MATLAB Functions

Eigenvalues and eigenvectors

d = eig(A)
[V,D] = eig(A)
[V,D] = eig(A,'nobalance')
d = eig(A,B)
[V,D] = eig(A,B)

Description

d = eig(A)

returns a vector of the eigenvalues of matrix A.

[V,D] = eig(A)

produces matrices of eigenvalues (D) and eigenvectors (V) of matrix A, so that A*V = V*D. Matrix D is the canonical form of A--a diagonal matrix with A's eigenvalues on the main diagonal. Matrix V is the modal matrix--its columns are the eigenvectors of A.

The eigenvectors are scaled so that the norm of each is 1.0. Use [W,D] = eig(A'); W = W' to compute the left eigenvectors, which satisfy
W*A = D*W.

[V,D] = eig(A,'nobalance')

finds eigenvalues and eigenvectors without a preliminary balancing step. Ordinarily, balancing improves the conditioning of the input matrix, enabling more accurate computation of the eigenvectors and eigenvalues. However, if a matrix contains small elements that are really due to roundoff error, balancing may scale them up to make them as significant as the other elements of the original matrix, leading to incorrect eigenvectors. Use the nobalance option in this event. See the balance function for more details.

d = eig(A,B)

returns a vector containing the generalized eigenvalues, if A and B are square matrices.

[V,D] = eig(A,B)

produces a diagonal matrix D of generalized eigenvalues and a full matrix V whose columns are the corresponding eigenvectors so that A*V = B*V*D. The eigenvectors are scaled so that the norm of each is 1.0.

Remarks

The eigenvalue problem is to determine the nontrivial solutions of the equation:

where A is an n-by-n matrix, x is a length n column vector, and

is a scalar. The n values of

that satisfy the equation are the eigenvalues, and the corresponding values of x are the right eigenvectors. In MATLAB, the function eig solves for the eigenvalues

, and optionally the eigenvectors x.

The generalized eigenvalue problem is to determine the nontrivial solutions of the equation

where both A and B are n-by-n matrices and

is a scalar. The values of

that satisfy the equation are the generalized eigenvalues and the corresponding values of x are the generalized right eigenvectors.

If B is nonsingular, the problem could be solved by reducing it to a standard eigenvalue problem

Because B can be singular, an alternative algorithm, called the QZ method, is necessary.

When a matrix has no repeated eigenvalues, the eigenvectors are always independent and the eigenvector matrix V diagonalizes the original matrix A if applied as a similarity transformation. However, if a matrix has repeated eigenvalues, it is not similar to a diagonal matrix unless it has a full (independent) set of eigenvectors. If the eigenvectors are not independent then the original matrix is said to be defective. Even if a matrix is defective, the solution from eig satisfies A*X = X*D.

Examples

The matrix

B = [3 -2 -.9 2*eps;-2 4 -1 -eps;-eps/4 eps/2 -1 0;-.5 -.5 .1 1];

has elements on the order of roundoff error. It is an example for which the nobalance option is necessary to compute the eigenvectors correctly. Try the statements

[VB,DB] = eig(B)
B*VB - VB*DB
[VN,DN] = eig(B,'nobalance')
B*VN - VN*DN

Algorithm

For real matrices, eig(X) uses the EISPACK routines BALANC, BALBAK, ORTHES, ORTRAN, and HQR2. BALANC and BALBAK balance the input matrix. ORTHES converts a real general matrix to Hessenberg form using orthogonal similarity transformations. ORTRAN accumulates the transformations used by ORTHES. HQR2 finds the eigenvalues and eigenvectors of a real upper Hessenberg matrix by the QR method. The EISPACK subroutine HQR2 is modified to make computation of eigenvectors optional.

When eig is used with two input arguments, the EISPACK routines QZHES, QZIT, QZVAL, and QZVEC solve for the generalized eigenvalues via the QZ algorithm. Modifications handle the complex case.

When eig is used with one complex argument, the solution is computed using the QZ algorithm as eig(X,eye(X)). Modifications to the QZ routines handle the special case B = I.

For detailed descriptions of these algorithms, see the EISPACK Guide.

Diagnostics

If the limit of 30n iterations is exhausted while seeking an eigenvalue:

Solution will not converge.