linear transformation. More precisely, an eigenvector $𝐯$ of a linear transformation $T$ is scaled by a constant factor $\lambda$ when the linear transformation is applied to it: Template:Tmath. The corresponding eigenvalue, characteristic value, or characteristic root is the multiplying factor $\lambda$ (possibly a negative or complex number).

Geometrically, vectors are multi-dimensional quantities with magnitude and direction, often pictured as arrows. A linear transformation rotates, stretches, or 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, then the eigenvector's direction is reversed.Template:Sfn

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.

Matrices

For an $n\times n$ matrix Template:Tmath and a nonzero Template:Tmath-vector Template:Tmath, if multiplying Template:Tmath by $𝐯$ (denoted Template:Tmath) simply scales $𝐯$ by a factor Template:Tmath, where Template:Tmath is a scalar, then $𝐯$ is called an eigenvector of Template:Tmath, and Template:Tmath is the corresponding eigenvalue. This relationship can be expressed as: Template:Tmath.

Given an [[Dimension (vector space)|Template:Tmath-dimensional vector space]] and a choice of basis, there is a direct correspondence between linear transformations from the vector space into itself and Template:Tmath 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 matrices.Template:SfnTemplate:Sfn

Overview

Eigenvalues and eigenvectors feature prominently in the analysis of linear transformations. The prefix eigen- is adopted from the German [[wikt:eigen#German|Template:Lang]] (cognate with the English word own) for 'proper', 'characteristic', 'own'.Template:Sfn Originally used to study principal axes of the rotational motion of rigid bodies, eigenvalues and eigenvectors have a wide range of applications, for example in stability analysis, vibration analysis, atomic orbitals, facial recognition, and matrix diagonalization.

In essence, an eigenvector Template:Math of a linear transformation Template:Mvar is a nonzero vector that, when Template:Mvar is applied to it, does not change direction. Applying Template:Mvar to the eigenvector only scales the eigenvector by the scalar value Template:Mvar, called an eigenvalue. This condition can be written as the equation $$T(𝐯)=\lambda 𝐯,$$ referred to as the eigenvalue equation or eigenequation. In general, Template:Mvar may be any scalar. For example, Template:Mvar may be negative, in which case the eigenvector reverses direction as part of the scaling, or it may be zero, or complex.

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 Template:Math.

A Template:Resx 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.

Animated examples of eigenvectors and real eigenvalues for several non-symmetric 2D linear transformations

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 Template:Tmath, in which case the eigenvectors are functions called eigenfunctions that are scaled by that differential operator, such as $$\frac{d}{dx}{e}^{\lambda x}=\lambda e^{\lambda x}.$$ Alternatively, the linear transformation could take the form of an Template:Resx matrix, in which case the eigenvectors are Template:Resx matrices.

If the linear transformation is expressed in the form of an Template:Resx matrix Template:Mvar, then the eigenvalue equation for a linear transformation above can be rewritten as the matrix multiplication $$A𝐯=\lambda 𝐯,$$ where the eigenvector Template:Math is an Template:Resx matrix. For a matrix, eigenvalues and eigenvectors can be used to decompose the matrix; for example, by 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.Template:SfnTemplate:Sfn
• The set of all eigenvectors of Template:Mvar corresponding to the same eigenvalue, together with the zero vector, is called an eigenspace, or the characteristic space of Template:Mvar associated with that eigenvalue.Template:Sfn
• If a set of eigenvectors of Template:Mvar forms a basis of the domain of Template:Mvar, then this basis is called an eigenbasis.

Additional animations of eigenvectors and eigenvalues in two dimensions, including symmetric transformations and transformations with complex eigenvalues, are available at Wikimedia Commons.

History

Eigenvalues are often introduced in the context of linear algebra or matrix theory. Historically, however, they arose in the study of quadratic forms and differential equations.

In the 18th century, Leonhard Euler studied the rotational motion of a rigid body, and discovered the importance of the principal axes.Template:Efn Joseph-Louis Lagrange realized that the principal axes are the eigenvectors of the inertia matrix.Template:Sfn

In the early 19th century, Augustin-Louis Cauchy saw how their work could be used to classify the quadric surfaces, and generalized it to arbitrary dimensions.Template:Sfn Cauchy also coined the term racine caractéristique (characteristic root), for what is now called eigenvalue; his term survives in characteristic equation.Template:Efn

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 The Analytic Theory of Heat (Théorie analytique de la chaleur).Template:Sfn 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 matrices have real eigenvalues.Template:Sfn This was extended by Charles Hermite in 1855 to what are now called Hermitian matrices.Template:Sfn

Around the same time, Francesco Brioschi proved that the eigenvalues of orthogonal matrices lie on the unit circle,Template:Sfn and Alfred Clebsch found the corresponding result for skew-symmetric matrices.Template:Sfn Finally, Karl Weierstrass clarified an important aspect in the stability theory started by Laplace, by realizing that defective matrices can cause instability.Template:Sfn

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.Template:Sfn Schwarz studied the first eigenvalue of Laplace's equation on general domains towards the end of the 19th century, while Poincaré studied Poisson's equation a few years later.Template:Sfn

At the start of the 20th century, David Hilbert studied the eigenvalues of integral operators by viewing the operators as infinite matrices.Template:Sfn He was the first to use the German word eigen, which means "own", to denote eigenvalues and eigenvectors in 1904,Template:Efn 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.Template:Sfn

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. FrancisTemplate:Sfn and Vera KublanovskayaTemplate:Sfn in 1961.Template:SfnTemplate:Sfn

Eigenvalues and eigenvectors of a matrix

Eigenvalues and eigenvectors are often introduced to students in the context of linear algebra courses focused on matrices. Furthermore, linear transformations over a finite-dimensional vector space can be represented using matrices,Template:SfnTemplate:Sfn which is especially common in numerical and computational applications.Template:Sfn

Matrix Template:Mvar acts by stretching the vector Template:Math, not changing its direction, so Template:Math is an eigenvector of Template:Mvar.

Consider two Template:Tmath-dimensional vectors that are formed as a list of Template:Tmath scalars, such as the three-dimensional vectors $$𝐱=\left[\begin{array}{c}1\\ -3\\ 4\end{array}\right]\text{and}𝐲=\left[\begin{array}{c}-20\\ 60\\ -80\end{array}\right].$$ These vectors are said to be scalar multiples of each other, or parallel, or collinear, if there is a scalar Template:Tmath such that $$𝐲=\lambda 𝐱.$$ In this example, Template:Tmath.

