Английская Википедия:Eigendecomposition of a matrix

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Шаблон:Short description In linear algebra, eigendecomposition is the factorization of a matrix into a canonical form, whereby the matrix is represented in terms of its eigenvalues and eigenvectors. Only diagonalizable matrices can be factorized in this way. When the matrix being factorized is a normal or real symmetric matrix, the decomposition is called "spectral decomposition", derived from the spectral theorem.

Fundamental theory of matrix eigenvectors and eigenvalues

A (nonzero) vector Шаблон:Math of dimension Шаблон:Mvar is an eigenvector of a square Шаблон:Math matrix Шаблон:Math if it satisfies a linear equation of the form

<math>\mathbf{A} \mathbf{v} = \lambda \mathbf{v}</math>

for some scalar Шаблон:Mvar. Then Шаблон:Mvar is called the eigenvalue corresponding to Шаблон:Math. Geometrically speaking, the eigenvectors of Шаблон:Math are the vectors that Шаблон:Math merely elongates or shrinks, and the amount that they elongate/shrink by is the eigenvalue. The above equation is called the eigenvalue equation or the eigenvalue problem.

This yields an equation for the eigenvalues

<math> p\left(\lambda\right) = \det\left(\mathbf{A} - \lambda \mathbf{I}\right)= 0. </math>

We call Шаблон:Math the characteristic polynomial, and the equation, called the characteristic equation, is an Шаблон:Mvarth-order polynomial equation in the unknown Шаблон:Mvar. This equation will have Шаблон:Mvar distinct solutions, where Шаблон:Math. The set of solutions, that is, the eigenvalues, is called the spectrum of Шаблон:Math.

If the field of scalars is algebraically closed, then we can factor Шаблон:Mvar as

<math>p\left(\lambda\right) = \left(\lambda - \lambda_1\right)^{n_1}\left(\lambda - \lambda_2\right)^{n_2} \cdots \left(\lambda-\lambda_{N_{\lambda}}\right)^{n_{N_{\lambda}}} = 0. </math>

The integer Шаблон:Mvar is termed the algebraic multiplicity of eigenvalue Шаблон:Mvar. The algebraic multiplicities sum to Шаблон:Mvar: <math display="inline">\sum_{i=1}^{N_\lambda}{n_i} = N.</math>

For each eigenvalue Шаблон:Mvar, we have a specific eigenvalue equation

<math>\left(\mathbf{A} - \lambda_i \mathbf{I}\right)\mathbf{v} = 0. </math>

There will be Шаблон:Math linearly independent solutions to each eigenvalue equation. The linear combinations of the Шаблон:Math solutions (except the one which gives the zero vector) are the eigenvectors associated with the eigenvalue Шаблон:Math. The integer Шаблон:Math is termed the geometric multiplicity of Шаблон:Math. It is important to keep in mind that the algebraic multiplicity Шаблон:Math and geometric multiplicity Шаблон:Math may or may not be equal, but we always have Шаблон:Math. The simplest case is of course when Шаблон:Math. The total number of linearly independent eigenvectors, Шаблон:Math, can be calculated by summing the geometric multiplicities

<math>\sum_{i=1}^{N_{\lambda}}{m_i} = N_{\mathbf{v}}.</math>

The eigenvectors can be indexed by eigenvalues, using a double index, with Шаблон:Math being the Шаблон:Mvarth eigenvector for the Шаблон:Mvarth eigenvalue. The eigenvectors can also be indexed using the simpler notation of a single index Шаблон:Math, with Шаблон:Math.

