Eigenvalues and eigenvectors

Overview

Eigenvalues and eigenvectors feature prominently in the analysis of linear transformations. The prefix eigen- is adopted from the German eigen (cognate with the English word own) for 'proper', 'characteristic', 'own'. 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 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 $${\displaystyle T(\mathbf{v})=\lambda \mathbf{v},}$$ referred to as the eigenvalue equation or eigenequation. In general, λ may be any 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.

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 ⁠${\displaystyle \frac{d}{dx}}$⁠, in which case the eigenvectors are functions called eigenfunctions that are scaled by that differential operator, such as $${\displaystyle \frac{d}{dx}{e}^{\lambda x}=\lambda e^{\lambda x}.}$$ Alternatively, the linear transformation could take the form of an n × n matrix, in which case the eigenvectors are n × 1 matrices.

If the linear transformation is expressed in the form of an n × n matrix A, then the eigenvalue equation for a linear transformation above can be rewritten as the matrix multiplication $${\displaystyle A\mathbf{v}=\lambda \mathbf{v},}$$ where the eigenvector v is an n × 1 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.
• 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.
• If a set of eigenvectors of T forms a basis of the domain of T, 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. Joseph-Louis Lagrange realized that the principal axes are the eigenvectors of the inertia matrix.

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. Cauchy also coined the term racine caractéristique (characteristic root), for what is now called eigenvalue; his term survives in characteristic equation.

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). 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. This was extended by Charles Hermite in 1855 to what are now called Hermitian matrices.

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

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. 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.

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

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 and Vera Kublanovskaya in 1961.

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, which is especially common in numerical and computational applications.

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

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

If it occurs that ⁠${\displaystyle \mathbf{v}}$⁠ and ⁠${\displaystyle \mathbf{w}}$⁠ are scalar multiples, that is, if

then ⁠${\displaystyle \mathbf{v}}$⁠ is an eigenvector of the linear transformation ⁠${\displaystyle A}$⁠ and the scale factor ⁠${\displaystyle \lambda }$⁠ is the eigenvalue corresponding to that eigenvector. Equation (1) is the eigenvalue equation for the matrix ⁠${\displaystyle A}$⁠.

Equation (1) can be stated equivalently as

where ⁠${\displaystyle I}$⁠ is the ⁠${\displaystyle n\times n}$⁠ identity matrix and ⁠${\displaystyle \mathbf{0}}$⁠ is the zero vector.

Eigenvalues and characteristic polynomial

Equation (2) has a nonzero solution v if and only if the determinant of the matrix (A − λI) is zero. Therefore, the eigenvalues of A are values of λ that satisfy the equation

Using the Leibniz formula for determinants, the left-hand side of equation (3) is a polynomial function of the variable λ and the degree of this polynomial is n, the order of the matrix A. Its coefficients depend on the entries of A, except that its term of degree n is always (−1)ⁿλⁿ. This polynomial is called the characteristic polynomial of A. Equation (3) is called the characteristic equation or secular equation of A.

The characteristic polynomial of an n-by-n matrix A, being a polynomial of degree n, has at most n 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 n linear terms:

where the complex numbers λ₁, λ₂, ..., λ_n, 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 $${\displaystyle A=\left[\begin{array}{cc}2& 1\\ 1& 2\end{array}\right].}$$ Taking the determinant of (A − λI), the characteristic polynomial of A is $${\displaystyle det(A-\lambda I)=\left|\begin{array}{cc}2-\lambda & 1\\ 1& 2-\lambda \end{array}\right|=3-4\lambda +{\lambda }^2.}$$ Setting the characteristic polynomial equal to zero, it has roots at λ = 1 and λ = 3, which are the two eigenvalues of A. The eigenvectors corresponding to each eigenvalue λ can be found by solving for the components of v in the equation (A − λI)v = 0. In this example, the eigenvectors are any nonzero scalar multiples of $${\displaystyle {\mathbf{v}}_{\lambda =1}=\left[\begin{array}{c}1\\ -1\end{array}\right],{\mathbf{v}}_{\lambda =3}=\left[\begin{array}{c}1\\ 1\end{array}\right].}$$

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 numbers even if all the entries of A are rational numbers or even if they are all integers. However, if the entries of A 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 λ_i be an eigenvalue of an n-by-n matrix A. The algebraic multiplicity μ_A(λ_i) of the eigenvalue is its multiplicity as a root of the characteristic polynomial, that is, the largest integer k such that (λ_i − λ)^k evenly divides that polynomial.

