Английская Википедия:Eigenvalue algorithm

Содержание

Текст

Текст

Текст

Текст

Текст

Текст

Текст

Текст

Текст

Шаблон:Short description In numerical analysis, one of the most important problems is designing efficient and stable algorithms for finding the eigenvalues of a matrix. These eigenvalue algorithms may also find eigenvectors.

Eigenvalues and eigenvectors

Шаблон:Main Given an Шаблон:Math square matrix Шаблон:Math of real or complex numbers, an eigenvalue Шаблон:Math and its associated generalized eigenvector Шаблон:Math are a pair obeying the relation

<math>\left(A - \lambda I\right)^k {\mathbf v} = 0,</math>

where Шаблон:Math is a nonzero Шаблон:Math column vector, Шаблон:Math is the Шаблон:Math identity matrix, Шаблон:Math is a positive integer, and both Шаблон:Math and Шаблон:Math are allowed to be complex even when Шаблон:Math is real. When Шаблон:Math, the vector is called simply an eigenvector, and the pair is called an eigenpair. In this case, Шаблон:Math. Any eigenvalue Шаблон:Math of Шаблон:Math has ordinary eigenvectors associated to it, for if Шаблон:Math is the smallest integer such that Шаблон:Math for a generalized eigenvector Шаблон:Math, then Шаблон:Math is an ordinary eigenvector. The value Шаблон:Math can always be taken as less than or equal to Шаблон:Math. In particular, Шаблон:Math for all generalized eigenvectors Шаблон:Math associated with Шаблон:Math.

For each eigenvalue Шаблон:Math of Шаблон:Math, the kernel Шаблон:Math consists of all eigenvectors associated with Шаблон:Math (along with 0), called the eigenspace of Шаблон:Math, while the vector space Шаблон:Math consists of all generalized eigenvectors, and is called the generalized eigenspace. The geometric multiplicity of Шаблон:Math is the dimension of its eigenspace. The algebraic multiplicity of Шаблон:Math is the dimension of its generalized eigenspace. The latter terminology is justified by the equation

<math>p_A\left(z\right) = \det\left( zI - A \right) = \prod_{i=1}^k (z - \lambda_i)^{\alpha_i},</math>

where Шаблон:Math is the determinant function, the Шаблон:Math are all the distinct eigenvalues of Шаблон:Math and the Шаблон:Math are the corresponding algebraic multiplicities. The function Шаблон:Math is the characteristic polynomial of Шаблон:Math. So the algebraic multiplicity is the multiplicity of the eigenvalue as a zero of the characteristic polynomial. Since any eigenvector is also a generalized eigenvector, the geometric multiplicity is less than or equal to the algebraic multiplicity. The algebraic multiplicities sum up to Шаблон:Math, the degree of the characteristic polynomial. The equation Шаблон:Math is called the characteristic equation, as its roots are exactly the eigenvalues of Шаблон:Math. By the Cayley–Hamilton theorem, Шаблон:Math itself obeys the same equation: Шаблон:Math. As a consequence, the columns of the matrix <math display="inline">\prod_{i \ne j} (A - \lambda_iI)^{\alpha_i}</math> must be either 0 or generalized eigenvectors of the eigenvalue Шаблон:Math, since they are annihilated by <math>(A - \lambda_jI)^{\alpha_j}</math>. In fact, the column space is the generalized eigenspace of Шаблон:Math.

Any collection of generalized eigenvectors of distinct eigenvalues is linearly independent, so a basis for all of Шаблон:Math can be chosen consisting of generalized eigenvectors. More particularly, this basis Шаблон:Math can be chosen and organized so that

• if Шаблон:Math and Шаблон:Math have the same eigenvalue, then so does Шаблон:Math for each Шаблон:Math between Шаблон:Math and Шаблон:Math, and
• if Шаблон:Math is not an ordinary eigenvector, and if Шаблон:Math is its eigenvalue, then Шаблон:Math (in particular, Шаблон:Math must be an ordinary eigenvector).

