Eigendecomposition of a matrix

In the mathematical discipline of linear algebra, eigendecomposition or sometimes spectral decomposition is the factorization of a matrix into a canonical form, whereby the matrix is represented in terms of its eigenvalues and eigenvectors.

Fundamental theory of matrix eigenvectors and eigenvalues

Main article: Eigenvalue, eigenvector and eigenspace

A (non-zero) vector v of dimension N is an eigenvector of a square (N×N) matrix A if and only if it satisfies the linear equation

$\mathbf {A} \mathbf {v} =\lambda \mathbf {v}$

where λ is a scalar, termed the eigenvalue corresponding to v. The above equation is called the eigenvalue equation or the eigenvalue problem.

This yields an equation for the eigenvalues

$p\left(\lambda \right):=\det \left(\mathbf {A} -\lambda \mathbf {I} \right)=0.\!\$

We call p(λ) the characteristic polynomial, and the equation, called the characteristic equation, is an Nth order polynomial equation in the unknown λ. This equation will have N_λ distinct solutions, where 1 ≤ N_λ ≤ N . The set of solutions, i.e. the eigenvalues, is sometimes called the spectrum of A.

We can factor p as

$p\left(\lambda \right)=(\lambda -\lambda _{1})^{n_{1}}(\lambda -\lambda _{2})^{n_{2}}\cdots (\lambda -\lambda _{k})^{n_{k}}=0\!\$

where

$\sum \limits _{i=1}^{N_{\lambda }}{n_{i}}=N.$

For each eigenvalue, λ_i, we have a specific eigenvalue equation

$\left(\mathbf {A} -\lambda _{i}\mathbf {I} \right)\mathbf {v} =0.\!\$

There will be 1 ≤ m_i ≤ n_i linearly independent solutions to each eigenvalue equation. The m_i solutions are the eigenvectors associated with the eigenvalue λ_i. The integer m_i is termed the geometric multiplicity of λ_i. It is important to keep in mind that the algebraic multiplicity n_i and geometric multiplicity m_i may or may not be equal, but we always have m_i ≤ n_i. The simplest case is of course when m_i = n_i = 1. The total number of linearly independent eigenvectors, N_v, can be calculated by summing the geometric multiplicities

$\sum \limits _{i=1}^{N_{\lambda }}{m_{i}}=N_{\mathbf {v} }.$

The eigenvectors can be indexed by eigenvalues, i.e. using a double index, with v_i,j being the j^th eigenvector for the i^th eigenvalue. The eigenvectors can also be indexed using the simpler notation of a single index v_k, with k = 1, 2, ... , N_v.

Eigendecomposition of a matrix

Let A be a square (N×N) matrix with N linearly independent eigenvectors, $q_{i}\,\,(i=1,\dots ,N).$ Then A can be factorized as

$\mathbf {A} =\mathbf {Q} \mathbf {\Lambda } \mathbf {Q} ^{-1}$

where Q is the square (N×N) matrix whose i^th column is the eigenvector $q_{i}$ of A and Λ is the diagonal matrix whose diagonal elements are the corresponding eigenvalues, i.e., $\Lambda _{ii}=\lambda _{i}$.

The eigenvectors $q_{i}\,\,(i=1,\dots ,N)$ are usually normalized, but they need not be. A non-normalized set of eigenvectors, $v_{i}\,\,(i=1,\dots ,N),$ can also be used as the columns of Q. That this is true can be understood by noting that the magnitude of the eigenvectors in Q gets canceled in the decomposition by the presence of Q⁻¹.

Matrix inverse via eigendecomposition

Main article: Inverse matrix

If matrix A can be eigendecomposed and if none of its eigenvalues are zero, then A is nonsingular and its inverse is given by

$\mathbf {A} ^{-1}=\mathbf {Q} \mathbf {\Lambda } ^{-1}\mathbf {Q} ^{-1}$

Because Λ is a diagonal matrix, its inverse is easy to calculate:

$\left[\Lambda ^{-1}\right]_{ii}={\frac {1}{\lambda _{i}}}$

Practical implications

When eigendecomposition is used on a matrix of measured, real data, the inverse may be less valid when all eigenvalues are used unmodified in the form above. This is because as eigenvalues become relatively small, their contribution to the inversion is large. Those near zero or at the "noise" of the measurement system will have undue influence and could hamper solutions (detection) using the inverse.

Two mitigations have been proposed: 1) truncating small/zero eigenvalues, 2) extending the lowest reliable eigenvalue to those below it.

The first mitigation method is similar to a sparse sample of the original matrix, removing components that are not considered valuable. However, if the solution or detection process is near the noise level, truncating may remove components that influence the desired solution.

The second mitigation extends the eigenvalue so that lower values have much less influence over inversion, but do still contribute, such that solutions near the noise will still be found.

The reliable eigenvalue can be found by assuming that eigenvalues of extremely similar and low value are a good representation of measurement noise (which is assumed low for most systems).