Eigendecomposition of a matrix

Let Шаблон:Math be a square Шаблон:Math matrix with Шаблон:Mvar linearly independent eigenvectors Шаблон:Mvar (where Шаблон:Math). Then Шаблон:Math can be factorized as

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

where Шаблон:Math is the square Шаблон:Math matrix whose Шаблон:Mvarth column is the eigenvector Шаблон:Mvar of Шаблон:Math, and Шаблон:Math is the diagonal matrix whose diagonal elements are the corresponding eigenvalues, Шаблон:Math. Note that only diagonalizable matrices can be factorized in this way. For example, the defective matrix <math>\left[ \begin{smallmatrix} 1 & 1 \\ 0 & 1 \end{smallmatrix} \right]</math> (which is a shear matrix) cannot be diagonalized.

The Шаблон:Mvar eigenvectors Шаблон:Mvar are usually normalized, but they need not be. A non-normalized set of Шаблон:Mvar eigenvectors, Шаблон:Mvar can also be used as the columns of Шаблон:Math. That can be understood by noting that the magnitude of the eigenvectors in Шаблон:Math gets canceled in the decomposition by the presence of Шаблон:Math. If one of the eigenvalues Шаблон:Math has multiple linearly independent eigenvectors (that is, the geometric multiplicity of Шаблон:Math is greater than 1), then these eigenvectors for this eigenvalue Шаблон:Math can be chosen to be mutually orthogonal; however, if two eigenvectors belong to two different eigenvalues, it may be impossible for them to be orthogonal to each other (see Example below). One special case is that if Шаблон:Math is a normal matrix, then by the spectral theorem, it's always possible to diagonalize Шаблон:Math in an orthonormal basis Шаблон:Mvar.

The decomposition can be derived from the fundamental property of eigenvectors:

<math>\begin{align}

\mathbf{A} \mathbf{v} &= \lambda \mathbf{v} \\ \mathbf{A} \mathbf{Q} &= \mathbf{Q} \mathbf{\Lambda} \\ \mathbf{A} &= \mathbf{Q}\mathbf{\Lambda}\mathbf{Q}^{-1} . \end{align}</math> The linearly independent eigenvectors Шаблон:Mvar with nonzero eigenvalues form a basis (not necessarily orthonormal) for all possible products Шаблон:Math, for Шаблон:Math, which is the same as the image (or range) of the corresponding matrix transformation, and also the column space of the matrix Шаблон:Math. The number of linearly independent eigenvectors Шаблон:Mvar with nonzero eigenvalues is equal to the rank of the matrix Шаблон:Math, and also the dimension of the image (or range) of the corresponding matrix transformation, as well as its column space.

The linearly independent eigenvectors Шаблон:Mvar with an eigenvalue of zero form a basis (which can be chosen to be orthonormal) for the null space (also known as the kernel) of the matrix transformation Шаблон:Math.

Example

The 2 × 2 real matrix Шаблон:Math

<math>\mathbf{A} = \begin{bmatrix} 1 & 0 \\ 1 & 3 \\ \end{bmatrix}</math>

may be decomposed into a diagonal matrix through multiplication of a non-singular matrix Шаблон:Math

<math>\mathbf{B} = \begin{bmatrix}

   a & b \\
   c & d
 \end{bmatrix} \in \mathbb{R}^{2\times2}.

</math>

Then

<math>\begin{bmatrix}

   a & b \\
   c & d 
 \end{bmatrix}^{-1}\begin{bmatrix}
   1 & 0 \\
   1 & 3
 \end{bmatrix}\begin{bmatrix}
   a & b \\
   c & d
 \end{bmatrix} =
 \begin{bmatrix}
   x & 0 \\
   0 & y
 \end{bmatrix},

</math> for some real diagonal matrix <math>\left[ \begin{smallmatrix} x & 0 \\ 0 & y \end{smallmatrix} \right]</math>.

