A comparison of eigenvalue condition numbers for matrix polynomials

TL;DR

This paper compares various eigenvalue condition numbers for matrix polynomials, highlighting their relationships, differences, and extensions to finite, infinite, and zero eigenvalues, with a focus on geometric interpretation.

Contribution

It unifies the understanding of different eigenvalue condition numbers, showing their equivalence for moderate-degree polynomials and extending the Wilkinson-like number to all eigenvalues.

Findings

Homogeneous condition numbers are essentially the same for moderate-degree polynomials.

Homogeneous numbers provide a geometric interpretation of each other.

Comparison shows how these numbers extend the Wilkinson-like condition number to all eigenvalues.

Abstract

In this paper, we consider the different eigenvalue condition numbers for matrix polynomials used in the literature and we compare them. One of these condition numbers is a generalization of the Wilkinson condition number for the standard eigenvalue problem. This number has the disadvantage of only being defined for finite eigenvalues. In order to give a unified approach to all the eigenvalues of a matrix polynomial, both finite and infinite, two (homogeneous) condition numbers have been defined in the literature. In their definition, very different approaches are used. One of the main goals of this note is to show that, when the matrix polynomial has a moderate degree, both homogeneous numbers are essentially the same and one of them provides a geometric interpretation of the other. We also show how the homogeneous condition numbers compare with the "Wilkinson-like" eigenvalue condition number and how they extend this condition number to zero and infinite eigenvalues.