If these basis vectors are placed as the column vectors of a matrix Шаблон:Math, then Шаблон:Math can be used to convert Шаблон:Math to its Jordan normal form:

<math>V^{-1}AV = \begin{bmatrix} \lambda_1 & \beta_1 & 0 & \ldots & 0 \\ 0 & \lambda_2 & \beta_2 & \ldots & 0 \\ 0 & 0 & \lambda_3 & \ldots & 0 \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & 0 & \ldots & \lambda_n \end{bmatrix},</math>

where the Шаблон:Math are the eigenvalues, Шаблон:Math if Шаблон:Math and Шаблон:Math otherwise.

More generally, if Шаблон:Math is any invertible matrix, and Шаблон:Math is an eigenvalue of Шаблон:Math with generalized eigenvector Шаблон:Math, then Шаблон:Math. Thus Шаблон:Math is an eigenvalue of Шаблон:Math with generalized eigenvector Шаблон:Math. That is, similar matrices have the same eigenvalues.

Normal, Hermitian, and real-symmetric matrices

Шаблон:Main The adjoint Шаблон:Math of a complex matrix Шаблон:Math is the transpose of the conjugate of Шаблон:Math: Шаблон:Math. A square matrix Шаблон:Math is called normal if it commutes with its adjoint: Шаблон:Math. It is called Hermitian if it is equal to its adjoint: Шаблон:Math. All Hermitian matrices are normal. If Шаблон:Math has only real elements, then the adjoint is just the transpose, and Шаблон:Math is Hermitian if and only if it is symmetric. When applied to column vectors, the adjoint can be used to define the canonical inner product on Шаблон:Math: Шаблон:Math. Normal, Hermitian, and real-symmetric matrices have several useful properties:

• Every generalized eigenvector of a normal matrix is an ordinary eigenvector.
• Any normal matrix is similar to a diagonal matrix, since its Jordan normal form is diagonal.
• Eigenvectors of distinct eigenvalues of a normal matrix are orthogonal.
• The null space and the image (or column space) of a normal matrix are orthogonal to each other.
• For any normal matrix Шаблон:Math, Шаблон:Math has an orthonormal basis consisting of eigenvectors of Шаблон:Math. The corresponding matrix of eigenvectors is unitary.
• The eigenvalues of a Hermitian matrix are real, since Шаблон:Math for a non-zero eigenvector Шаблон:Math.
• If Шаблон:Math is real, there is an orthonormal basis for Шаблон:Math consisting of eigenvectors of Шаблон:Math if and only if Шаблон:Math is symmetric.

It is possible for a real or complex matrix to have all real eigenvalues without being Hermitian. For example, a real triangular matrix has its eigenvalues along its diagonal, but in general is not symmetric.

Condition number

Any problem of numeric calculation can be viewed as the evaluation of some function Шаблон:Math for some input Шаблон:Math. The condition number Шаблон:Math of the problem is the ratio of the relative error in the function's output to the relative error in the input, and varies with both the function and the input. The condition number describes how error grows during the calculation. Its base-10 logarithm tells how many fewer digits of accuracy exist in the result than existed in the input. The condition number is a best-case scenario. It reflects the instability built into the problem, regardless of how it is solved. No algorithm can ever produce more accurate results than indicated by the condition number, except by chance. However, a poorly designed algorithm may produce significantly worse results. For example, as mentioned below, the problem of finding eigenvalues for normal matrices is always well-conditioned. However, the problem of finding the roots of a polynomial can be very ill-conditioned. Thus eigenvalue algorithms that work by finding the roots of the characteristic polynomial can be ill-conditioned even when the problem is not.

For the problem of solving the linear equation Шаблон:Math where Шаблон:Math is invertible, the matrix condition number Шаблон:Math is given by Шаблон:Math, where Шаблон:Nowrap is the operator norm subordinate to the normal Euclidean norm on Шаблон:Math. Since this number is independent of Шаблон:Math and is the same for Шаблон:Math and Шаблон:Math, it is usually just called the condition number Шаблон:Math of the matrix Шаблон:Math. This value Шаблон:Math is also the absolute value of the ratio of the largest eigenvalue of Шаблон:Math to its smallest. If Шаблон:Math is unitary, then Шаблон:Math, so Шаблон:Math. For general matrices, the operator norm is often difficult to calculate. For this reason, other matrix norms are commonly used to estimate the condition number.