If the eigenvalues are rank-sorted by value, then the reliable eigenvalue can be found by minimization of the Laplacian of the sorted eigenvalues:

$\min |\nabla ^{2}\lambda _{s}|$

where the eigenvalues are subscripted with an 's' to denote being sorted. The position of the minimization is the lowest reliable eigenvalue. In measurement systems, the square root of this reliable eigenvalue is the average noise over the components of the system.

Functional calculus

The eigendecomposition allows for much easier computation of power series of matrices. If f(x) is given by

$f(x)=a_{0}+a_{1}x+a_{2}x^{2}+\cdots$

then we know that

$f\left(\mathbf {A} \right)=\mathbf {Q} f\left(\mathbf {\Lambda } \right)\mathbf {Q} ^{-1}$

Because Λ is a diagonal matrix, functions of Λ are very easy to calculate:

$\left[f\left(\mathbf {\Lambda } \right)\right]_{ii}=f\left(\lambda _{i}\right)$

The off-diagonal elements of f(Λ) are zero; that is, f(Λ) is also a diagonal matrix. Therefore, calculating f(A) reduces to just calculating the function on each of the eigenvalues .

A similar technique works more generally with the holomorphic functional calculus, using

$\mathbf {A} ^{-1}=\mathbf {Q} \mathbf {\Lambda } ^{-1}\mathbf {Q} ^{-1}$

from above. Once again, we find that

$\left[f\left(\mathbf {\Lambda } \right)\right]_{ii}=f\left(\lambda _{i}\right)$

Examples

$\mathbf {A} ^{2}=(\mathbf {Q} \mathbf {\Lambda } \mathbf {Q} ^{-1})(\mathbf {Q} \mathbf {\Lambda } \mathbf {Q} ^{-1})=\mathbf {Q} \mathbf {\Lambda } (\mathbf {Q} ^{-1}\mathbf {Q} )\mathbf {\Lambda } \mathbf {Q} ^{-1}=\mathbf {Q} \mathbf {\Lambda } ^{2}\mathbf {Q} ^{-1}$

$\mathbf {A} ^{n}=\mathbf {Q} \mathbf {\Lambda } ^{n}\mathbf {Q} ^{-1}$

Decomposition for special matrices

Symmetric matrices

Every N×N real symmetric matrix has N linearly independent eigenvectors. Moreover, these eigenvectors can be chosen such that they are orthogonal to each other and have norm one. Thus a real symmetric matrix A can be decomposed as

$\mathbf {A} =\mathbf {Q} \mathbf {\Lambda } \mathbf {Q} ^{T}$

where Q is an orthogonal matrix, and Λ is real and diagonal.

Normal matrices

Similarly, a complex normal matrix has an orthogonal eigenvector basis, so a normal matrix can be decomposed as

$\mathbf {A} =\mathbf {U} \mathbf {\Lambda } \mathbf {U} ^{H}$

where U is a unitary matrix. Further, if A is Hermitian, the diagonal matrix Λ has only real values, and if A is unitary, Λ takes all its values on the complex unit circle.

Useful facts

Useful facts regarding eigenvalues

• The product of the eigenvalues is equal to the determinant of A

$\det \left(\mathbf {A} \right)=\prod \limits _{i=1}^{N_{\lambda }}{\lambda _{i}^{n_{i}}}\!\$

Note that each eigenvalue is raised to the power n_i, the algebraic multiplicity.

• The sum of the eigenvalues is equal to the trace of A

$\operatorname {tr} \left(\mathbf {A} \right)=\sum \limits _{i=1}^{N_{\lambda }}{{n_{i}}\lambda _{i}}\!\$

Note that each eigenvalue is multiplied by n_i, the algebraic multiplicity.

• If the eigenvalues of A are λ_i, and A is invertible, then the eigenvalues of A^-1 are simply λ_i^-1.
• If the eigenvalues of A are λ_i, then the eigenvalues of f(A) are simply f(λ_i), for any holomorphic function f.

Useful facts regarding eigenvectors

• If A is (real) symmetric, then N_v=N, the eigenvectors are real, mutually orthogonal and provide a basis for $\mathbb {R} ^{N}$.
• The eigenvectors of A^-1 are the same as the eigenvectors of A
• The eigenvectors of f(A) are the same as the eigenvectors of A

Useful facts regarding eigendecomposition

• A can be eigendecomposed if and only if

$N_{\mathbf {v} }=N$

• If p(λ) has no repeated roots, i.e. N_λ=N, then A can be eigendecomposed.

• The statement "A can be eigendecomposed" does not imply that A has an inverse.

• The statement "A has an inverse" does not imply that A can be eigendecomposed.

