Computing invariant subspaces of a general matrix when the eigensystem is poorly conditioned

Abstract:

The problem of calculating the eigensystem of a general complex matrix is well known. In many cases, however, the eigensystem is poorly determined numerically in the sense that small changes in the matrix can cause large changes in the eigensystem. For these matrices, a decomposition into higher-dimensional invariant subspaces is desirable. In this paper we define a class of matrices where this is true, and propose a technique for calculating bases for these invariant subspaces. We show that for this class the technique provides basis vectors which are accurate and span the subspaces well.