For the eigenvalue problem, Bauer and Fike proved that if Шаблон:Math is an eigenvalue for a diagonalizable Шаблон:Math matrix Шаблон:Math with eigenvector matrix Шаблон:Math, then the absolute error in calculating Шаблон:Math is bounded by the product of Шаблон:Math and the absolute error in Шаблон:Math. As a result, the condition number for finding Шаблон:Math is Шаблон:Math. If Шаблон:Math is normal, then Шаблон:Math is unitary, and Шаблон:Math. Thus the eigenvalue problem for all normal matrices is well-conditioned.

The condition number for the problem of finding the eigenspace of a normal matrix Шаблон:Math corresponding to an eigenvalue Шаблон:Math has been shown to be inversely proportional to the minimum distance between Шаблон:Math and the other distinct eigenvalues of Шаблон:Math. In particular, the eigenspace problem for normal matrices is well-conditioned for isolated eigenvalues. When eigenvalues are not isolated, the best that can be hoped for is to identify the span of all eigenvectors of nearby eigenvalues.

Algorithms

The most reliable and most widely used algorithm for computing eigenvalues is John G. F. Francis' QR algorithm, considered one of the top ten algorithms of 20th century.

Any monic polynomial is the characteristic polynomial of its companion matrix. Therefore, a general algorithm for finding eigenvalues could also be used to find the roots of polynomials. The Abel–Ruffini theorem shows that any such algorithm for dimensions greater than 4 must either be infinite, or involve functions of greater complexity than elementary arithmetic operations and fractional powers. For this reason algorithms that exactly calculate eigenvalues in a finite number of steps only exist for a few special classes of matrices. For general matrices, algorithms are iterative, producing better approximate solutions with each iteration.

Some algorithms produce every eigenvalue, others will produce a few, or only one. However, even the latter algorithms can be used to find all eigenvalues. Once an eigenvalue Шаблон:Math of a matrix Шаблон:Math has been identified, it can be used to either direct the algorithm towards a different solution next time, or to reduce the problem to one that no longer has Шаблон:Math as a solution.

Redirection is usually accomplished by shifting: replacing Шаблон:Math with Шаблон:Math for some constant Шаблон:Math. The eigenvalue found for Шаблон:Math must have Шаблон:Math added back in to get an eigenvalue for Шаблон:Math. For example, for power iteration, Шаблон:Math. Power iteration finds the largest eigenvalue in absolute value, so even when Шаблон:Math is only an approximate eigenvalue, power iteration is unlikely to find it a second time. Conversely, inverse iteration based methods find the lowest eigenvalue, so Шаблон:Math is chosen well away from Шаблон:Math and hopefully closer to some other eigenvalue.

Reduction can be accomplished by restricting Шаблон:Math to the column space of the matrix Шаблон:Math, which Шаблон:Math carries to itself. Since Шаблон:Math is singular, the column space is of lesser dimension. The eigenvalue algorithm can then be applied to the restricted matrix. This process can be repeated until all eigenvalues are found.

If an eigenvalue algorithm does not produce eigenvectors, a common practice is to use an inverse iteration based algorithm with Шаблон:Math set to a close approximation to the eigenvalue. This will quickly converge to the eigenvector of the closest eigenvalue to Шаблон:Math. For small matrices, an alternative is to look at the column space of the product of Шаблон:Math for each of the other eigenvalues Шаблон:Math.

A formula for the norm of unit eigenvector components of normal matrices was discovered by Robert Thompson in 1966 and rediscovered independently by several others. If Шаблон:Math is an <math display="inline"> n \times n</math> normal matrix with eigenvalues Шаблон:Math and corresponding unit eigenvectors Шаблон:Math whose component entries are Шаблон:Math, let Шаблон:Math be the <math display="inline"> n - 1 \times n - 1</math> matrix obtained by removing the Шаблон:Math-th row and column from Шаблон:Math, and let Шаблон:Math be its Шаблон:Math-th eigenvalue. Then <math display="block"> |v_{i,j}|^2 \prod_{k=1,k\ne i}^n (\lambda_i(A) - \lambda_k(A)) = \prod_{k=1}^{n-1}(\lambda_i(A) - \lambda_k(A_j))</math>

If <math>p, p_j</math> are the characteristic polynomials of <math>A</math> and <math>A_j</math>, the formula can be re-written as <math display="block"> |v_{i,j}|^2 = \frac{p_j(\lambda_i(A))}{p'(\lambda_i(A))}</math> assuming the derivative <math>p'</math> is not zero at <math>\lambda_i(A)</math>.

