Spectral problem for polynomial pencils of matrices

Abstract

Let

$$D(\lambda ) = \lambda ^t A_0 + \lambda ^{t - 1} A_1 + ...A_t $$

((1))

be a polynomial pencil of m×n matrices of rank r.

The spectral problem for the pencil (1) is the problem to solve the equations

$$D(\lambda )u = 0 and D^T (\lambda )v = 0.$$

((2))

We propose an algorithm which allows to reduce the spectral problem for an arbitrary polynomial pencil of degree t⩾1 to the spectral problem for a linear pencil of larger dimension but of the same type as the initial pencil.

In the case of a linear pencil of full column rank we indicate a new algorithm for the isolation of regular blocks.

For the solution of the partial eigenvalue problem of a polynomial pencil (1) of full column rank we propose an algorithm which allows the computation of eigenvalues by means of scalar equations, using the methods of Muller, Newton, et al. We also indicate a method to compute the eigenvectors of (1) corresponding to isolated eigenvalues.