Suppose a matrix A has order n and d ≤ n distinct eigenvalues. Whereas equation (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: $${\displaystyle det(A-\lambda I)=({\lambda }_1-\lambda {)}^{{\mu }_A({\lambda }_1)}({\lambda }_2-\lambda {)}^{{\mu }_A({\lambda }_2)}\cdots ({\lambda }_d-\lambda {)}^{{\mu }_A({\lambda }_d)}.}$$ If d = n, then the right-hand side is the product of n linear terms, and this is the same as equation (4). The size of each eigenvalue's algebraic multiplicity is related to the dimension n as $${\displaystyle \begin{array}{cc}1& \le {\mu }_A({\lambda }_i)\le n,\\ {\mu }_A& =\sum _{i=1}^d{\mu }_A\left({\lambda }_i\right)=n.\end{array}}$$ If μ_A(λ_i) = 1, then λ_i is said to be a simple eigenvalue. If μ_A(λ_i) equals the geometric multiplicity of λ_i (denoted by γ_A(λ_i) and defined in the next section), then λ_i is said to be a semisimple eigenvalue.

Eigenspaces, geometric multiplicities, and eigenbasis for a matrix

Given a particular eigenvalue ⁠${\displaystyle \lambda }$⁠ of the ⁠${\displaystyle n\times n}$⁠ matrix ⁠${\displaystyle A}$⁠, define the set ⁠${\displaystyle E}$⁠ to be all vectors ⁠${\displaystyle \mathbf{v}}$⁠ that satisfy equation (2): $${\displaystyle E=\left\{\mathbf{v}:\left(A-\lambda I\right)\mathbf{v}=\mathbf{0}\right\}.}$$ On one hand, ⁠${\displaystyle E}$⁠ is precisely the kernel or nullspace of the matrix ⁠${\displaystyle A-\lambda I}$⁠. On the other hand, by definition, any nonzero vector that satisfies this condition is an eigenvector of ⁠${\displaystyle A}$⁠ associated with ⁠${\displaystyle \lambda }$⁠; so ⁠${\displaystyle E}$⁠ is the union of the zero vector with the set of all eigenvectors of ⁠${\displaystyle A}$⁠ associated with ⁠${\displaystyle \lambda }$⁠. The space ⁠${\displaystyle E}$⁠ is called the eigenspace or characteristic space of ⁠${\displaystyle A}$⁠ associated with ⁠${\displaystyle \lambda }$⁠. In general, ⁠${\displaystyle \lambda }$⁠ is a complex number and the eigenvectors are complex ⁠${\displaystyle n\times 1}$⁠ matrices (column vectors). Because every nullspace is a linear subspace of the domain, ⁠${\displaystyle E}$⁠ is a linear subspace of ⁠${\displaystyle {\mathbb{C}}^n}$⁠.

Because the eigenspace ⁠${\displaystyle E}$⁠ is a linear subspace, it is closed under addition. That is, if two vectors ⁠${\displaystyle \mathbf{u}}$⁠ and ⁠${\displaystyle \mathbf{v}}$⁠ belong to the set ⁠${\displaystyle E}$⁠, written ⁠${\displaystyle \mathbf{u},\mathbf{v}\in E}$⁠, then ⁠${\displaystyle \mathbf{u}+\mathbf{v}\in E}$⁠, or equivalently ⁠${\displaystyle A(\mathbf{u}+\mathbf{v})=\lambda (\mathbf{u}+\mathbf{v})}$⁠. This can be checked using the distributive property of matrix multiplication. Similarly, because ⁠${\displaystyle E}$⁠ is a linear subspace, it is closed under scalar multiplication. That is, if ⁠${\displaystyle \mathbf{v}\in E}$⁠ and ⁠${\displaystyle \alpha \in \mathbb{C}}$⁠, then ⁠${\displaystyle \alpha \mathbf{v}\in E}$⁠, or equivalently ⁠${\displaystyle A(\alpha \mathbf{v})=\lambda (\alpha \mathbf{v})}$⁠. This can be checked by noting that multiplication of complex matrices by complex numbers is commutative. As long as ⁠${\displaystyle \mathbf{u}+\mathbf{v}}$⁠ and ⁠${\displaystyle \alpha \mathbf{v}}$⁠ are not zero, they are also eigenvectors of ⁠${\displaystyle A}$⁠ associated with ⁠${\displaystyle \lambda }$⁠.

