# eig (Eigenvalue and Eigenvector)

Posted 2011.10.27 00:31
[V,D] = eig(A)
V: Eigenvector
D: Eigenvalue

# eig-Eigenvalues and eigenvectors

## Syntax

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

## Description

d = eig(A) returns a vector of the eigenvalues of matrix A.

d = eig(A,B) returns a vector containing the generalized eigenvalues, if A and B are square matrices.

 Note   If S is sparse and symmetric, you can use d = eig(S) to return the eigenvalues of S. If S is sparse but not symmetric, or if you want to return the eigenvectors of S, use the function eigs instead of eig.

[V,D] = eig(A) produces matrices of eigenvalues (D) and eigenvectors (V) of matrix A, so that A*V = V*D. Matrix D is thecanonical 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.

If W is a matrix such that W'*A = D*W', the columns of W are the left eigenvectors of A. Use [W,D] = eig(A.'); W = conj(W) to compute the left eigenvectors.

[V,D] = eig(A,'nobalance') finds eigenvalues and eigenvectors without a preliminary balancing step. This may give more accurate results for certain problems with unusual scaling. 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.

[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 .

[V,D] = eig(A,B,flag) specifies the algorithm used to compute eigenvalues and eigenvectors. flag can be:

 'chol' Computes the generalized eigenvalues of A and B using the Cholesky factorization of B. This is the default for symmetric (Hermitian) A and symmetric (Hermitian) positive definite B. 'qz' Ignores the symmetry, if any, and uses the QZ algorithm as it would for nonsymmetric (non-Hermitian) A and B.

 Note   For eig(A), the eigenvectors are scaled so that the norm of each is 1.0. For eig(A,B), eig(A,'nobalance'), andeig(A,B,flag), the eigenvectors are not normalized. Also note that if A is symmetric, eig(A,'nobalance') ignores the nobalance option since A is already balanced.

## Tips

The eigenvalue problem is to determine the nontrivial solutions of the equation Ax = λx

where A is an n-by-n matrix, x is a length n column vector, and λ is a scalar. The n values of λ that satisfy the equation are theeigenvalues, and the corresponding values of x are the right eigenvectors. TheMATLAB function eig solves for the eigenvalues λ, and optionally the eigenvectors x.

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

where both A and B are n-by-n matrices and λ is a scalar. The values of λ that satisfy the equation are the generalized eigenvalues and the corresponding values of  are the generalized right eigenvectors.

If B is nonsingular, the problem could be solved by reducing it to a standard eigenvalue problem B–1Ax = λx

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 diagonalizesthe 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