Now consider the linear transformation of Template:Tmath-dimensional vectors defined by an Template:Tmath matrix Template:Tmath: $$A𝐯=𝐰,$$ or $$\left[\begin{array}{cccc}{A}_{11}& A_{12}& \cdots & A_{1n}\\ A_{21}& A_{22}& \cdots & A_{2n}\\ ⋮& ⋮& \ddots & ⋮\\ A_{n1}& A_{n2}& \cdots & A_{nn}\end{array}\right]\left[\begin{array}{c}{v}_1\\ v_2\\ ⋮\\ v_n\end{array}\right]=\left[\begin{array}{c}{w}_1\\ w_2\\ ⋮\\ w_n\end{array}\right]$$ where, for each row, $$w_i=A_{i1}{v}_1+A_{i2}{v}_2+\cdots +A_{in}{v}_n=\sum _{j=1}^n{A}_{ij}{v}_j.$$

If it occurs that Template:Tmath and Template:Tmath are scalar multiples, that is, if Template:NumBlk then Template:Tmath is an eigenvector of the linear transformation Template:Tmath and the scale factor Template:Tmath is the eigenvalue corresponding to that eigenvector. Equation (Template:EquationNote) is the eigenvalue equation for the matrix Template:Tmath.

Equation (Template:EquationNote) can be stated equivalently as Template:NumBlk where Template:Tmath is the Template:Tmath identity matrix and Template:Tmath is the zero vector.

Eigenvalues and characteristic polynomial

Equation (Template:EquationNote) has a nonzero solution Template:Mvar if and only if the determinant of the matrix Template:Math is zero. Therefore, the eigenvalues of Template:Mvar are values of Template:Mvar that satisfy the equation Template:NumBlk

Using the Leibniz formula for determinants, the left-hand side of equation (Template:EquationNote) is a polynomial function of the variable Template:Mvar and the degree of this polynomial is Template:Mvar, the order of the matrix Template:Mvar. Its coefficients depend on the entries of Template:Mvar, except that its term of degree Template:Mvar is always Template:Math. This polynomial is called the characteristic polynomial of Template:Mvar. Equation (Template:EquationNote) is called the characteristic equation or secular equation of Template:Mvar.

The characteristic polynomial of an Template:Mvar-by-Template:Mvar matrix Template:Mvar, being a polynomial of degree Template:Mvar, has at most Template:Mvar complex number roots, which can be found by factoring the characteristic polynomial, or numerically by root finding. The characteristic polynomial can be factored into the product of Template:Mvar linear terms: Template:NumBlk where the complex numbers Template:Math, Template:Math, ..., Template:Math, each of which is an eigenvalue, may repeat. (The number of times an eigenvalue appears in the characteristic polynomial is known as its algebraic multiplicity.)

As a brief example, which is described in more detail in the examples section later, consider the matrix $$A=\left[\begin{array}{cc}2& 1\\ 1& 2\end{array}\right].$$ Taking the determinant of Template:Math, the characteristic polynomial of Template:Mvar is $$det(A-\lambda I)=\left|\begin{array}{cc}2-\lambda & 1\\ 1& 2-\lambda \end{array}\right|=3-4\lambda +{\lambda }^2.$$ Setting the characteristic polynomial equal to zero, it has roots at Template:Math and Template:Math, which are the two eigenvalues of Template:Mvar. The eigenvectors corresponding to each eigenvalue Template:Mvar can be found by solving for the components of Template:Math in the equation Template:Math. In this example, the eigenvectors are any nonzero scalar multiples of $${𝐯}_{\lambda =1}=\left[\begin{array}{c}1\\ -1\end{array}\right],{𝐯}_{\lambda =3}=\left[\begin{array}{c}1\\ 1\end{array}\right].$$

If the entries of the matrix Template:Mvar 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 numbers even if all the entries of Template:Mvar are rational numbers or even if they are all integers. However, if the entries of Template:Mvar are all algebraic numbers, which include the rationals, then 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 conjugates, 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 is the list of its eigenvalues, repeated according to their multiplicities; in a shorter notation, the set of its eigenvalues with their multiplicities indicated.

An important quantity associated with the spectrum of a matrix is the maximum absolute value of all of its eigenvalues. This is known as the spectral radius of the considered matrix.

Algebraic multiplicity

Let Template:Math be an eigenvalue of an Template:Mvar-by-Template:Mvar matrix Template:Mvar. The algebraic multiplicity Template:Math of the eigenvalue is its multiplicity as a root of the characteristic polynomial, that is, the largest integer Template:Mvar such that Template:Math evenly divides that polynomial.Template:SfnTemplate:SfnTemplate:Sfn

Suppose a matrix Template:Mvar has order Template:Mvar and Template:Math distinct eigenvalues. Whereas equation (Template:EquationNote) factors the characteristic polynomial of Template:Mvar into the product of Template:Mvar linear terms with some terms potentially repeating, the characteristic polynomial can also be written as the product of Template:Mvar terms each corresponding to a distinct eigenvalue and raised to the power of the algebraic multiplicity: $$det(A-\lambda I)=({\lambda }_1-\lambda {)}^{{\mu }_A({\lambda }_1)}({\lambda }_2-\lambda {)}^{{\mu }_A({\lambda }_2)}\cdots ({\lambda }_d-\lambda {)}^{{\mu }_A({\lambda }_d)}.$$ If Template:Math, then the right-hand side is the product of Template:Mvar linear terms, and this is the same as equation (Template:EquationNote). The size of each eigenvalue's algebraic multiplicity is related to the dimension Template:Mvar as $$\begin{array}{cc}1& \le {\mu }_A({\lambda }_i)\le n,\\ {\mu }_A& =\sum _{i=1}^d{\mu }_A\left({\lambda }_i\right)=n.\end{array}$$ If Template:Math, then Template:Mvar is said to be a Template:Dfni.Template:Sfn If Template:Math equals the geometric multiplicity of Template:Mvar (denoted by Template:Math and defined in the next section), then Template:Mvar is said to be a Template:Dfni.

Eigenspaces, geometric multiplicities, and eigenbasis for a matrix

Given a particular eigenvalue Template:Tmath of the Template:Tmath matrix Template:Tmath, define the set Template:Tmath to be all vectors Template:Tmath that satisfy equation (Template:EquationNote): $$E=\{𝐯:(A-\lambda I)𝐯=\mathrm{𝟎}\}.$$ On one hand, Template:Tmath is precisely the kernel or nullspace of the matrix Template:Tmath. On the other hand, by definition, any nonzero vector that satisfies this condition is an eigenvector of Template:Tmath associated with Template:Tmath; so Template:Tmath is the union of the zero vector with the set of all eigenvectors of Template:Tmath associated with Template:Tmath. The space Template:Tmath is called the eigenspace or characteristic space of Template:Tmath associated with Template:Tmath.Template:SfnTemplate:Sfn In general, Template:Tmath is a complex number and the eigenvectors are complex Template:Tmath matrices (column vectors). Because every nullspace is a linear subspace of the domain, Template:Tmath is a linear subspace of Template:Tmath.

