Generalized Eigenvalue Problem for Symmetric Matrices
The program calculates the eigenvalues λi and corresponding eigenvectorsxi
for the generalized eigenvalue problem:
Ax = λ Bx.
Alternatively the standard eigenvalue problem
Ax = λ x
can also be solved.
For the algorithms used here to work on the generalized eigenvalue problem is required that B is positive definite or B is regular and A is positive definite.
A matrix is positive definite if all diagonal elements are greater than the sum of the amounts of the other elements of the respective row.
An application for the generalized eigenvalue problem is vibration analysis of dynamical systems with multiple degrees of freedom.
For dynamical systems with multiple degrees of freedom A is the stiffness matrix and B is the mass matrix. B must be positive definite and matrix A must be positive semi-definite to give nonnegative values for the λi.
The eigenfrequencies ωi of the modes of vibration can be calculated by means of ωi=λi1/2.
The eigenvectors describe the relative strength of the movement that points of the system take part in the vibration mode.