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

Proof of inequality ⁠${\displaystyle {\gamma }_A(\lambda )\le {\mu }_A(\lambda )}$⁠: Let B = A − λI, where λ is a fixed complex number, and the eigenspace associated with λ is the nullspace of B. Let the dimension of that eigenspace be ⁠${\displaystyle k={\gamma }_A(\lambda )}$⁠. This means that the last k rows of the echelon form of B are zero. Thus, there is an invertible matrix E coming from Gauss-Jordan reduction, such that $${\displaystyle EB=\left[\begin{array}{cc}\ast & \ast \\ {\mathbf{0}}_{k\times (n-k)}& {\mathbf{0}}_{k\times k}\end{array}\right].}$$ Therefore the last k rows of EB − tE are (−t) times the last k rows of E. Therefore the polynomial t^k evenly divides the polynomial det(EB − tE), because of basic properties of determinants (homogeneity). On the other hand, det(EB − tE) = det E det(B − tI) = p_A(t + λ) det E, so (t − λ)^k divides p_A(t), and so the algebraic multiplicity of λ is at least ⁠${\displaystyle k}$⁠. Q.E.D.

Suppose A has d ≤ n distinct eigenvalues λ₁, ..., λ_d, where the geometric multiplicity of λ_i is γ_A(λ_i). The total geometric multiplicity of A, $${\displaystyle {\gamma }_A=\sum _{i=1}^d{\gamma }_A({\lambda }_i),}$$ is the dimension of the sum of all the eigenspaces of A's eigenvalues, or equivalently the maximum number of linearly independent eigenvectors of A. By construction, ⁠${\displaystyle d\le {\gamma }_A\le n}$⁠. If ⁠${\displaystyle {\gamma }_A=n}$⁠, then:

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

Additional properties

Let A be an arbitrary n × n matrix of complex numbers with eigenvalues λ₁, ..., λ_n. Each eigenvalue appears μ_A(λ_i) times in this list, where μ_A(λ_i) is the eigenvalue's algebraic multiplicity. The following are properties of this matrix and its eigenvalues:

• The trace of ⁠${\displaystyle A}$⁠, defined as the sum of its diagonal elements, is also the sum of all its eigenvalues: $${\displaystyle \mathrm{tr}(A)=\sum _{i=1}^n{a}_{ii}=\sum _{i=1}^n{\lambda }_i={\lambda }_1+{\lambda }_2+\cdots +{\lambda }_n.}$$
• The determinant of ⁠${\displaystyle A}$⁠ is the product of all its eigenvalues: $${\displaystyle det(A)=\prod _{i=1}^n{\lambda }_i={\lambda }_1{\lambda }_2\cdots {\lambda }_n.}$$
• For any positive integer ⁠${\displaystyle k}$⁠, the eigenvalues of the ⁠${\displaystyle k}$⁠th power of ⁠${\displaystyle A}$⁠, that is, ⁠${\displaystyle A^k}$⁠, are ⁠${\displaystyle {\lambda }_1^k,\dots ,{\lambda }_n^k}$⁠.
• The eigenvalues of matrix ⁠${\displaystyle A+I}$⁠ (where ⁠${\displaystyle I}$⁠ is the identity matrix) are ⁠${\displaystyle {\lambda }_1+1,\dots ,{\lambda }_n+1}$⁠. Moreover, for any ⁠${\displaystyle \alpha \in \mathbb{C}}$⁠, the eigenvalues of matrix ⁠${\displaystyle A+\alpha I}$⁠ are ⁠${\displaystyle {\lambda }_1+\alpha ,\dots ,{\lambda }_n+\alpha }$⁠.
• More generally, for any polynomial ⁠${\displaystyle P}$⁠, the eigenvalues of matrix ⁠${\displaystyle P(A)}$⁠ are ⁠${\displaystyle P({\lambda }_1),\dots ,P({\lambda }_n)}$⁠.
• ⁠${\displaystyle A}$⁠ is invertible if and only if every eigenvalue ⁠${\displaystyle {\lambda }_i}$⁠ is nonzero.
• If ⁠${\displaystyle A}$⁠ is invertible, then the eigenvalues of ⁠${\displaystyle A^{-1}}$⁠ are $\frac{1}{{\lambda }_1},\dots ,\frac{1}{{\lambda }_n}$ and for each pair of corresponding eigenvalues, the geometric multiplicities ⁠${\displaystyle {\gamma }_A({\lambda }_i)}$⁠ 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 ⁠${\displaystyle {\mu }_A({\lambda }_i)}$⁠ and ${\mu }_{A^{-1}}(\frac{1}{{\lambda }_i})$ coincide.
• If ⁠${\displaystyle A}$⁠ is equal to its conjugate transpose ⁠${\displaystyle A^{\ast }}$⁠, that is, ⁠${\displaystyle A}$⁠ is Hermitian, then every eigenvalue ⁠${\displaystyle {\lambda }_i}$⁠ is real. The same is true of any symmetric real matrix.
• If ⁠${\displaystyle A}$⁠ is not only Hermitian but also positive-definite, positive-semidefinite, negative-definite, or negative-semidefinite, then every eigenvalue ⁠${\displaystyle {\lambda }_i}$⁠ is positive, non-negative, negative, or non-positive, respectively.
• If ⁠${\displaystyle A}$⁠ is unitary, then every eigenvalue ⁠${\displaystyle {\lambda }_i}$⁠ has absolute value ⁠${\displaystyle |{\lambda }_i|=1}$⁠.

