An algorithm for matrix polynomial eigenproblems

Abstract

When a system is described by higher order differential equations with time as the independent variable, the solution for the homogeneous system is defined by a matrix polynomial eigenproblem. A method alternative to the classical companion matrix method is introduced to expand the determinant algebraically to result in a scalar polynomial equation for the eigenvalues. The eigenvectors are obtained by inverse iteration. It is shown that the new method is computationally more advantageous than the conventional companion matrix method. A computer program is given for general matrix polynomials as well as Hermitian matrix polynomials. Defective eigenproblems can be handled without special attention.