Hessenberg and tridiagonal matrices

Because the eigenvalues of a triangular matrix are its diagonal elements, for general matrices there is no finite method like gaussian elimination to convert a matrix to triangular form while preserving eigenvalues. But it is possible to reach something close to triangular. An upper Hessenberg matrix is a square matrix for which all entries below the subdiagonal are zero. A lower Hessenberg matrix is one for which all entries above the superdiagonal are zero. Matrices that are both upper and lower Hessenberg are tridiagonal. Hessenberg and tridiagonal matrices are the starting points for many eigenvalue algorithms because the zero entries reduce the complexity of the problem. Several methods are commonly used to convert a general matrix into a Hessenberg matrix with the same eigenvalues. If the original matrix was symmetric or Hermitian, then the resulting matrix will be tridiagonal.

When only eigenvalues are needed, there is no need to calculate the similarity matrix, as the transformed matrix has the same eigenvalues. If eigenvectors are needed as well, the similarity matrix may be needed to transform the eigenvectors of the Hessenberg matrix back into eigenvectors of the original matrix.

Method	Applies to	Produces	Cost without similarity matrix	Cost with similarity matrix	Description
Householder transformations	General	Hessenberg	Шаблон:MathШаблон:Rp	Шаблон:MathШаблон:Rp	Reflect each column through a subspace to zero out its lower entries.
Givens rotations	General	Hessenberg	Шаблон:MathШаблон:Rp		Apply planar rotations to zero out individual entries. Rotations are ordered so that later ones do not cause zero entries to become non-zero again.
Arnoldi iteration	General	Hessenberg			Perform Gram–Schmidt orthogonalization on Krylov subspaces.
Lanczos algorithm	Hermitian	Tridiagonal			Arnoldi iteration for Hermitian matrices, with shortcuts.

For symmetric tridiagonal eigenvalue problems all eigenvalues (without eigenvectors) can be computed numerically in time O(n log(n)), using bisection on the characteristic polynomial.

Iterative algorithms

Iterative algorithms solve the eigenvalue problem by producing sequences that converge to the eigenvalues. Some algorithms also produce sequences of vectors that converge to the eigenvectors. Most commonly, the eigenvalue sequences are expressed as sequences of similar matrices which converge to a triangular or diagonal form, allowing the eigenvalues to be read easily. The eigenvector sequences are expressed as the corresponding similarity matrices.

Method	Applies to	Produces	Cost per step	Convergence	Description
Lanczos algorithm	Hermitian	Шаблон:Math largest/smallest eigenpairs
Power iteration	general	eigenpair with largest value	Шаблон:Math	linear	Repeatedly applies the matrix to an arbitrary starting vector and renormalizes.
Inverse iteration	general	Шаблон:Nowrap		linear	Power iteration for Шаблон:Math
Rayleigh quotient iteration	Hermitian	any eigenpair		cubic	Power iteration for Шаблон:Math, where Шаблон:Math for each iteration is the Rayleigh quotient of the previous iteration.
Preconditioned inverse iteration or LOBPCG algorithm	positive-definite real symmetric	eigenpair with value closest to μ			Inverse iteration using a preconditioner (an approximate inverse to Шаблон:Math).
Bisection method	real symmetric tridiagonal	any eigenvalue		linear	Uses the bisection method to find roots of the characteristic polynomial, supported by the Sturm sequence.
Laguerre iteration	real symmetric tridiagonal	any eigenvalue		cubic	Uses Laguerre's method to find roots of the characteristic polynomial, supported by the Sturm sequence.
QR algorithm	Hessenberg	all eigenvalues	Шаблон:Math	cubic	Factors A = QR, where Q is orthogonal and R is triangular, then applies the next iteration to RQ.
		all eigenpairs	Шаблон:Math
Jacobi eigenvalue algorithm	real symmetric	all eigenvalues	Шаблон:Math	quadratic	Uses Givens rotations to attempt clearing all off-diagonal entries. This fails, but strengthens the diagonal.
Divide-and-conquer	Hermitian tridiagonal	all eigenvalues	Шаблон:Math		Divides the matrix into submatrices that are diagonalized then recombined.
		all eigenpairs	Шаблон:Math
Homotopy method	real symmetric tridiagonal	all eigenpairs	Шаблон:Math		Constructs a computable homotopy path from a diagonal eigenvalue problem.
Folded spectrum method	real symmetric	eigenpair with value closest to μ			Preconditioned inverse iteration applied to Шаблон:Math
MRRR algorithm	real symmetric tridiagonal	some or all eigenpairs	Шаблон:Math		"Multiple relatively robust representations" – performs inverse iteration on a LDLT decomposition of the shifted matrix.
Gram iteration	general	Eigenpair with largest eigenvalue		super-linear	Repeatedly computes the Gram product and rescales, deterministically.

