On the location of eigenvalues of matrix polynomials

TL;DR

This paper derives bounds for the eigenvalues of matrix polynomials using the norms of their coefficients, comparing these bounds to existing results in the literature.

Contribution

It introduces new upper and lower bounds for eigenvalues of matrix polynomials based on coefficient matrix norms, enhancing previous bounds.

Findings

Established new bounds for eigenvalues of matrix polynomials.

Compared bounds with those by Higham, Tisseur, Maroulas, and Psarrakos.

Results improve understanding of eigenvalue localization for matrix polynomials.

Abstract

A number $\lambda \in \mathbb C $ is called an {\it eigenvalue} of the matrix polynomial $P(z)$ if there exists a nonzero vector $x \in \mathbb C^n$ such that $P(\lambda)x = 0$. Note that each finite eigenvalue of $P(z)$ is a zero of the characteristic polynomial $\det(P(z))$. In this paper we establish some (upper and lower) bounds for eigenvalues of matrix polynomials based on the norm of their coefficient matrices and compare these bounds to those given by N.J. Higham and F. Tisseur, J. Maroulas and P. Psarrakos.