Because the eigenspace Template:Tmath is a linear subspace, it is closed under addition. That is, if two vectors Template:Tmath and Template:Tmath belong to the set Template:Tmath, written Template:Tmath, then Template:Tmath, or equivalently Template:Tmath. This can be checked using the distributive property of matrix multiplication. Similarly, because Template:Tmath is a linear subspace, it is closed under scalar multiplication. That is, if Template:Tmath and Template:Tmath, then Template:Tmath, or equivalently Template:Tmath. This can be checked by noting that multiplication of complex matrices by complex numbers is commutative. As long as Template:Tmath and Template:Tmath are not zero, they are also eigenvectors of Template:Tmath associated with Template:Tmath.

The dimension of the eigenspace Template:Tmath associated with Template:Tmath, or equivalently the maximum number of linearly independent eigenvectors associated with Template:Tmath, is referred to as the eigenvalue's geometric multiplicity and denoted by Template:Tmath. Because Template:Tmath is also the nullspace of Template:Tmath, the geometric multiplicity of Template:Tmath is the dimension of the nullspace of Template:Tmath, also called the Template:Dfni of Template:Tmath. This quantity is related to the size and rank of Template:Tmath by the equation: $${\gamma }_A(\lambda )=n-\mathrm{rank}(A-\lambda I).$$ 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 Template:Tmath. In summary, $$1\le {\gamma }_A(\lambda )\le {\mu }_A(\lambda )\le n.$$

Proof of inequality Template:Tmath: Let Template:Math, where Template:Mvar is a fixed complex number, and the eigenspace associated with Template:Mvar is the nullspace of Template:Mvar. Let the dimension of that eigenspace be Template:Tmath. This means that the last Template:Mvar rows of the echelon form of Template:Mvar are zero. Thus, there is an invertible matrix Template:Mvar coming from Gauss-Jordan reduction, such that $$EB=\left[\begin{array}{cc}\ast & \ast \\ {\mathrm{𝟎}}_{k\times (n-k)}& {\mathrm{𝟎}}_{k\times k}\end{array}\right].$$ Therefore the last Template:Mvar rows of Template:Math are Template:Math times the last Template:Mvar rows of Template:Mvar. Therefore the polynomial Template:Mvar evenly divides the polynomial Template:Math, because of basic properties of determinants (homogeneity). On the other hand, Template:Math Template:Math Template:Math, so Template:Math divides Template:Math, and so the algebraic multiplicity of Template:Mvar is at least Template:Tmath. Q.E.D.

Suppose Template:Mvar has Template:Math distinct eigenvalues Template:Math, where the geometric multiplicity of Template:Mvar is Template:Math. The total geometric multiplicity of Template:Mvar, $${\gamma }_A=\sum _{i=1}^d{\gamma }_A({\lambda }_i),$$ is the dimension of the sum of all the eigenspaces of Template:Mvar's eigenvalues, or equivalently the maximum number of linearly independent eigenvectors of Template:Mvar. By construction, Template:Tmath. If Template:Tmath, then:

• The direct sum of the eigenspaces of all of Template:Mvar's eigenvalues is the entire vector space Template:Tmath.
• A basis of ${ℂ}^n$ can be formed from Template:Mvar linearly independent eigenvectors of Template:Mvar; such a basis is called an eigenbasis.
• Any vector in ${ℂ}^n$ can be written as a linear combination of eigenvectors of Template:Mvar.

Additional properties

Let Template:Mvar be an arbitrary Template:Resx matrix of complex numbers with eigenvalues Template:Math. Each eigenvalue appears Template:Math times in this list, where Template:Math is the eigenvalue's algebraic multiplicity. The following are properties of this matrix and its eigenvalues:

• The trace of Template:Tmath, defined as the sum of its diagonal elements, is also the sum of all its eigenvalues:Template:SfnTemplate:SfnTemplate:Sfn $$\mathrm{tr}(A)=\sum _{i=1}^n{a}_{ii}=\sum _{i=1}^n{\lambda }_i={\lambda }_1+{\lambda }_2+\cdots +{\lambda }_n.$$
• The determinant of Template:Tmath is the product of all its eigenvalues:Template:SfnTemplate:SfnTemplate:Sfn $$det(A)=\prod _{i=1}^n{\lambda }_i={\lambda }_1{\lambda }_2\cdots {\lambda }_n.$$
• For any positive integer Template:Tmath, the eigenvalues of the Template:Tmathth power of Template:Tmath, that is, Template:Tmath, are Template:Tmath.
• The eigenvalues of matrix Template:Tmath (where Template:Tmath is the identity matrix) are Template:Tmath. Moreover, for any Template:Tmath, the eigenvalues of matrix Template:Tmath are Template:Tmath.
• More generally, for any polynomial Template:Tmath, the eigenvalues of matrix Template:Tmath are Template:Tmath.
• Template:Tmath is invertible if and only if every eigenvalue Template:Tmath is nonzero.
• If Template:Tmath is invertible, then the eigenvalues of Template:Tmath are $\frac{1}{{\lambda }_1},\dots ,\frac{1}{{\lambda }_n}$ and for each pair of corresponding eigenvalues, the geometric multiplicities Template:Tmath and ${\gamma }_{A^{-1}}(\frac{1}{{\lambda }_i})$ coincide. Moreover, since the characteristic polynomial of the inverse is the reciprocal polynomial of the original up to a scalar factor, for each pair of corresponding eigenvalues, the algebraic multiplicities Template:Tmath and ${\mu }_{A^{-1}}(\frac{1}{{\lambda }_i})$ coincide.
• If Template:Tmath is equal to its conjugate transpose Template:Tmath, that is, Template:Tmath is Hermitian, then every eigenvalue Template:Tmath is real. The same is true of any symmetric real matrix.
• If Template:Tmath is not only Hermitian but also positive-definite, positive-semidefinite, negative-definite, or negative-semidefinite, then every eigenvalue Template:Tmath is positive, non-negative, negative, or non-positive, respectively.
• If Template:Tmath is unitary, then every eigenvalue Template:Tmath has absolute value Template:Tmath.

Left and right eigenvectors

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 'Template:Dfni', namely a Template:Em vector that Template:Em multiplies the Template:Resx matrix Template:Mvar in the defining equation, equation (Template:EquationNote), $$A𝐯=\lambda 𝐯.$$ The eigenvalue and eigenvector problem can also be defined for Template:Em vectors that Template:Em multiply matrix Template:Mvar. In this formulation, the defining equation is $$𝐮A=\kappa 𝐮,$$ where Template:Mvar is a scalar and Template:Math is a Template:Resx matrix. Any row vector Template:Math satisfying this equation is called a 'Template:Dfni' of Template:Mvar, and Template:Mvar is still called its associated eigenvalue. Taking the transpose of this equation, $$A^{𝖳}{𝐮}^{𝖳}=\kappa {𝐮}^{𝖳}.$$

Comparing this equation to equation (Template:EquationNote), it follows immediately that a left eigenvector of Template:Tmath is the same as the transpose of a right eigenvector of Template:Tmath, with the same eigenvalue. Furthermore, since the characteristic polynomial of Template:Tmath is the same as the characteristic polynomial of Template:Tmath, the left and right eigenvectors of Template:Tmath are associated with the same eigenvalues.