Direct calculation

While there is no simple algorithm to directly calculate eigenvalues for general matrices, there are numerous special classes of matrices where eigenvalues can be directly calculated. These include:

Triangular matrices

Since the determinant of a triangular matrix is the product of its diagonal entries, if T is triangular, then <math display="inline">\det(\lambda I - T) = \prod_i (\lambda - T_{ii})</math>. Thus the eigenvalues of T are its diagonal entries.

Factorable polynomial equations

If Шаблон:Math is any polynomial and Шаблон:Math then the eigenvalues of Шаблон:Math also satisfy the same equation. If Шаблон:Math happens to have a known factorization, then the eigenvalues of Шаблон:Math lie among its roots.

For example, a projection is a square matrix Шаблон:Math satisfying Шаблон:Math. The roots of the corresponding scalar polynomial equation, Шаблон:Math, are 0 and 1. Thus any projection has 0 and 1 for its eigenvalues. The multiplicity of 0 as an eigenvalue is the nullity of Шаблон:Math, while the multiplicity of 1 is the rank of Шаблон:Math.

Another example is a matrix Шаблон:Math that satisfies Шаблон:Math for some scalar Шаблон:Math. The eigenvalues must be Шаблон:Math. The projection operators

<math>P_+=\frac{1}{2}\left(I+\frac{A}{\alpha}\right)</math>

<math>P_-=\frac{1}{2}\left(I-\frac{A}{\alpha}\right)</math>

satisfy

<math>AP_+=\alpha P_+ \quad AP_-=-\alpha P_-</math>

and

<math>P_+P_+=P_+ \quad P_-P_-=P_- \quad P_+P_-=P_-P_+=0.</math>

The column spaces of Шаблон:Math and Шаблон:Math are the eigenspaces of Шаблон:Math corresponding to Шаблон:Math and Шаблон:Math, respectively.

2×2 matrices

For dimensions 2 through 4, formulas involving radicals exist that can be used to find the eigenvalues. While a common practice for 2×2 and 3×3 matrices, for 4×4 matrices the increasing complexity of the root formulas makes this approach less attractive.

For the 2×2 matrix

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

the characteristic polynomial is

<math>\det \begin{bmatrix} \lambda - a & -b \\ -c & \lambda - d \end{bmatrix} = \lambda^2\, -\, \left( a + d \right )\lambda\, +\, \left ( ad - bc \right ) = \lambda^2\, -\, \lambda\, {\rm tr}(A)\, +\, \det(A).</math>

Thus the eigenvalues can be found by using the quadratic formula:

<math>\lambda = \frac{{\rm tr}(A) \pm \sqrt{{\rm tr}^2 (A) - 4 \det(A)}}{2}.</math>

Defining <math display="inline"> {\rm gap}\left ( A \right ) = \sqrt{{\rm tr}^2 (A) - 4 \det(A)}</math> to be the distance between the two eigenvalues, it is straightforward to calculate

<math>\frac{\partial\lambda}{\partial a} = \frac{1}{2}\left ( 1 \pm \frac{a - d}{{\rm gap}(A)} \right ),\qquad \frac{\partial\lambda}{\partial b} = \frac{\pm c}{{\rm gap}(A)}</math>

with similar formulas for Шаблон:Math and Шаблон:Math. From this it follows that the calculation is well-conditioned if the eigenvalues are isolated.