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 'right eigenvector', namely a column vector that right multiplies the n × n matrix A in the defining equation, equation (1), $${\displaystyle A\mathbf{v}=\lambda \mathbf{v}.}$$ The eigenvalue and eigenvector problem can also be defined for row vectors that left multiply matrix A. In this formulation, the defining equation is $${\displaystyle \mathbf{u}A=\kappa \mathbf{u},}$$ where κ is a scalar and u is a 1 × n matrix. Any row vector u satisfying this equation is called a 'left eigenvector' of A, and κ is still called its associated eigenvalue. Taking the transpose of this equation, $${\displaystyle A^{\mathsf{T}}{\mathbf{u}}^{\mathsf{T}}=\kappa {\mathbf{u}}^{\mathsf{T}}.}$$

Comparing this equation to equation (1), it follows immediately that a left eigenvector of ⁠${\displaystyle A}$⁠ is the same as the transpose of a right eigenvector of ⁠${\displaystyle A^{\mathsf{T}}}$⁠, with the same eigenvalue. Furthermore, since the characteristic polynomial of ⁠${\displaystyle A^{\mathsf{T}}}$⁠ is the same as the characteristic polynomial of ⁠${\displaystyle A}$⁠, the left and right eigenvectors of ⁠${\displaystyle A}$⁠ 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 ⁠${\displaystyle \lambda }$⁠ is an eigenvalue of an ⁠${\displaystyle n\times n}$⁠ matrix ⁠${\displaystyle A}$⁠ with eigenvector ⁠${\displaystyle x}$⁠. Then ⁠${\displaystyle Ax=\lambda x}$⁠; equivalently, ⁠${\displaystyle (A-\lambda I)x=0}$⁠.

Thus, the columns of ⁠${\displaystyle (A-\lambda I)}$⁠ are linearly dependent. Equivalently, the rank of the matrix is less than ⁠${\displaystyle n}$⁠.

But as column rank = row rank, the rows are also linearly dependent. Hence, there are numbers ⁠${\displaystyle y_1,y_2,\dots ,y_n}$⁠, not all zero, such that $\sum _{i=1}^{i=n}{y}_i{R}_i=0,$ where the ⁠${\displaystyle R_i}$⁠'s are the rows of ⁠${\displaystyle (A-\lambda I)}$⁠. Let the row vector ⁠${\displaystyle y=(y_1,y_2,\dots ,y_n)}$⁠; then ⁠${\displaystyle y(A-\lambda I)=0}$⁠. Taking the transpose, ⁠${\displaystyle (A^{\mathsf{T}}-\lambda I)y^{\mathsf{T}}=0}$⁠. Moreover, ⁠${\displaystyle y^{\mathsf{T}}}$⁠ is not the zero vector; so ⁠${\displaystyle \lambda }$⁠ is also an eigenvalue of ⁠${\displaystyle A^{\mathsf{T}}}$⁠.

