Perturbation of monic matrix polynomials

TL;DR

This paper investigates how the spectral properties of monic matrix polynomials change continuously with parameters, showing that eigenvalues and related sets depend Hölder continuously and analytically on parameters within certain submanifolds.

Contribution

It establishes Hölder continuity of spectral sets and analytic dependence of eigenvalues on parameters for monic matrix polynomials, extending to locally nonsingular cases.

Findings

Spectral sets are Hölder continuous with respect to parameters.

Eigenvalues and Jordan pairs depend analytically on parameters within submanifolds.

Results hold under local nonsingularity of leading coefficient matrices.

Abstract

In this paper, we study the stability of matrix polynomials under structured perturbations of their coefficients. More precisely, we consider a family of matrix polynomials \[ P_u(\lambda)=A_d(u)\lambda^d+A_{d-1}(u)\lambda^{d-1}+\cdots+A_0(u), \] whose matrix coefficients depend continuously and semialgebraically on a parameter vector $u\in\mathbb{C}^p$. Assuming that the matrix polynomial is monic, we show that the spectrum, the $\varepsilon$-pseudospectrum, the numerical range, and the joint numerical range associated with $P_u(\lambda)$ define set-valued maps that are H\"older continuous with respect to the parameter $u$. Moreover, the parameter space $\mathbb{C}^p$ can be decomposed into a finite union of analytic semialgebraic submanifolds such that, on each submanifold, the eigenvalues and the Jordan pairs of $P_u(\lambda)$ depend analytically on $u$. We also note that most of the results remain valid if the monicity assumption is replaced by the local nonsingularity of the leading coefficient matrix $A_d(u)$. However, the monic setting is adopted throughout the paper in order to simplify the exposition and to avoid additional technical assumptions, which are required in particular for results concerning numerical ranges.