Eigenvectors can be found by exploiting the Cayley–Hamilton theorem. If Шаблон:Math are the eigenvalues, then Шаблон:Math, so the columns of Шаблон:Math are annihilated by Шаблон:Math and vice versa. Assuming neither matrix is zero, the columns of each must include eigenvectors for the other eigenvalue. (If either matrix is zero, then Шаблон:Math is a multiple of the identity and any non-zero vector is an eigenvector.)

For example, suppose

<math>A = \begin{bmatrix} 4 & 3 \\ -2 & -3 \end{bmatrix},</math>

then Шаблон:Math and Шаблон:Math, so the characteristic equation is

<math> 0 = \lambda^2 - \lambda - 6 = (\lambda - 3)(\lambda + 2),</math>

and the eigenvalues are 3 and -2. Now,

<math>A - 3I = \begin{bmatrix} 1 & 3 \\ -2 & -6 \end{bmatrix}, \qquad A + 2I = \begin{bmatrix} 6 & 3 \\ -2 & -1 \end{bmatrix}.</math>

In both matrices, the columns are multiples of each other, so either column can be used. Thus, Шаблон:Math can be taken as an eigenvector associated with the eigenvalue -2, and Шаблон:Math as an eigenvector associated with the eigenvalue 3, as can be verified by multiplying them by Шаблон:Math.

Symmetric 3×3 matrices

The characteristic equation of a symmetric 3×3 matrix Шаблон:Math is:

<math>\det \left( \alpha I - A \right) = \alpha^3 - \alpha^2 {\rm tr}(A) - \alpha \frac{1}{2}\left( {\rm tr}(A^2) - {\rm tr}^2(A) \right) - \det(A) = 0.</math>

This equation may be solved using the methods of Cardano or Lagrange, but an affine change to Шаблон:Math will simplify the expression considerably, and lead directly to a trigonometric solution. If Шаблон:Math, then Шаблон:Math and Шаблон:Math have the same eigenvectors, and Шаблон:Math is an eigenvalue of Шаблон:Math if and only if Шаблон:Math is an eigenvalue of Шаблон:Math. Letting <math display="inline"> q = {\rm tr}(A)/3</math> and <math display="inline"> p =\left({\rm tr}\left((A - qI)^2\right)/ 6\right)^{1/2}</math>, gives

<math>\det \left( \beta I - B \right) = \beta^3 - 3 \beta - \det(B) = 0.</math>

The substitution Шаблон:Math and some simplification using the identity Шаблон:Math reduces the equation to Шаблон:Math. Thus

<math>\beta = 2{\cos}\left(\frac{1}{3}{\arccos}\left( \det(B)/2 \right) + \frac{2k\pi}{3}\right), \quad k = 0, 1, 2.</math>

If Шаблон:Math is complex or is greater than 2 in absolute value, the arccosine should be taken along the same branch for all three values of Шаблон:Math. This issue doesn't arise when Шаблон:Math is real and symmetric, resulting in a simple algorithm:

% Given a real symmetric 3x3 matrix A, compute the eigenvalues
% Note that acos and cos operate on angles in radians

p1 = A(1,2)^2 + A(1,3)^2 + A(2,3)^2
if (p1 == 0) 
   % A is diagonal.
   eig1 = A(1,1)
   eig2 = A(2,2)
   eig3 = A(3,3)
else
   q = trace(A)/3               % trace(A) is the sum of all diagonal values
   p2 = (A(1,1) - q)^2 + (A(2,2) - q)^2 + (A(3,3) - q)^2 + 2 * p1
   p = sqrt(p2 / 6)
   B = (1 / p) * (A - q * I)    % I is the identity matrix
   r = det(B) / 2