Multiplying both sides of the equation on the left by Шаблон:Math:

<math>\begin{bmatrix} 1 & 0 \\ 1 & 3 \end{bmatrix} \begin{bmatrix} a & b \\ c & d \end{bmatrix} = \begin{bmatrix} a & b \\ c & d \end{bmatrix} \begin{bmatrix} x & 0 \\ 0 & y \end{bmatrix}.</math>

The above equation can be decomposed into two simultaneous equations:

<math> \begin{cases} \begin{bmatrix} 1 & 0\\ 1 & 3 \end{bmatrix} \begin{bmatrix} a \\ c \end{bmatrix} = \begin{bmatrix} ax \\ cx \end{bmatrix} \\ \begin{bmatrix} 1 & 0\\ 1 & 3 \end{bmatrix} \begin{bmatrix} b \\ d \end{bmatrix} = \begin{bmatrix} by \\ dy \end{bmatrix} \end{cases} .</math>

Factoring out the eigenvalues Шаблон:Mvar and Шаблон:Mvar:

<math> \begin{cases} \begin{bmatrix} 1 & 0\\ 1 & 3 \end{bmatrix} \begin{bmatrix} a \\ c \end{bmatrix} = x\begin{bmatrix} a \\ c \end{bmatrix} \\ \begin{bmatrix} 1 & 0\\ 1 & 3 \end{bmatrix} \begin{bmatrix} b \\ d \end{bmatrix} = y\begin{bmatrix} b \\ d \end{bmatrix} \end{cases} </math>

Letting

<math>\mathbf{a} = \begin{bmatrix} a \\ c \end{bmatrix}, \quad \mathbf{b} = \begin{bmatrix} b \\ d \end{bmatrix},</math>

this gives us two vector equations:

<math> \begin{cases}

\mathbf{A} \mathbf{a} = x \mathbf{a} \\ \mathbf{A} \mathbf{b} = y \mathbf{b} \end{cases}</math>

And can be represented by a single vector equation involving two solutions as eigenvalues:

<math>\mathbf{A} \mathbf{u} = \lambda \mathbf{u}</math>

where Шаблон:Mvar represents the two eigenvalues Шаблон:Mvar and Шаблон:Mvar, and Шаблон:Math represents the vectors Шаблон:Math and Шаблон:Math.

Shifting Шаблон:Math to the left hand side and factoring Шаблон:Math out

<math>(\mathbf{A} - \lambda \mathbf{I}) \mathbf{u} = \mathbf{0}</math>

Since Шаблон:Math is non-singular, it is essential that Шаблон:Math is nonzero. Therefore,

<math>\det(\mathbf{A} - \lambda \mathbf{I}) = 0</math>

Thus

<math>(1- \lambda)(3 - \lambda) = 0</math>

giving us the solutions of the eigenvalues for the matrix Шаблон:Math as Шаблон:Math or Шаблон:Math, and the resulting diagonal matrix from the eigendecomposition of Шаблон:Math is thus <math>\left[ \begin{smallmatrix} 1 & 0 \\ 0 & 3 \end{smallmatrix} \right]</math>.

Putting the solutions back into the above simultaneous equations

<math> \begin{cases} \begin{bmatrix} 1 & 0 \\ 1 & 3 \end{bmatrix} \begin{bmatrix} a \\ c \end{bmatrix} = 1\begin{bmatrix} a \\ c \end{bmatrix} \\ \begin{bmatrix} 1 & 0\\ 1 & 3 \end{bmatrix} \begin{bmatrix} b \\ d \end{bmatrix} = 3\begin{bmatrix} b \\ d \end{bmatrix} \end{cases} </math>

Solving the equations, we have

<math>a = -2c \quad\text{and} \quad b = 0, \qquad c,d \in \mathbb{R}.</math>

Thus the matrix Шаблон:Math required for the eigendecomposition of Шаблон:Math is

<math>\mathbf{B} = \begin{bmatrix} -2c & 0 \\ c & d \end{bmatrix},\qquad c, d\in \mathbb{R}, </math>

that is:

<math>\begin{bmatrix}

-2c & 0 \\ c & d \end{bmatrix}^{-1} \begin{bmatrix} 1 & 0 \\ 1 & 3 \end{bmatrix} \begin{bmatrix} -2c & 0 \\ c & d \end{bmatrix} = \begin{bmatrix} 1 & 0 \\ 0 & 3 \end{bmatrix},\qquad c, d\in \mathbb{R}</math>