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

Diagonalization and eigendecomposition

Suppose the eigenvectors of A form a basis of ⁠${\displaystyle {\mathbb{C}}^n}$⁠, or equivalently A has n linearly independent eigenvectors v₁, v₂, ..., v_n (with associated eigenvalues λ₁, λ₂, ..., λ_n). The eigenvectors need not be orthogonal to one another, and the eigenvalues need not be distinct. Define the square matrix Q whose columns are the n linearly independent eigenvectors of A, $${\displaystyle Q=\left[\begin{array}{cccc}{\mathbf{v}}_1& {\mathbf{v}}_2& \cdots & {\mathbf{v}}_n\end{array}\right].}$$ Since each column of Q is an eigenvector of A, right multiplying A by Q scales each column of Q by its associated eigenvalue: $${\displaystyle AQ=\left[\begin{array}{cccc}{\lambda }_1{\mathbf{v}}_1& {\lambda }_2{\mathbf{v}}_2& \cdots & {\lambda }_n{\mathbf{v}}_n\end{array}\right].}$$

With this in mind, define the diagonal matrix Λ where each diagonal element Λ_ii is the eigenvalue associated with the ith column of Q. Then $${\displaystyle AQ=Q\Lambda .}$$ Because the columns of Q are linearly independent, Q is invertible. Right multiplying both sides of the equation by Q⁻¹, $${\displaystyle A=Q\Lambda Q^{-1};}$$ or instead left multiplying both sides by Q⁻¹, $${\displaystyle Q^{-1}AQ=\Lambda .}$$ 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; it is a similarity transformation. Such a matrix A is said to be similar to the diagonal matrix Λ, or 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⁻¹AP is some diagonal matrix D. Left multiplying both by P yields 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.

In conclusion, the eigenvectors of ⁠${\displaystyle A}$⁠ form a basis of ⁠${\displaystyle {\mathbb{C}}^n}$⁠ if and only if ⁠${\displaystyle A}$⁠ 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 A 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 H is the maximum value of the quadratic form x^THx/x^Tx. A value of x that realizes that maximum is an eigenvector.

Matrix examples

Two-dimensional matrix example

Consider the matrix $${\displaystyle A=\left[\begin{array}{cc}2& 1\\ 1& 2\end{array}\right].}$$ The figure on the right shows the effect of this transformation on point coordinates in the plane. The eigenvectors v of this transformation satisfy equation (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, $${\displaystyle \begin{array}{cc}det(A-\lambda I)& =\left|\left[\begin{array}{cc}2& 1\\ 1& 2\end{array}\right]-\lambda \left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]\right|=\left|\begin{array}{cc}2-\lambda & 1\\ 1& 2-\lambda \end{array}\right|\\ & =3-4\lambda +{\lambda }^2\\ & =(\lambda -3)(\lambda -1).\end{array}}$$ Setting the characteristic polynomial equal to zero, it has roots at λ = 1 and λ = 3, which are the two eigenvalues of A.

For λ = 1, equation (2) becomes, $${\displaystyle \begin{array}{cc}(A-I){\mathbf{v}}_{\lambda =1}& =\left[\begin{array}{cc}1& 1\\ 1& 1\end{array}\right]\left[\begin{array}{c}{v}_1\\ v_2\end{array}\right]=\left[\begin{array}{c}0\\ 0\end{array}\right]\\ 1v_1+1v_2& =0\end{array}}$$ Any nonzero vector with v₁ = −v₂ solves this equation. Therefore, $${\displaystyle {\mathbf{v}}_{\lambda =1}=\left[\begin{array}{c}{v}_1\\ -v_1\end{array}\right]=\left[\begin{array}{c}1\\ -1\end{array}\right]}$$ is an eigenvector of A corresponding to λ = 1, as is any scalar multiple of this vector.

For λ = 3, equation (2) becomes $${\displaystyle \begin{array}{cc}(A-3I){\mathbf{v}}_{\lambda =3}& =\left[\begin{array}{cc}-1& \phantom{-}1\\ \phantom{-}1& -1\end{array}\right]\left[\begin{array}{c}{v}_1\\ v_2\end{array}\right]=\left[\begin{array}{c}0\\ 0\end{array}\right]\\ -1v_1+1v_2& =0;\\ 1v_1-1v_2& =0\end{array}}$$ Any nonzero vector with v₁ = v₂ solves this equation. Therefore, $${\displaystyle {\mathbf{v}}_{\lambda =3}=\left[\begin{array}{c}{v}_1\\ v_1\end{array}\right]=\left[\begin{array}{c}1\\ 1\end{array}\right]}$$ is an eigenvector of A corresponding to λ = 3, as is any scalar multiple of this vector. Thus, the vectors v_λ=1 and v_λ=3 are eigenvectors of A associated with the eigenvalues λ = 1 and λ = 3, respectively.

