Eigenvalues of Graded Matrices and the Condition Numbers of a Multiple Eigenvalue

Abstract

This paper concerns two closely related topics: the behavior of the eigenvalues of graded matrices and the perturbation of a nondefective multiple eigenvalue. We will show that the eigenvalues of a graded matrix tend to share the graded structure of the matrix and give precise conditions insuring that this tendency is realized. These results are then applied to show that the secants of the canonical angles between the left and right invariant subspaces of a multiple eigenvalue tend to characterize its behavior when its matrix is slightly perturbed. (Also cross-referenced as UMIACS-TR-90-31) Appeared in Numerische Mathematik 58 (1991) 703--712.