Matrix inverse via eigendecomposition

If a matrix Шаблон:Math can be eigendecomposed and if none of its eigenvalues are zero, then Шаблон:Math is invertible and its inverse is given by

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

If <math>\mathbf{A}</math> is a symmetric matrix, since <math>\mathbf{Q}</math> is formed from the eigenvectors of <math>\mathbf{A}</math>, <math>\mathbf{Q}</math> is guaranteed to be an orthogonal matrix, therefore <math>\mathbf{Q}^{-1} = \mathbf{Q}^\mathrm{T}</math>. Furthermore, because Шаблон:Math is a diagonal matrix, its inverse is easy to calculate:

<math>\left[\Lambda^{-1}\right]_{ii} = \frac{1}{\lambda_i}</math>

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: truncating small or zero eigenvalues, and extending the lowest reliable eigenvalue to those below it. See also Tikhonov regularization as a statistically motivated but biased method for rolling off eigenvalues as they become dominated by noise.

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:

<math>\min\left|\nabla^2 \lambda_\mathrm{s}\right|</math>

where the eigenvalues are subscripted with an Шаблон:Math 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 Шаблон:Math is given by

<math>f(x) = a_0 + a_1 x + a_2 x^2 + \cdots</math>

then we know that

<math>f\!\left(\mathbf{A}\right) = \mathbf{Q}\,f\!\left(\mathbf{\Lambda}\right)\mathbf{Q}^{-1}</math>

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

<math>\left[f\left(\mathbf{\Lambda}\right)\right]_{ii} = f\left(\lambda_i\right)</math>

The off-diagonal elements of Шаблон:Math are zero; that is, Шаблон:Math is also a diagonal matrix. Therefore, calculating Шаблон:Math reduces to just calculating the function on each of the eigenvalues.

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

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

from above. Once again, we find that

<math>\left[f\left(\mathbf{\Lambda}\right)\right]_{ii} = f\left(\lambda_i\right)</math>

Examples

<math>\begin{align}

 \mathbf{A}^2 &= \left(\mathbf{Q}\mathbf{\Lambda}\mathbf{Q}^{-1}\right)\left(\mathbf{Q}\mathbf{\Lambda}\mathbf{Q}^{-1}\right)
               = \mathbf{Q}\mathbf{\Lambda}\left(\mathbf{Q}^{-1}\mathbf{Q}\right)\mathbf{\Lambda}\mathbf{Q}^{-1}
               = \mathbf{Q}\mathbf{\Lambda}^2\mathbf{Q}^{-1} \\
 \mathbf{A}^n &= \mathbf{Q}\mathbf{\Lambda}^n\mathbf{Q}^{-1}
 \\
 \exp \mathbf{A}
 &= \mathbf{Q} \exp(\mathbf{\Lambda}) \mathbf{Q}^{-1}

\end{align}</math> which are examples for the functions <math> f(x)=x^2, \; f(x)=x^n, \; f(x)=\exp{x} </math>. Furthermore, <math> \exp{\mathbf{A}} </math> is the matrix exponential.

Decomposition for special matrices

Файл:Taxonomy of Complex Matrices.svg

Subsets of important classes of matrices

When Шаблон:Math is normal or real symmetric matrix, the decomposition is called "spectral decomposition", derived from the spectral theorem.

Normal matrices

A complex-valued square matrix Шаблон:Math is normal (meaning Шаблон:Math, where Шаблон:Math is the conjugate transpose) if and only if it can be decomposed as

<math>\mathbf{A} = \mathbf{U}\mathbf{\Lambda}\mathbf{U}^*</math>

where Шаблон:Math is a unitary matrix (meaning Шаблон:Math) and Шаблон:Math is a diagonal matrix. The columns u₁, ..., u_n of Шаблон:Math form an orthonormal basis and are eigenvectors of Шаблон:Math with corresponding eigenvalues λ₁, ..., λ_n.