Three-dimensional matrix example

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

The roots of the characteristic polynomial are 2, 1, and 11, which are the only three eigenvalues of A. These eigenvalues correspond to the eigenvectors [1 0 0]^T, [0 −2 1]^T, and [0 1 2]^T, or any nonzero multiple thereof.

Three-dimensional matrix example with complex eigenvalues

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

This matrix shifts the coordinates of the vector up by one position and moves the first coordinate to the bottom. Its characteristic polynomial is 1 − λ³, whose roots are $${\displaystyle \begin{array}{cc}{\lambda }_1& =1\\ {\lambda }_2& =-\frac{1}{2}+i\frac{\sqrt{3}}{2}\\ {\lambda }_3& ={\lambda }_2^{\ast }=-\frac{1}{2}-i\frac{\sqrt{3}}{2}\end{array}}$$ where i is an imaginary unit with i² = −1.

For the real eigenvalue λ₁ = 1, any vector with three equal nonzero entries is an eigenvector. For example, $${\displaystyle A\left[\begin{array}{c}5\\ 5\\ 5\end{array}\right]=\left[\begin{array}{c}5\\ 5\\ 5\end{array}\right]=1\cdot \left[\begin{array}{c}5\\ 5\\ 5\end{array}\right].}$$

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

Therefore, the other two eigenvectors of A are complex and are v_λ2 = [1 λ₂ λ₃]^T and v_λ3 = [1 λ₃ λ₂]^T with eigenvalues λ₂ and λ₃, respectively. The two complex eigenvectors also appear in a complex conjugate pair, $${\displaystyle {\mathbf{v}}_{{\lambda }_2}={\mathbf{v}}_{{\lambda }_3}^{\ast }.}$$

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 $${\displaystyle A=\left[\begin{array}{ccc}1& 0& 0\\ 0& 2& 0\\ 0& 0& 3\end{array}\right].}$$ The characteristic polynomial of A is $${\displaystyle det(A-\lambda I)=(1-\lambda )(2-\lambda )(3-\lambda ),}$$ which has the roots λ₁ = 1, λ₂ = 2, and λ₃ = 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, $${\displaystyle {\mathbf{v}}_{{\lambda }_1}=\left[\begin{array}{c}1\\ 0\\ 0\end{array}\right],{\mathbf{v}}_{{\lambda }_2}=\left[\begin{array}{c}0\\ 1\\ 0\end{array}\right],{\mathbf{v}}_{{\lambda }_3}=\left[\begin{array}{c}0\\ 0\\ 1\end{array}\right],}$$ 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, $${\displaystyle A=\left[\begin{array}{ccc}1& 0& 0\\ 1& 2& 0\\ 2& 3& 3\end{array}\right].}$$

The characteristic polynomial of A is $${\displaystyle det(A-\lambda I)=(1-\lambda )(2-\lambda )(3-\lambda ),}$$ which has the roots λ₁ = 1, λ₂ = 2, and λ₃ = 3. These roots are the diagonal elements as well as the eigenvalues of A.

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

Matrix with repeated eigenvalues example

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

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

On the other hand, the geometric multiplicity of the eigenvalue 2 is only 1, because its eigenspace is spanned by just one vector [0 1 −1 1]^T and is therefore 1-dimensional. Similarly, the geometric multiplicity of the eigenvalue 3 is 1 because its eigenspace is spanned by just one vector [0 0 0 1]^T. The total geometric multiplicity γ_A is 2, which is the smallest it could be for a matrix with two distinct eigenvalues. Geometric multiplicities are defined in a later section.

Eigenvector-eigenvalue identity

For a Hermitian matrix A, the norm squared of the αth component of a normalized eigenvector can be calculated using only the matrix eigenvalues and the eigenvalues of the corresponding minor matrix, $${\displaystyle |v_{i\alpha }{|}^2=\frac{\prod _k({\lambda }_i(A)-{\lambda }_k(A_{\alpha }))}{\prod _{k\ne i}({\lambda }_i(A)-{\lambda }_k(A))},}$$ where ⁠${\displaystyle A_{\alpha }}$⁠ is the submatrix formed by removing the αth row and column from the original matrix. This identity also extends to diagonalizable matrices. It has been discovered in and rediscovered many times in the literature (see, e.g., ).

