eigen¶

Purpose:

Solve the generalized eigenvalue problem.

Syntax:

L = eigen(K, M)
L = eigen(K, M, b)
[L, X] = eigen(K, M)
[L, X] = eigen(K, M, b)

Description:

eigen solves the eigenvalue problem

\[| \mathbf{K} - \lambda \mathbf{M} | = 0\]

where \(\mathbf{K}\) and \(\mathbf{M}\) are square matrices. The eigenvalues \(\lambda\) are stored in the vector \(\mathbf{L}\) and the corresponding eigenvectors in the matrix \(\mathbf{X}\).

If certain rows and columns in matrices \(\mathbf{K}\) and \(\mathbf{M}\) are to be eliminated in computing the eigenvalues, b must be given in the function. The rows (and columns) that are to be eliminated are described in the vector b defined as

b\(=\begin{bmatrix} dof_1 \\ dof_2 \\ \vdots \\ dof_{nb} \end{bmatrix}\)

The computed eigenvalues are given in order ranging from the smallest to the largest. The eigenvectors are normalized so that

\[\mathbf{X}^T \mathbf{M} \mathbf{X} = \mathbf{I}\]

where \(\mathbf{I}\) is the identity matrix.