If Шаблон:Math is restricted to be a Hermitian matrix (Шаблон:Math), then Шаблон:Math has only real valued entries. If Шаблон:Math is restricted to a unitary matrix, then Шаблон:Math takes all its values on the complex unit circle, that is, Шаблон:Math.

Real symmetric matrices

As a special case, for every Шаблон:Math real symmetric matrix, the eigenvalues are real and the eigenvectors can be chosen real and orthonormal. Thus a real symmetric matrix Шаблон:Math can be decomposed as

<math>\mathbf{A} = \mathbf{Q}\mathbf{\Lambda}\mathbf{Q}^\mathsf{T}</math>

where Шаблон:Math is an orthogonal matrix whose columns are the real, orthonormal eigenvectors of Шаблон:Math, and Шаблон:Math is a diagonal matrix whose entries are the eigenvalues of Шаблон:Math.

Useful facts

Useful facts regarding eigenvalues

• The product of the eigenvalues is equal to the determinant of Шаблон:Math <math display="block">\det\left(\mathbf{A}\right) = \prod_{i=1}^{N_\lambda}{\lambda_i^{n_i} }</math> Note that each eigenvalue is raised to the power Шаблон:Math, the algebraic multiplicity.
• The sum of the eigenvalues is equal to the trace of Шаблон:Math <math display="block"> \operatorname{tr}\left(\mathbf{A}\right) = \sum_{i=1}^{N_\lambda}{ {n_i}\lambda_i}</math> Note that each eigenvalue is multiplied by Шаблон:Math, the algebraic multiplicity.
• If the eigenvalues of Шаблон:Math are Шаблон:Math, and Шаблон:Math is invertible, then the eigenvalues of Шаблон:Math are simply Шаблон:Math.
• If the eigenvalues of Шаблон:Math are Шаблон:Math, then the eigenvalues of Шаблон:Math are simply Шаблон:Math, for any holomorphic function Шаблон:Mvar.

Useful facts regarding eigenvectors

• If Шаблон:Math is Hermitian and full-rank, the basis of eigenvectors may be chosen to be mutually orthogonal. The eigenvalues are real.
• The eigenvectors of Шаблон:Math are the same as the eigenvectors of Шаблон:Math.
• Eigenvectors are only defined up to a multiplicative constant. That is, if Шаблон:Math then Шаблон:Math is also an eigenvector for any scalar Шаблон:Math. In particular, Шаблон:Math and Шаблон:Math (for any θ) are also eigenvectors.
• In the case of degenerate eigenvalues (an eigenvalue having more than one eigenvector), the eigenvectors have an additional freedom of linear transformation, that is to say, any linear (orthonormal) combination of eigenvectors sharing an eigenvalue (in the degenerate subspace) is itself an eigenvector (in the subspace).

Useful facts regarding eigendecomposition

• Шаблон:Math can be eigendecomposed if and only if the number of linearly independent eigenvectors, Шаблон:Math, equals the dimension of an eigenvector: Шаблон:Math
• If the field of scalars is algebraically closed and if Шаблон:Math has no repeated roots, that is, if <math>N_\lambda = N,</math> then Шаблон:Math can be eigendecomposed.
• The statement "Шаблон:Math can be eigendecomposed" does not imply that Шаблон:Math has an inverse as some eigenvalues may be zero, which is not invertible.
• The statement "Шаблон:Math has an inverse" does not imply that Шаблон:Math can be eigendecomposed. A counterexample is <math>\left[ \begin{smallmatrix} 1 & 1 \\ 0 & 1 \end{smallmatrix} \right]</math>, which is an invertible defective matrix.