Eigenvalues of transpose

A matrix has the same eigenvalues as its transpose, as can be directly seen as follows. Assume Template:Tmath is an eigenvalue of an Template:Tmath matrix Template:Tmath with eigenvector Template:Tmath. Then Template:Tmath; equivalently, Template:Tmath.

Thus, the columns of Template:Tmath are linearly dependent. Equivalently, the rank of the matrix is less than Template:Tmath.

But as column rank = row rank, the rows are also linearly dependent. Hence, there are numbers Template:Tmath, not all zero, such that ${\sum }_{i=1}^{i=n}{y}_i{R}_i=0,$ where the Template:Tmath's are the rows of Template:Tmath. Let the row vector Template:Tmath; then Template:Tmath. Taking the transpose, Template:Tmath. Moreover, Template:Tmath is not the zero vector; so Template:Tmath is also an eigenvalue of Template:Tmath.

Furthermore, this argument shows that the eigenvalues of $A$ and $A^{𝖳}$ have the same geometric multiplicity (since column nullity = row nullity).

Diagonalization and eigendecomposition

Suppose the eigenvectors of Template:Mvar form a basis of Template:Tmath, or equivalently Template:Mvar has Template:Mvar linearly independent eigenvectors Template:Math, Template:Math, ..., Template:Math (with associated eigenvalues Template:Math, Template:Math, ..., Template:Math). The eigenvectors need not be orthogonal to one another, and the eigenvalues need not be distinct. Define the square matrix Template:Mvar whose columns are the Template:Mvar linearly independent eigenvectors of Template:Mvar, $$Q=\left[\begin{array}{cccc}{𝐯}_1& {𝐯}_2& \cdots & {𝐯}_n\end{array}\right].$$ Since each column of Template:Mvar is an eigenvector of Template:Mvar, right multiplying Template:Mvar by Template:Mvar scales each column of Template:Mvar by its associated eigenvalue: $$AQ=\left[\begin{array}{cccc}{\lambda }_1{𝐯}_1& {\lambda }_2{𝐯}_2& \cdots & {\lambda }_n{𝐯}_n\end{array}\right].$$

With this in mind, define the diagonal matrix Template:Mvar where each diagonal element Template:Mvar is the eigenvalue associated with the Template:Mvarth column of Template:Mvar. Then $$AQ=Q𝛬.$$ Because the columns of Template:Mvar are linearly independent, Template:Mvar is invertible. Right multiplying both sides of the equation by Template:Math, $$A=Q𝛬Q^{-1};$$ or instead left multiplying both sides by Template:Math, $$Q^{-1}AQ=𝛬.$$ Template:Mvar 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; it is a similarity transformation. Such a matrix Template:Mvar is said to be Template:Dfni to the diagonal matrix Template:Mvar, or diagonalizable. The matrix Template:Mvar is the change of basis matrix of the similarity transformation. Essentially, the matrices Template:Mvar and Template:Mvar represent the same linear transformation expressed in two different bases. The eigenvectors are used as the basis when representing the linear transformation as Template:Mvar.

Conversely, suppose a matrix Template:Mvar is diagonalizable. Let Template:Mvar be a non-singular square matrix such that Template:Math is some diagonal matrix Template:Mvar. Left multiplying both by Template:Mvar yields Template:Math. Each column of Template:Mvar must therefore be an eigenvector of Template:Mvar whose eigenvalue is the corresponding diagonal element of Template:Mvar. Since the columns of Template:Mvar must be linearly independent for Template:Mvar to be invertible, there exist Template:Mvar linearly independent eigenvectors of Template:Mvar.

In conclusion, the eigenvectors of Template:Tmath form a basis of Template:Tmath if and only if Template:Tmath is diagonalizable.

A matrix that is not diagonalizable is said to be defective. For defective matrices, the notion of eigenvectors generalizes to generalized eigenvectors and the diagonal matrix of eigenvalues generalizes to the Jordan normal form. Over an algebraically closed field, any matrix Template:Mvar has a Jordan normal form and therefore admits a basis of generalized eigenvectors and a decomposition into generalized eigenspaces.

Variational characterization

In the Hermitian case, eigenvalues can be given a variational characterization. The largest eigenvalue of Template:Mvar is the maximum value of the quadratic form Template:Math. A value of Template:Math that realizes that maximum is an eigenvector.

Matrix examples

Two-dimensional matrix example

The transformation matrix Template:Math preserves the direction of magenta vectors parallel to Template:Math Template:Math and blue vectors parallel to Template:Math Template:Math. 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 Template:Math), while blue vectors are three times the length of the original (due to their eigenvalue of Template:Math). See also: An extended version, showing all four quadrants.

Consider the matrix $$A=\left[\begin{array}{cc}2& 1\\ 1& 2\end{array}\right].$$ The figure on the right shows the effect of this transformation on point coordinates in the plane. The eigenvectors Template:Mvar of this transformation satisfy equation (Template:EquationNote), and the values of Template:Mvar for which the determinant of the matrix Template:Math equals zero are the eigenvalues.

Taking the determinant to find characteristic polynomial of Template:Mvar, $$\begin{array}{cc}det(A-\lambda I)& =|\left[\begin{array}{cc}2& 1\\ 1& 2\end{array}\right]-\lambda \left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]|=\left|\begin{array}{cc}2-\lambda & 1\\ 1& 2-\lambda \end{array}\right|\\ & =3-4\lambda +{\lambda }^2\\ & =(\lambda -3)(\lambda -1).\end{array}$$ Setting the characteristic polynomial equal to zero, it has roots at Template:Math and Template:Math, which are the two eigenvalues of Template:Mvar.

For Template:Math, equation (Template:EquationNote) becomes, $$\begin{array}{cc}(A-I){𝐯}_{\lambda =1}& =\left[\begin{array}{cc}1& 1\\ 1& 1\end{array}\right]\left[\begin{array}{c}{v}_1\\ v_2\end{array}\right]=\left[\begin{array}{c}0\\ 0\end{array}\right]\\ 1v_1+1v_2& =0\end{array}$$ Any nonzero vector with Template:Math solves this equation. Therefore, $${𝐯}_{\lambda =1}=\left[\begin{array}{c}{v}_1\\ -v_1\end{array}\right]=\left[\begin{array}{c}1\\ -1\end{array}\right]$$ is an eigenvector of Template:Mvar corresponding to Template:Math, as is any scalar multiple of this vector.

For Template:Math, equation (Template:EquationNote) becomes $$\begin{array}{cc}(A-3I){𝐯}_{\lambda =3}& =\left[\begin{array}{cc}-1& \phantom{-}1\\ \phantom{-}1& -1\end{array}\right]\left[\begin{array}{c}{v}_1\\ v_2\end{array}\right]=\left[\begin{array}{c}0\\ 0\end{array}\right]\\ -1v_1+1v_2& =0;\\ 1v_1-1v_2& =0\end{array}$$ Any nonzero vector with Template:Math solves this equation. Therefore, $${𝐯}_{\lambda =3}=\left[\begin{array}{c}{v}_1\\ v_1\end{array}\right]=\left[\begin{array}{c}1\\ 1\end{array}\right]$$ is an eigenvector of Template:Mvar corresponding to Template:Math, as is any scalar multiple of this vector. Thus, the vectors Template:Math and Template:Math are eigenvectors of Template:Mvar associated with the eigenvalues Template:Math and Template:Math, respectively.