Useful facts regarding matrix inverse

• $\mathbf {A}$ can be inverted if and only if

$\lambda _{i}\neq 0\;\forall \,i$

• If $\lambda _{i}\neq 0\;\forall \,i$ and $N_{\mathbf {v} }=N$, the inverse is given by

$\mathbf {A} ^{-1}=\mathbf {Q} \mathbf {\Lambda } ^{-1}\mathbf {Q} ^{-1}$

Numerical computations

Further information: eigenvalue algorithm

Numerical computation of eigenvalues

Suppose that we want to compute the eigenvalues of a given matrix. If the matrix is small, we can compute them symbolically using the characteristic polynomial. However, this is often impossible for larger matrices, in which case we must use a numerical method.

In practice, eigenvalues of large matrices are not computed using the characteristic polynomial. Computing the polynomial becomes expensive in itself, and exact (symbolic) roots of a high-degree polynomial can be difficult to compute and express: the Abel–Ruffini theorem implies that the roots of high-degree (5 and above) polynomials cannot in general be expressed simply using nth roots. Effective numerical algorithms for approximating roots of polynomials exist, but small errors in the eigenvalues can lead to large errors in the eigenvectors. Therefore, general algorithms to find eigenvectors and eigenvalues are iterative. The easiest method is the power method: a random vector $v$ is chosen and a sequence of unit vectors is computed as

${\frac {Av}{\|Av\|}},{\frac {A^{2}v}{\|A^{2}v\|}},{\frac {A^{3}v}{\|A^{3}v\|}},\dots$

This sequence will almost always converge to an eigenvector corresponding to the eigenvalue of greatest magnitude. This algorithm is simple, but not very useful by itself. However, popular methods such as the QR algorithm are based on it.

Numerical computation of eigenvectors

Once the eigenvalues are computed, the eigenvectors can be calculated by solving the equation

$\left(\mathbf {A} -\lambda _{i}\mathbf {I} \right)\mathbf {v} _{i,j}=0\!\$

using Gaussian elimination or any other method for solving matrix equations.

Additional topics

Generalized eigenspaces

Recall that the geometric multiplicity of an eigenvalue can be described as the dimension of the associated eigenspace, the nullspace of λI − A. The algebraic multiplicity can also be thought of as a dimension: it is the dimension of the associated generalized eigenspace (1st sense), which is the nullspace of the matrix (λI − A)^k for any sufficiently large k. That is, it is the space of generalized eigenvectors (1st sense), where a generalized eigenvector is any vector which eventually becomes 0 if λI − A is applied to it enough times successively. Any eigenvector is a generalized eigenvector, and so each eigenspace is contained in the associated generalized eigenspace. This provides an easy proof that the geometric multiplicity is always less than or equal to the algebraic multiplicity.

This usage should not be confused with the generalized eigenvalue problem described below.

Conjugate eigenvector

A conjugate eigenvector or coneigenvector is a vector sent after transformation to a scalar multiple of its conjugate, where the scalar is called the conjugate eigenvalue or coneigenvalue of the linear transformation. The coneigenvectors and coneigenvalues represent essentially the same information and meaning as the regular eigenvectors and eigenvalues, but arise when an alternative coordinate system is used. The corresponding equation is

$Av=\lambda v^{*}.\,$

For example, in coherent electromagnetic scattering theory, the linear transformation A represents the action performed by the scattering object, and the eigenvectors represent polarization states of the electromagnetic wave. In optics, the coordinate system is defined from the wave's viewpoint, known as the Forward Scattering Alignment (FSA), and gives rise to a regular eigenvalue equation, whereas in radar, the coordinate system is defined from the radar's viewpoint, known as the Back Scattering Alignment (BSA), and gives rise to a coneigenvalue equation.

Generalized eigenvalue problem

A generalized eigenvalue problem (2nd sense) is of the form

$Av=\lambda Bv\quad \quad$

where A and B are matrices. The generalized eigenvalues (2nd sense) λ can be obtained by solving the equation

$\det(A-\lambda B)=0.\,$

The set of matrices of the form $A-\lambda B$, where $\lambda$ is a complex number, is called a pencil. If B is invertible, then the original problem can be written in the form

$B^{-1}Av=\lambda v\quad \quad$

which is a standard eigenvalue problem. However, in most situations it is preferable not to perform the inversion, but rather to solve the generalized eigenvalue problem as stated originally.

See also

• Matrix decomposition
• List of matrices
• Eigenvalue, eigenvector and eigenspace
• Spectral theorem

Bibliography

• Golub, G. H. and Van Loan, C. F. (1996). Matrix Computations. 3rd ed., Johns Hopkins University Press, Baltimore. ISBN 0-8018-5414-8.
• Horn, Roger A. and Johnson, Charles R (1985). Matrix Analysis. Cambridge University Press. ISBN 0-521-38632-2.
• Horn, Roger A. and Johnson, Charles R (1991). Topics in Matrix Analysis. Cambridge University Press. ISBN 0-521-46713-6.
• Strang G (1998). Introduction to Linear Algebra. 3rd ed., Wellesley-Cambridge Press. ISBN 0-9614088-5-5.