Useful facts regarding matrix inverse

• Шаблон:Math can be inverted if and only if all eigenvalues are nonzero: <math display="block">\lambda_i \ne 0 \quad \forall \,i</math>
• If Шаблон:Math and Шаблон:Math, the inverse is given by <math display="block">\mathbf{A}^{-1} = \mathbf{Q}\mathbf{\Lambda}^{-1}\mathbf{Q}^{-1}</math>

Numerical computations

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 or above) polynomials cannot in general be expressed simply using Шаблон:Mvarth roots. Therefore, general algorithms to find eigenvectors and eigenvalues are iterative.

Iterative numerical algorithms for approximating roots of polynomials exist, such as Newton's method, but in general it is impractical to compute the characteristic polynomial and then apply these methods. One reason is that small round-off errors in the coefficients of the characteristic polynomial can lead to large errors in the eigenvalues and eigenvectors: the roots are an extremely ill-conditioned function of the coefficients.

A simple and accurate iterative method is the power method: a random vector Шаблон:Math is chosen and a sequence of unit vectors is computed as

<math>\frac{\mathbf{A}\mathbf{v}}{\left\|\mathbf{A}\mathbf{v}\right\|}, \frac{\mathbf{A}^2\mathbf{v}}{\left\|\mathbf{A}^2\mathbf{v}\right\|}, \frac{\mathbf{A}^3\mathbf{v}}{\left\|\mathbf{A}^3\mathbf{v}\right\|}, \ldots</math>

This sequence will almost always converge to an eigenvector corresponding to the eigenvalue of greatest magnitude, provided that Шаблон:Math has a nonzero component of this eigenvector in the eigenvector basis (and also provided that there is only one eigenvalue of greatest magnitude). This simple algorithm is useful in some practical applications; for example, Google uses it to calculate the page rank of documents in their search engine. Also, the power method is the starting point for many more sophisticated algorithms. For instance, by keeping not just the last vector in the sequence, but instead looking at the span of all the vectors in the sequence, one can get a better (faster converging) approximation for the eigenvector, and this idea is the basis of Arnoldi iteration. Alternatively, the important QR algorithm is also based on a subtle transformation of a power method.

Numerical computation of eigenvectors

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

<math>\left(\mathbf{A} - \lambda_i \mathbf{I}\right)\mathbf{v}_{i,j} = \mathbf{0} </math>

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

However, in practical large-scale eigenvalue methods, the eigenvectors are usually computed in other ways, as a byproduct of the eigenvalue computation. In power iteration, for example, the eigenvector is actually computed before the eigenvalue (which is typically computed by the Rayleigh quotient of the eigenvector). In the QR algorithm for a Hermitian matrix (or any normal matrix), the orthonormal eigenvectors are obtained as a product of the Шаблон:Math matrices from the steps in the algorithm. (For more general matrices, the QR algorithm yields the Schur decomposition first, from which the eigenvectors can be obtained by a backsubstitution procedure.) For Hermitian matrices, the Divide-and-conquer eigenvalue algorithm is more efficient than the QR algorithm if both eigenvectors and eigenvalues are desired.

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 Шаблон:Math. 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 Шаблон:Math for any sufficiently large Шаблон:Mvar. That is, it is the space of generalized eigenvectors (first sense), where a generalized eigenvector is any vector which eventually becomes 0 if Шаблон:Math 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

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

For example, in coherent electromagnetic scattering theory, the linear transformation Шаблон:Math 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 (second sense) is the problem of finding a (nonzero) vector Шаблон:Math that obeys

<math> \mathbf{A}\mathbf{v} = \lambda \mathbf{B} \mathbf{v}</math>

where Шаблон:Math and Шаблон:Math are matrices. If Шаблон:Math obeys this equation, with some Шаблон:Mvar, then we call Шаблон:Math the generalized eigenvector of Шаблон:Math and Шаблон:Math (in the second sense), and Шаблон:Mvar is called the generalized eigenvalue of Шаблон:Math and Шаблон:Math (in the second sense) which corresponds to the generalized eigenvector Шаблон:Math. The possible values of Шаблон:Mvar must obey the following equation