Three-dimensional matrix example

Consider the matrix $$A=\left[\begin{array}{ccc}2& 0& 0\\ 0& 3& 4\\ 0& 4& 9\end{array}\right].$$ The characteristic polynomial of Template:Mvar is $$\begin{array}{cc}det(A-\lambda I)& =|\left[\begin{array}{ccc}2& 0& 0\\ 0& 3& 4\\ 0& 4& 9\end{array}\right]-\lambda \left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]|=\left|\begin{array}{ccc}2-\lambda & 0& 0\\ 0& 3-\lambda & 4\\ 0& 4& 9-\lambda \end{array}\right|,\\ & =(2-\lambda )[(3-\lambda )(9-\lambda )-16]=-{\lambda }^3+14{\lambda }^2-35\lambda +22.\end{array}$$

The roots of the characteristic polynomial are 2, 1, and 11, which are the only three eigenvalues of Template:Mvar. These eigenvalues correspond to the eigenvectors Template:Math, Template:Math, and Template:Math, or any nonzero multiple thereof.

Three-dimensional matrix example with complex eigenvalues

Consider the cyclic permutation matrix $$A=\left[\begin{array}{ccc}0& 1& 0\\ 0& 0& 1\\ 1& 0& 0\end{array}\right].$$

This matrix shifts the coordinates of the vector up by one position and moves the first coordinate to the bottom. Its characteristic polynomial is Template:Math, whose roots are $$\begin{array}{cc}{\lambda }_1& =1\\ {\lambda }_2& =-\frac{1}{2}+i\frac{\sqrt{3}}{2}\\ {\lambda }_3& ={\lambda }_2^{\ast }=-\frac{1}{2}-i\frac{\sqrt{3}}{2}\end{array}$$ where Template:Mvar is an imaginary unit with Template:Math.

For the real eigenvalue Template:Math, any vector with three equal nonzero entries is an eigenvector. For example, $$A\left[\begin{array}{c}5\\ 5\\ 5\end{array}\right]=\left[\begin{array}{c}5\\ 5\\ 5\end{array}\right]=1\cdot \left[\begin{array}{c}5\\ 5\\ 5\end{array}\right].$$

For the complex conjugate pair of imaginary eigenvalues, $${\lambda }_2{\lambda }_3=1,{\lambda }_2^2={\lambda }_3,{\lambda }_3^2={\lambda }_2.$$ Then $$A\left[\begin{array}{c}1\\ {\lambda }_2\\ {\lambda }_3\end{array}\right]=\left[\begin{array}{c}{\lambda }_2\\ {\lambda }_3\\ 1\end{array}\right]={\lambda }_2\cdot \left[\begin{array}{c}1\\ {\lambda }_2\\ {\lambda }_3\end{array}\right],$$ and $$A\left[\begin{array}{c}1\\ {\lambda }_3\\ {\lambda }_2\end{array}\right]=\left[\begin{array}{c}{\lambda }_3\\ {\lambda }_2\\ 1\end{array}\right]={\lambda }_3\cdot \left[\begin{array}{c}1\\ {\lambda }_3\\ {\lambda }_2\end{array}\right].$$

Therefore, the other two eigenvectors of Template:Mvar are complex and are Template:Math and Template:Math with eigenvalues Template:Math and Template:Math, respectively. The two complex eigenvectors also appear in a complex conjugate pair, $${𝐯}_{{\lambda }_2}={𝐯}_{{\lambda }_3}^{\ast }.$$

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 $$A=\left[\begin{array}{ccc}1& 0& 0\\ 0& 2& 0\\ 0& 0& 3\end{array}\right].$$ The characteristic polynomial of Template:Mvar is $$det(A-\lambda I)=(1-\lambda )(2-\lambda )(3-\lambda ),$$ which has the roots Template:Math, Template:Math, and Template:Math. These roots are the diagonal elements as well as the eigenvalues of Template:Mvar.

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, $${𝐯}_{{\lambda }_1}=\left[\begin{array}{c}1\\ 0\\ 0\end{array}\right],{𝐯}_{{\lambda }_2}=\left[\begin{array}{c}0\\ 1\\ 0\end{array}\right],{𝐯}_{{\lambda }_3}=\left[\begin{array}{c}0\\ 0\\ 1\end{array}\right],$$ 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, $$A=\left[\begin{array}{ccc}1& 0& 0\\ 1& 2& 0\\ 2& 3& 3\end{array}\right].$$

The characteristic polynomial of Template:Mvar is $$det(A-\lambda I)=(1-\lambda )(2-\lambda )(3-\lambda ),$$ which has the roots Template:Math, Template:Math, and Template:Math. These roots are the diagonal elements as well as the eigenvalues of Template:Mvar.

These eigenvalues correspond to the eigenvectors, $${𝐯}_{{\lambda }_1}=\left[\begin{array}{c}1\\ -1\\ \frac{1}{2}\end{array}\right],{𝐯}_{{\lambda }_2}=\left[\begin{array}{c}0\\ 1\\ -3\end{array}\right],{𝐯}_{{\lambda }_3}=\left[\begin{array}{c}0\\ 0\\ 1\end{array}\right],$$ respectively, as well as scalar multiples of these vectors.

Matrix with repeated eigenvalues example

As in the previous example, the lower triangular matrix $$A=\left[\begin{array}{cccc}2& 0& 0& 0\\ 1& 2& 0& 0\\ 0& 1& 3& 0\\ 0& 0& 1& 3\end{array}\right],$$ has a characteristic polynomial that is the product of its diagonal elements, $$det(A-\lambda I)=\left|\begin{array}{cccc}2-\lambda & 0& 0& 0\\ 1& 2-\lambda & 0& 0\\ 0& 1& 3-\lambda & 0\\ 0& 0& 1& 3-\lambda \end{array}\right|=(2-\lambda {)}^2(3-\lambda {)}^2.$$

The roots of this polynomial, and hence the eigenvalues, are 2 and 3. The Template:Dfni of each eigenvalue is 2; in other words they are both double roots. The sum of the algebraic multiplicities of all distinct eigenvalues is Template:Math Template:Math, the order of the characteristic polynomial and the dimension of Template:Mvar.

