Floating-point perturbations of Hermitian matrices

Abstract

We consider the perturbation properties of the eigensolution of Hermitian matrices. For the matrix entries and the eigenvalues we use the realistic “floating-point” error measure |δ a/ a|. Recently, Demmel and Veselić considered the same problem for a positive definite matrix H, showing that the floating-point perturbation theory holds with constants depending on the condition number of the matrix A= DHD, where A ii =1 and D is a diagonal scaling. We study the general Hermitian case along the same lines, thus obtaining new classes of well-behaved matrices and matrix pairs. Our theory is applicable to the already known class of scaled diagonally dominant matrices as well as to matrices given by factors—like those in symmetric indefinite decompositions. We also obtain norm estimates for the perturbations of the eigenprojections, and show that some of our techniques extend to non-Hermitian matrices. However, unlike in the positive definite case, we are still unable to describe simply the set of all well-behaved Hermitian matrices.