Eigenvalue location of certain matrix polynomials

TL;DR

This paper extends known eigenvalue bounds for matrix polynomials with unitary coefficients to those with doubly stochastic or Schur stable coefficients, showing eigenvalues lie within specific annular regions.

Contribution

It proves that matrix polynomials with doubly stochastic or Schur stable coefficients have eigenvalues confined to certain annular regions, generalizing previous results.

Findings

Eigenvalues of such matrix polynomials are in specific annuli.

Eigenvalue bounds apply to doubly stochastic coefficient matrices.

Eigenvalues are also confined for Schur stable coefficient matrices.

Abstract

It is known that a matrix polynomial with unitary matrix coefficients has its eigenvalues in the annular region $\frac{1}{2} < |\lambda| < 2$. We prove in this short note that under certain assumptions, matrix polynomials with either doubly stochastic matrix coefficients or Schur stable matrix coefficients also have eigenvalues in similar annular regions.