On the other hand, the Template:Dfni of the eigenvalue 2 is only 1, because its eigenspace is spanned by just one vector Template: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 Template:Math. The total geometric multiplicity Template:Math 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 Template:Mvar, the norm squared of the Template:Mvarth component of a normalized eigenvector can be calculated using only the matrix eigenvalues and the eigenvalues of the corresponding minor matrix, $$|v_{i\alpha }{|}^2=\frac{\prod _k({\lambda }_i(A)-{\lambda }_k(A_{\alpha }))}{\prod _{k\ne i}({\lambda }_i(A)-{\lambda }_k(A))},$$ where Template:Tmath is the submatrix formed by removing the Template:Mvarth row and column from the original matrix.Template:SfnTemplate:Sfn This identity also extends to diagonalizable matrices. It has been discovered in and rediscovered many times in the literature (see, e.g., Template:SfnTemplate:Sfn).

Eigenvalues and eigenfunctions of differential operators

The definitions of eigenvalue and eigenvectors of a linear transformation Template:Mvar remains valid even if the underlying vector space is an infinite-dimensional Hilbert or Banach space. A widely used class of linear transformations acting on infinite-dimensional spaces are the differential operators on function spaces. Let Template:Mvar be a linear differential operator on the space $C^{\mathrm{\infty }}(ℝ)$ of infinitely differentiable real functions of a real argument Template:Mvar. The eigenvalue equation for Template:Mvar is the differential equation $$Df(t)=\lambda f(t)$$ The functions that satisfy this equation are eigenvectors of Template:Mvar and are commonly called 'Template:Dfni'.

Derivative operator example

Consider the derivative operator $\frac{d}{dt}$ with eigenvalue equation $$\frac{d}{dt}f(t)=\lambda f(t).$$ This differential equation can be solved by multiplying both sides by Template:Math and integrating. Its solution, the exponential function $$f(t)=f(0)e^{\lambda t},$$ is the eigenfunction of the derivative operator. In this case the eigenfunction is itself a function of its associated eigenvalue. In particular, for Template:Math the eigenfunction Template:Math is a constant.

General definition

The concept of eigenvalues and eigenvectors extends naturally to arbitrary linear transformations on arbitrary vector spaces. Let Template:Mvar be any vector space over some field Template:Mvar of scalars, and let Template:Mvar be a linear transformation mapping Template:Mvar into Template:Mvar, $$T:V\to V.$$

We say that a nonzero vector Template:Math is an 'Template:Dfni' of Template:Mvar if and only if there exists a scalar Template:Math such that Template:NumBlk

This equation is called the eigenvalue equation for Template:Mvar, and the scalar Template:Mvar is the eigenvalue of Template:Mvar corresponding to the eigenvector Template:Math. Template:Math is the result of applying the transformation Template:Mvar to the vector Template:Math, while Template:Math is the product of the scalar Template:Mvar with Template:Math.Template:SfnTemplate:Sfn

Eigenspaces, geometric multiplicity, and the eigenbasis

Given an eigenvalue Template:Mvar, consider the set $$E=\{𝐯:T(𝐯)=\lambda 𝐯\},$$ which is the union of the zero vector with the set of all eigenvectors associated with Template:Mvar. Template:Mvar is called the Template:Dfni or Template:Dfni of Template:Mvar associated with Template:Mvar.Template:Sfn It is the kernel of the linear transformation Template:Math.

By definition of a linear transformation, $$\begin{array}{cc}T(𝐱+𝐲)& =T(𝐱)+T(𝐲),\\ T(\alpha 𝐱)& =\alpha T(𝐱),\end{array}$$ for Template:Math and Template:Math. Therefore, if Template:Math and Template:Math are eigenvectors of Template:Mvar associated with eigenvalue Template:Mvar, namely Template:Math, then $$\begin{array}{cc}T(𝐮+𝐯)& =\lambda (𝐮+𝐯),\\ T(\alpha 𝐯)& =\lambda (\alpha 𝐯).\end{array}$$ So, both Template:Math and Template:Math are either zero or eigenvectors of Template:Mvar associated with Template:Mvar, namely Template:Math, Template:Math, and Template:Mvar is closed under addition and scalar multiplication. The eigenspace Template:Mvar associated with Template:Mvar is therefore a linear subspace of Template:Mvar.Template:Sfnm If that subspace has dimension 1, it is sometimes called an Template:Dfni.Template:Sfn

The Template:Dfni Template:Math of an eigenvalue Template:Mvar is the dimension of the eigenspace associated with Template:Mvar; that is, the maximum number of linearly independent eigenvectors associated with that eigenvalue.Template:Sfnm By the definition of eigenvalues and eigenvectors, Template:Math because every eigenvalue has at least one eigenvector.

The eigenspaces of Template:Mvar always form a direct sum. As a consequence, eigenvectors of Template:Em eigenvalues are always linearly independent. Therefore, the sum of the dimensions of the eigenspaces cannot exceed the dimension Template:Mvar of the vector space on which Template:Mvar operates, and there cannot be more than Template:Mvar distinct eigenvalues.Template:Efn

Any subspace spanned by eigenvectors of Template:Mvar is an invariant subspace of Template:Mvar, and the restriction of Template:Mvar to such a subspace is diagonalizable. Moreover, if the entire vector space Template:Mvar can be spanned by the eigenvectors of Template:Mvar, or equivalently if the direct sum of the eigenspaces associated with all the eigenvalues of Template:Mvar is the entire vector space Template:Mvar, then a basis of Template:Mvar called an Template:Dfni can be formed from linearly independent eigenvectors of Template:Mvar. When Template:Mvar admits an eigenbasis, Template:Mvar is diagonalizable.

Spectral theory

If Template:Mvar is an eigenvalue of Template:Mvar, then the operator Template:Math is not one-to-one, and therefore its inverse Template:Math does not exist. The converse is true for finite-dimensional vector spaces, but not for infinite-dimensional vector spaces. In general, the operator Template:Math may not have an inverse even if Template:Mvar is not an eigenvalue.

For this reason, in functional analysis eigenvalues can be generalized to the spectrum of a linear operator Template:Mvar as the set of all scalars Template:Mvar for which the operator Template:Math has no bounded inverse. The spectrum of an operator always contains all its eigenvalues but is not limited to them.

Associative algebras and 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 representationTemplate:Snd an associative algebra acting on a module. The study of such actions is the field of representation theory.

