Stability Of Matrix Polynomials In One And Several Variables

TL;DR

This paper investigates the stability of matrix polynomials in multiple variables, introducing hyperstability, and extending classical complex analysis tools to analyze eigenvalue localization and stability properties.

Contribution

It introduces a stronger notion of hyperstability for matrix polynomials and extends classical theorems like Gauss-Lucas and Szász inequality to matrix settings.

Findings

Matrix versions of Gauss-Lucas theorem and Szász inequality are established.

Several second- and third-order matrix polynomials are shown to be stable under certain conditions.

Methods for eigenvalue localization of matrix polynomials are developed.

Abstract

The paper presents methods of eigenvalue localisation of regular matrix polynomials, in particular, stability of matrix polynomials is investigated. For this aim a stronger notion of hyperstability is introduced and widely discussed. Matrix versions of the Gauss-Lucas theorem and Sz\'asz inequality are shown. Further, tools for investigating (hyper)stability by multivariate complex analysis methods are provided. Several second- and third-order matrix polynomials with particular semi-definiteness assumptions on coefficients are shown to be stable.