Eigenvalues and eigenfunctions of differential operators

The definitions of eigenvalue and eigenvectors of a linear transformation T 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 D be a linear differential operator on the space ${\displaystyle C^{\mathrm{\infty }}(\mathbb{R})}$ of infinitely differentiable real functions of a real argument t. The eigenvalue equation for D is the differential equation $${\displaystyle Df(t)=\lambda f(t)}$$ The functions that satisfy this equation are eigenvectors of D and are commonly called 'eigenfunctions'.

Derivative operator example

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

General definition

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

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

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 λv is the product of the scalar λ with v.

Eigenspaces, geometric multiplicity, and the eigenbasis

Given an eigenvalue λ, consider the set $${\displaystyle E=\left\{\mathbf{v}:T(\mathbf{v})=\lambda \mathbf{v}\right\},}$$ 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. It is the kernel of the linear transformation T − λI.

By definition of a linear transformation, $${\displaystyle \begin{array}{cc}T(\mathbf{x}+\mathbf{y})& =T(\mathbf{x})+T(\mathbf{y}),\\ T(\alpha \mathbf{x})& =\alpha T(\mathbf{x}),\end{array}}$$ 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 $${\displaystyle \begin{array}{cc}T(\mathbf{u}+\mathbf{v})& =\lambda (\mathbf{u}+\mathbf{v}),\\ T(\alpha \mathbf{v})& =\lambda (\alpha \mathbf{v}).\end{array}}$$ So, both u + v and αv are either zero or eigenvectors of T associated with λ, namely u + v, αv ∈ E, and E is closed under addition and scalar multiplication. The eigenspace E associated with λ is therefore a linear subspace of V. If that subspace has dimension 1, it is sometimes called an eigenline.

The geometric multiplicity γ_T(λ) of an eigenvalue λ is the dimension of the eigenspace associated with λ; that is, the maximum number of linearly independent eigenvectors associated with that eigenvalue. By the definition of eigenvalues and eigenvectors, γ_T(λ) ≥ 1 because every eigenvalue has at least one eigenvector.

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

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

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

For this reason, in functional analysis eigenvalues can be generalized to the spectrum of a linear operator T as the set of all scalars λ for which the operator (T − λI) 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 representation – 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 $${\displaystyle x_t=a_1{x}_{t-1}+a_2{x}_{t-2}+\cdots +a_k{x}_{t-k}.}$$ The solution of this equation for x in terms of t is found by using its characteristic equation $${\displaystyle {\lambda }^k-a_1{\lambda }^{k-1}-a_2{\lambda }^{k-2}-\cdots -a_{k-1}\lambda -a_k=0,}$$ which can be found by stacking into matrix form a set of equations consisting of the above difference equation and the k – 1 equations x_t–1 = x_t–1, ..., x_t–k+1 = x_t–k+1, giving a k-dimensional system of the first order in the stacked variable vector [x_t ⋅⋅⋅ x_t–k+1] in terms of its once-lagged value, and taking the characteristic equation of this system's matrix. This equation gives k characteristic roots λ₁, ... , λ_k, for use in the solution equation $${\displaystyle x_t=c_1{\lambda }_1^t+\cdots +c_k{\lambda }_k^t.}$$

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

Calculation

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 A can be determined by finding the roots of the characteristic polynomial. This is easy for 2 × 2 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. 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). 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 n × n matrix is a sum of n! different products.

Explicit algebraic formulas for the roots of a polynomial exist only if the degree n 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 n is the characteristic polynomial of some companion matrix of order n.) 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 $${\displaystyle A=\left[\begin{array}{cc}4& 1\\ 6& 3\end{array}\right]}$$ we can find its eigenvectors by solving the equation Av = 6v, that is $${\displaystyle \left[\begin{array}{cc}4& 1\\ 6& 3\end{array}\right]\left[\begin{array}{c}x\\ y\end{array}\right]=6\cdot \left[\begin{array}{c}x\\ y\end{array}\right]}$$ This matrix equation is equivalent to two linear equations $${\displaystyle \{\begin{array}{cc}4x+\phantom{3}y& =6x\\ 6x+3y& =6y\end{array}}$$ that is, $${\displaystyle \{\begin{array}{cc}-2x+\phantom{3}y& =0\\ 6x-3y& =0\end{array}}$$

Both equations reduce to the single linear equation y = 2x. Therefore, any vector of the form [a 2a]^T, for any nonzero real number a, is an eigenvector of A with eigenvalue λ = 6.