The 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 equations have the form $$x_t=a_1{x}_{t-1}+a_2{x}_{t-2}+\cdots +a_k{x}_{t-k}.$$ The solution of this equation for Template:Mvar in terms of Template:Mvar is found by using its characteristic equation $${\lambda }^k-a_1{\lambda }^{k-1}-a_2{\lambda }^{k-2}-\cdots -a_{k-1}\lambda -a_k=0,$$ which can be found by stacking into matrix form a set of equations consisting of the above difference equation and the Template:Math equations Template:Math, ..., Template:Math, giving a Template:Mvar-dimensional system of the first order in the stacked variable vector Template:Math in terms of its once-lagged value, and taking the characteristic equation of this system's matrix. This equation gives Template:Mvar characteristic roots Template:Math, for use in the solution equation $$x_t=c_1{\lambda }_1^t+\cdots +c_k{\lambda }_k^t.$$

A similar procedure is used for solving a differential equation of the form $$\frac{d^{k}x}{d{t}^k}+a_{k-1}\frac{d^{k-1}x}{d{t}^{k-1}}+\cdots +a_1\frac{dx}{dt}+a_{0}x=0.$$

Calculation

Template:Main 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 Template:Mvar can be determined by finding the roots of the characteristic polynomial. This is easy for Template:Resx 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.Template:Sfn However, this approach is not viable in practice because the coefficients would be contaminated by unavoidable round-off errors, and the roots of a polynomial can be an extremely sensitive function of the coefficients (as exemplified by Wilkinson's polynomial).Template:Sfn 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 Template:Resx matrix is a sum of Template:Math different products.Template:Efn

Explicit algebraic formulas for the roots of a polynomial exist only if the degree Template:Mvar 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 Template:Mvar is the characteristic polynomial of some companion matrix of order Template:Mvar.) 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 methods. Even the 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 system of linear equations with known coefficients. For example, once it is known that 6 is an eigenvalue of the matrix $$A=\left[\begin{array}{cc}4& 1\\ 6& 3\end{array}\right]$$ we can find its eigenvectors by solving the equation Template:Math, that is $$\left[\begin{array}{cc}4& 1\\ 6& 3\end{array}\right]\left[\begin{array}{c}x\\ y\end{array}\right]=6\cdot \left[\begin{array}{c}x\\ y\end{array}\right]$$ This matrix equation is equivalent to two linear equations $$\{\begin{array}{cc}4x+\phantom{3}y& =6x\\ 6x+3y& =6y\end{array}$$ that is, $$\{\begin{array}{cc}-2x+\phantom{3}y& =0\\ 6x-3y& =0\end{array}$$

Both equations reduce to the single linear equation Template:Math. Therefore, any vector of the form Template:Math, for any nonzero real number Template:Mvar, is an eigenvector of Template:Mvar with eigenvalue Template:Math.

The matrix Template:Mvar above has another eigenvalue Template:Math. A similar calculation shows that the corresponding eigenvectors are the nonzero solutions of Template:Math, that is, any vector of the form Template:Math, for any nonzero real number Template:Mvar.

Simple iterative methods

Template:Main 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. A variation is to instead multiply the vector by Template:Math; this causes it to converge to an eigenvector of the eigenvalue closest to Template:Nowrap

If Template:Math is (a good approximation of) an eigenvector of Template:Mvar, then the corresponding eigenvalue can be computed as $$\lambda =\frac{{𝐯}^{\ast }A𝐯}{{𝐯}^{\ast }𝐯}$$ where Template:Math denotes the conjugate transpose of Template:Math.

Modern methods

Efficient, accurate methods to compute eigenvalues and eigenvectors of arbitrary matrices were not known until the QR algorithm was designed in 1961.Template:Sfn Combining the Householder transformation with the LU decomposition results in an algorithm with better convergence than the QR algorithm.Template:Citation needed For large Hermitian sparse matrices, the Lanczos algorithm is one example of an efficient iterative method to compute eigenvalues and eigenvectors, among several other possibilities.Template:Sfn

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

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 Template:Resx matrices, eigenvalues, and eigenvectors.

Eigenvalues of geometric transformations

	Scaling	Unequal scaling	Rotation	Horizontal shear	Hyperbolic rotation
Illustration
Matrix	$\left[\begin{array}{cc}k& 0\\ 0& k\end{array}\right]$	$\left[\begin{array}{cc}{k}_1& 0\\ 0& k_2\end{array}\right]$	$\left[\begin{array}{cc}\cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{array}\right]$	$\left[\begin{array}{cc}1& k\\ 0& 1\end{array}\right]$	$\left[\begin{array}{cc}\cosh \phi & \sinh \phi \\ \sinh \phi & \cosh \phi \end{array}\right]$
Characteristic polynomial	$(\lambda -k{)}^2$	$(\lambda -k_1)(\lambda -k_2)$	${\lambda }^2-2\cos (\theta )\lambda +1$	$(\lambda -1{)}^2$	${\lambda }^2-2\cosh (\phi )\lambda +1$
Eigenvalues, ${\lambda }_i$	${\lambda }_1={\lambda }_2=k$	$\begin{array}{cc}{\lambda }_1& =k_1\\ {\lambda }_2& =k_2\end{array}$	$\begin{array}{cc}{\lambda }_1& =e^{i\theta }\\ & =\cos \theta +i\sin \theta \\ {\lambda }_2& =e^{-i\theta }\\ & =\cos \theta -i\sin \theta \end{array}$	${\lambda }_1={\lambda }_2=1$	$\begin{array}{cc}{\lambda }_1& =e^{\phi }\\ & =\cosh \phi +\sinh \phi \\ {\lambda }_2& =e^{-\phi }\\ & =\cosh \phi -\sinh \phi \end{array}$
Algebraic Template:Abbr, ${\mu }_i=\mu ({\lambda }_i)$	${\mu }_1=2$	$\begin{array}{cc}{\mu }_1& =1\\ {\mu }_2& =1\end{array}$	$\begin{array}{cc}{\mu }_1& =1\\ {\mu }_2& =1\end{array}$	${\mu }_1=2$	$\begin{array}{cc}{\mu }_1& =1\\ {\mu }_2& =1\end{array}$
Geometric Template:Abbr, ${\gamma }_i=\gamma ({\lambda }_i)$	${\gamma }_1=2$	$\begin{array}{cc}{\gamma }_1& =1\\ {\gamma }_2& =1\end{array}$	$\begin{array}{cc}{\gamma }_1& =1\\ {\gamma }_2& =1\end{array}$	${\gamma }_1=1$	$\begin{array}{cc}{\gamma }_1& =1\\ {\gamma }_2& =1\end{array}$
Eigenvectors	All nonzero vectors	$\begin{array}{cc}{𝐮}_1& =\left[\begin{array}{c}1\\ 0\end{array}\right]\\ {𝐮}_2& =\left[\begin{array}{c}0\\ 1\end{array}\right]\end{array}$	$\begin{array}{cc}{𝐮}_1& =\left[\begin{array}{c}1\\ -i\end{array}\right]\\ {𝐮}_2& =\left[\begin{array}{c}1\\ +i\end{array}\right]\end{array}$	${𝐮}_1=\left[\begin{array}{c}1\\ 0\end{array}\right]$	$\begin{array}{cc}{𝐮}_1& =\left[\begin{array}{c}1\\ 1\end{array}\right]\\ {𝐮}_2& =\left[\begin{array}{c}1\\ -1\end{array}\right]\end{array}$

The characteristic equation for a rotation is a quadratic equation with discriminant Template:Math, which is a negative number whenever Template:Mvar is not an integer multiple of Template:Pi (180°). Therefore, except for these special cases, the two eigenvalues are complex numbers, Template: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

PCA of the multivariate Gaussian distribution centered at Template:Math with a standard deviation of Template:Math in roughly the Template:Math direction and of Template:Math 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.