<math>\det(\mathbf{A} - \lambda \mathbf{B})=0. </math>

If Шаблон:Math linearly independent vectors Шаблон:Math can be found, such that for every Шаблон:Math, Шаблон:Math, then we define the matrices Шаблон:Math and Шаблон:Math such that

<math>P = \begin{bmatrix}

   | & & | \\
   \mathbf{v}_1 & \cdots & \mathbf{v}_n \\
   | &  & | 
 \end{bmatrix} \equiv
 \begin{bmatrix}
   (\mathbf{v}_1)_1 & \cdots & (\mathbf{v}_n)_1 \\
   \vdots &  & \vdots   \\
   (\mathbf{v}_1)_n & \cdots & (\mathbf{v}_n)_n 
 \end{bmatrix}

</math>

<math>(D)_{ij} = \begin{cases}

 \lambda_i,  & \text{if }i = j\\
 0,          & \text{otherwise}

\end{cases}</math>

Then the following equality holds

<math>\mathbf{A} = \mathbf{B}\mathbf{P}\mathbf{D}\mathbf{P}^{-1}</math>

And the proof is

<math>

 \mathbf{A}\mathbf{P}= \mathbf{A} \begin{bmatrix}
   | & & | \\
   \mathbf{v}_1 & \cdots & \mathbf{v}_n   \\
   | &  & | 
 \end{bmatrix} = \begin{bmatrix}
   | & & | \\
   A\mathbf{v}_1 & \cdots & A\mathbf{v}_n   \\
   | &  & | 
 \end{bmatrix} = \begin{bmatrix}
   | & & | \\
   \lambda_1B\mathbf{v}_1 & \cdots & \lambda_nB\mathbf{v}_n   \\
   | &  & | 
 \end{bmatrix} = \begin{bmatrix}
   | & & | \\
   B\mathbf{v}_1 & \cdots & B\mathbf{v}_n   \\
   | &  & | 
 \end{bmatrix}
 \mathbf{D} =
 \mathbf{B}\mathbf{P}\mathbf{D}

</math>

And since Шаблон:Math is invertible, we multiply the equation from the right by its inverse, finishing the proof.

The set of matrices of the form Шаблон:Math, where Шаблон:Mvar is a complex number, is called a pencil; the term matrix pencil can also refer to the pair Шаблон:Math of matrices.

If Шаблон:Math is invertible, then the original problem can be written in the form

<math>\mathbf{B}^{-1}\mathbf{A}\mathbf{v} = \lambda \mathbf{v}</math>

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. This is especially important if Шаблон:Math and Шаблон:Math are Hermitian matrices, since in this case Шаблон:Math is not generally Hermitian and important properties of the solution are no longer apparent.

If Шаблон:Math and Шаблон:Math are both symmetric or Hermitian, and Шаблон:Math is also a positive-definite matrix, the eigenvalues Шаблон:Math are real and eigenvectors Шаблон:Math and Шаблон:Math with distinct eigenvalues are Шаблон:Math-orthogonal (Шаблон:Math). In this case, eigenvectors can be chosen so that the matrix Шаблон:Math defined above satisfies

<math>\mathbf{P}^* \mathbf B \mathbf{P} = \mathbf{I}</math> or <math>\mathbf{P}\mathbf{P}^*\mathbf B = \mathbf{I}</math>,

and there exists a basis of generalized eigenvectors (it is not a defective problem). This case is sometimes called a Hermitian definite pencil or definite pencil.

See also

• Eigenvalue perturbation
• Frobenius covariant
• Householder transformation
• Jordan normal form
• List of matrices
• Matrix decomposition
• Singular value decomposition
• Sylvester's formula

References

• Шаблон:Cite book
• Шаблон:Citation
• Шаблон:Cite book
• Шаблон:Cite book
• Шаблон:Citation
• Шаблон:Citation
• Шаблон:Cite book

External links

• Interactive program & tutorial of Spectral Decomposition.