On condition numbers of polynomial eigenvalue problems with nonsingular leading coefficients

TL;DR

This paper analyzes the condition numbers of eigenvalues in matrix polynomial problems with nonsingular leading coefficients, establishing new bounds and relations to pseudospectral growth, and providing a novel eigenvalue condition number expression.

Contribution

It introduces new bounds and a vector-free expression for eigenvalue condition numbers in polynomial eigenvalue problems with nonsingular leading coefficients.

Findings

Condition numbers relate to pseudospectral growth rates.

Ill-conditioned eigenvalues are near multiple eigenvalues.

New Elsner-like perturbation bounds are derived.

Abstract

In this paper, we investigate condition numbers of eigenvalue problems of matrix polynomials with nonsingular leading coefficients, generalizing classical results of matrix perturbation theory. We provide a relation between the condition numbers of eigenvalues and the pseudospectral growth rate. We obtain that if a simple eigenvalue of a matrix polynomial is ill-conditioned in some respects, then it is close to be multiple, and we construct an upper bound for this distance (measured in the euclidean norm). We also derive a new expression for the condition number of a simple eigenvalue, which does not involve eigenvectors. Moreover, an Elsner-like perturbation bound for matrix polynomials is presented.