The eigendecomposition of a symmetric positive semidefinite (PSD) 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 analysis, where the sample covariance matrices are PSD. This orthogonal decomposition is called principal component analysis (PCA) in statistics. PCA studies linear relations 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 and the eigenvalues to the 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 sets, such as those encountered in bioinformatics. In Q methodology, the eigenvalues of the correlation matrix determine the Q-methodologist's judgment of Template:Em significance (which differs from the statistical significance of hypothesis testing; cf. criteria for determining the number of factors). More generally, principal component analysis can be used as a method of factor analysis in structural equation modeling.

Graphs

In spectral graph theory, an eigenvalue of a graph is defined as an eigenvalue of the graph's adjacency matrix Template:Mvar, or (increasingly) of the graph's Laplacian matrix due to its discrete Laplace operator, which is either Template:Math (sometimes called the Template:Dfni) or Template:Math (sometimes called the Template:Dfni), where Template:Mvar is a diagonal matrix with Template:Mvar equal to the degree of vertex Template:Mvar, and in Template:Math, the Template:Mathth diagonal entry is Template:Nowrap The Template:Mvarth principal eigenvector of a graph is defined as either the eigenvector corresponding to the Template:Mvarth largest or Template:Mvarth 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 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

Mode shape of a tuning fork at eigenfrequency Template:Val

Eigenvalue problems occur naturally in the vibration analysis of mechanical structures with many degrees of freedom. The eigenvalues are the natural frequencies (or 'Template:Dfni') of vibration, and the eigenvectors are the shapes of these vibrational modes. In particular, undamped vibration is governed by $$m\ddot{x}+kx=0$$ or $$m\ddot{x}=-kx$$

That is, acceleration is proportional to position (i.e., we expect Template:Mvar to be sinusoidal in time).

In Template:Mvar dimensions, Template:Mvar becomes a mass matrix and Template:Mvar a stiffness matrix. Admissible solutions are then a linear combination of solutions to the generalized eigenvalue problem $$kx={\omega }^{2}mx$$ where Template:Math is the eigenvalue and Template:Mvar is the (imaginary) angular frequency. The principal vibration modes are different from the principal compliance modes, which are the eigenvectors of Template:Mvar alone. Furthermore, damped vibration, governed by $$m\ddot{x}+c\dot{x}+kx=0$$ leads to a so-called quadratic eigenvalue problem, $$({\omega }^{2}m+\omega c+k)x=0.$$ This can be reduced to a generalized eigenvalue problem by algebraic manipulation at the cost of solving a larger system.

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.

Tensor of moment of inertia

In mechanics, the eigenvectors of the moment of inertia tensor define the 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 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 components; the components it does have are the principal components.

Schrödinger equation

The wavefunctions associated with the bound states of an electron in a hydrogen atom can be seen as the eigenvectors of the hydrogen atom Hamiltonian as well as of the angular momentum operator. They are associated with eigenvalues interpreted as their energies (increasing downward: Template:Math) and angular momentum (increasing across: s, p, d, ...). The illustration shows the square of the absolute value of the wavefunctions. Brighter areas correspond to higher probability density for a position measurement. The center of each figure is the atomic nucleus, a proton.

An example of an eigenvalue equation where the transformation Template:Mvar is represented in terms of a differential operator is the time-independent Schrödinger equation in quantum mechanics: $$H{\psi }_E=E{\psi }_E$$ where the Hamiltonian Template:Mvar is a second-order differential operator, and the wavefunction Template:Mvar is one of its eigenfunctions corresponding to the eigenvalue Template:Mvar, 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 Template:Mvar within the space of square integrable functions. Since this space is a Hilbert space with a well-defined scalar product, one can introduce a basis set in which Template:Mvar and Template:Mvar 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 Template:Math. In this notation, the Schrödinger equation is: $$H|{\mathrm{\Psi }}_E⟩=E|{\mathrm{\Psi }}_E⟩$$ where Template:Math is an 'Template:Dfni' of Template:Mvar, and Template:Mvar represents the eigenvalue. Template:Mvar is an observable self-adjoint operator, the infinite-dimensional analog of Hermitian matrices. As in the matrix case, in the equation above Template:Math is understood to be the vector obtained by application of the transformation Template:Mvar to Template:Math.

Wave transport

Light, acoustic waves, and microwaves are randomly 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 Template:Math.Template:SfnTemplate:Sfn The eigenvectors of the transmission operator Template: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, Template:Mvar, of Template: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 Template:Math and Template:Math.Template:Sfn Furthermore, one of the striking properties of open eigenchannels, beyond the perfect transmittance, is the statistically robust spatial profile of the eigenchannels.Template:Sfn

Molecular orbitals

In quantum mechanics, and in particular in atomic and molecular physics, within the Hartree–Fock theory, the atomic and molecular orbitals can be defined by the eigenvectors of the Fock operator. The corresponding eigenvalues are interpreted as ionization potentials 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. This particular representation is a generalized eigenvalue problem called Roothaan equations.

Geology and glaciology

In geology, especially in the study of glacial till, (roughly the rock below the ice), eigenvectors and eigenvalues are used to represent data about orientation. When the data is given in the form eigenvalue, eigenvector azimuth◦, eigenvector plunge◦, this allows understanding the eigenvector's orientation in a real world setting. The three different eigenvalues of the three eigenvectors describing a sample can be compared to determine a possible preferred direction of orientation.

Basic reproduction number

Template:Main The basic reproduction number (Template: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 Template:Math is the average number of people that one typical infectious person will infect. The generation time of an infection is the time, Template:Mvar, 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 Template:Mvar has passed. The value Template:Math is then the largest eigenvalue of the next generation matrix.Template:SfnTemplate:Sfn

Eigenfaces

Eigenfaces as examples of eigenvectors

Template:Main In image processing, processed images of faces can be seen as vectors whose components are the brightnesses of each pixel.Template:Sfn 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 eigenfaces; this is an example of principal component analysis. They are very useful for synthesizing images as a linear combination of vectors. In the facial recognition branch of biometrics, eigenfaces provide a means of applying data compression to faces for 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

Sources

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Further reading

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External links

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• What are Eigen Values? – non-technical introduction from PhysLink.com's "Ask the Experts"
• Eigen Values and Eigen Vectors Numerical Examples – Tutorial and Interactive Program from Revoledu.
• Introduction to Eigen Vectors and Eigen Values – lecture from Khan Academy
• Eigenvectors and eigenvalues | Essence of linear algebra, chapter 10 – A visual explanation with 3Blue1Brown
• Matrix Eigenvectors Calculator from Symbolab (Click on the bottom right button of the 2×12 grid to select a matrix size. Select an $n\times n$ size (for a square matrix), then fill out the entries numerically and click on the Go button. It can accept complex numbers as well.)

Theory

• Computation of Eigenvalues
• Numerical solution of eigenvalue problems Edited by Zhaojun Bai, James Demmel, Jack Dongarra, Axel Ruhe, and Henk van der Vorst