The matrix A above has another eigenvalue λ = 1. A similar calculation shows that the corresponding eigenvectors are the nonzero solutions of 3x + y = 0, that is, any vector of the form [b −3b]^T, for any nonzero real number b.

Simple iterative methods

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 (A − μI)⁻¹; this causes it to converge to an eigenvector of the eigenvalue closest to ${\displaystyle \mu \in \mathbb{C}}$.

If v is (a good approximation of) an eigenvector of A, then the corresponding eigenvalue can be computed as $${\displaystyle \lambda =\frac{{\mathbf{v}}^{\ast }A\mathbf{v}}{{\mathbf{v}}^{\ast }\mathbf{v}}}$$ where v^∗ denotes the conjugate transpose of v.

Modern methods

Efficient, accurate methods to compute eigenvalues and eigenvectors of arbitrary matrices were not known until the QR algorithm was designed in 1961. Combining the Householder transformation with the LU decomposition results in an algorithm with better convergence than the QR algorithm. 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.

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 2 × 2 matrices, eigenvalues, and eigenvectors.

The characteristic equation for a rotation is a quadratic equation with discriminant D = −4(sin θ)², which is a negative number whenever θ is not an integer multiple of π (180°). Therefore, except for these special cases, the two eigenvalues are complex numbers, cos θ ± isin θ; 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

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 practical 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 A, or (increasingly) of the graph's Laplacian matrix due to its discrete Laplace operator, which is either D − A (sometimes called the combinatorial Laplacian) or I − D^−1/2AD^−1/2 (sometimes called the normalized Laplacian), where D is a diagonal matrix with D_ii equal to the degree of vertex v_i, and in D^−1/2, the ith diagonal entry is $1/\sqrt{\mathrm{deg}(v_i)}$. The kth principal eigenvector of a graph is defined as either the eigenvector corresponding to the kth largest or kth 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

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

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

In n dimensions, m becomes a mass matrix and k a stiffness matrix. Admissible solutions are then a linear combination of solutions to the generalized eigenvalue problem $${\displaystyle kx={\omega }^{2}mx}$$ where ω² is the eigenvalue and ω is the (imaginary) angular frequency. The principal vibration modes are different from the principal compliance modes, which are the eigenvectors of k alone. Furthermore, damped vibration, governed by $${\displaystyle m\ddot{x}+c\dot{x}+kx=0}$$ leads to a so-called quadratic eigenvalue problem, $${\displaystyle \left({\omega }^{2}m+\omega c+k\right)x=0.}$$ 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

An example of an eigenvalue equation where the transformation T is represented in terms of a differential operator is the time-independent Schrödinger equation in quantum mechanics: $${\displaystyle H{\psi }_E=E{\psi }_E}$$ where the Hamiltonian H is a second-order differential operator, and the wavefunction ψ_E is one of its eigenfunctions corresponding to the eigenvalue E, 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 ψ_E 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 ψ_E and H 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 |Ψ_E⟩. In this notation, the Schrödinger equation is: $${\displaystyle H|{\mathrm{\Psi }}_E⟩=E|{\mathrm{\Psi }}_E⟩}$$ where |Ψ_E⟩ is an 'eigenstate' of H, and E represents the eigenvalue. H is an observable self-adjoint operator, the infinite-dimensional analog of Hermitian matrices. As in the matrix case, in the equation above H|Ψ_E⟩ is understood to be the vector obtained by application of the transformation H to |Ψ_E⟩.

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 t. The eigenvectors of the transmission operator t^†t 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, τ, of t^†t 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 τ_max = 1 and τ_min = 0. Furthermore, one of the striking properties of open eigenchannels, beyond the perfect transmittance, is the statistically robust spatial profile of the eigenchannels.

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

The basic reproduction number (R₀) 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 R₀ is the average number of people that one typical infectious person will infect. The generation time of an infection is the time, t_G, 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 t_G has passed. The value R₀ is then the largest eigenvalue of the next generation matrix.

Eigenfaces

In image processing, processed images of faces can be seen as vectors whose components are the brightnesses of each pixel. 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.