   % In exact arithmetic for a symmetric matrix  -1 <= r <= 1
   % but computation error can leave it slightly outside this range.
   if (r <= -1) 
      phi = pi / 3
   elseif (r >= 1)
      phi = 0
   else
      phi = acos(r) / 3
   end

   % the eigenvalues satisfy eig3 <= eig2 <= eig1
   eig1 = q + 2 * p * cos(phi)
   eig3 = q + 2 * p * cos(phi + (2*pi/3))
   eig2 = 3 * q - eig1 - eig3     % since trace(A) = eig1 + eig2 + eig3
end

Once again, the eigenvectors of Шаблон:Math can be obtained by recourse to the Cayley–Hamilton theorem. If Шаблон:Math are distinct eigenvalues of Шаблон:Math, then Шаблон:Math. Thus the columns of the product of any two of these matrices will contain an eigenvector for the third eigenvalue. However, if Шаблон:Math, then Шаблон:Math and Шаблон:Math. Thus the generalized eigenspace of Шаблон:Math is spanned by the columns of Шаблон:Math while the ordinary eigenspace is spanned by the columns of Шаблон:Math. The ordinary eigenspace of Шаблон:Math is spanned by the columns of Шаблон:Math.

For example, let

<math>A = \begin{bmatrix} 3 & 2 & 6 \\ 2 & 2 & 5 \\ -2 & -1 & -4 \end{bmatrix}.</math>

The characteristic equation is

<math> 0 = \lambda^3 - \lambda^2 - \lambda + 1 = (\lambda - 1)^2(\lambda + 1),</math>

with eigenvalues 1 (of multiplicity 2) and -1. Calculating,

<math>A - I = \begin{bmatrix} 2 & 2 & 6 \\ 2 & 1 & 5 \\ -2 & -1 & -5 \end{bmatrix}, \qquad A + I = \begin{bmatrix} 4 & 2 & 6 \\ 2 & 3 & 5 \\ -2 & -1 & -3 \end{bmatrix}</math>

and

<math>(A - I)^2 = \begin{bmatrix} -4 & 0 & -8 \\ -4 & 0 & -8 \\ 4 & 0 & 8 \end{bmatrix}, \qquad (A - I)(A + I) = \begin{bmatrix} 0 & 4 & 4 \\ 0 & 2 & 2 \\ 0 & -2 & -2 \end{bmatrix}</math>

Thus Шаблон:Math is an eigenvector for −1, and Шаблон:Math is an eigenvector for 1. Шаблон:Math and Шаблон:Math are both generalized eigenvectors associated with 1, either one of which could be combined with Шаблон:Math and Шаблон:Math to form a basis of generalized eigenvectors of Шаблон:Math. Once found, the eigenvectors can be normalized if needed.

Eigenvectors of normal 3×3 matrices

If a 3×3 matrix <math>A</math> is normal, then the cross-product can be used to find eigenvectors. If <math>\lambda</math> is an eigenvalue of <math>A</math>, then the null space of <math>A - \lambda I</math> is perpendicular to its column space. The cross product of two independent columns of <math>A - \lambda I</math> will be in the null space. That is, it will be an eigenvector associated with <math>\lambda</math>. Since the column space is two dimensional in this case, the eigenspace must be one dimensional, so any other eigenvector will be parallel to it.

If <math>A - \lambda I</math> does not contain two independent columns but is not Шаблон:Math, the cross-product can still be used. In this case <math>\lambda</math> is an eigenvalue of multiplicity 2, so any vector perpendicular to the column space will be an eigenvector. Suppose <math>\mathbf v</math> is a non-zero column of <math>A - \lambda I</math>. Choose an arbitrary vector <math>\mathbf u</math> not parallel to <math>\mathbf v</math>. Then <math>\mathbf v\times \mathbf u</math> and <math>(\mathbf v\times \mathbf u)\times \mathbf v</math> will be perpendicular to <math>\mathbf v</math> and thus will be eigenvectors of <math>\lambda</math>.

This does not work when <math>A</math> is not normal, as the null space and column space do not need to be perpendicular for such matrices.

See also

• List of eigenvalue algorithms

Further reading

• Шаблон:Cite journal