Polynomial eigenvalue bounds from companion forms

TL;DR

This paper introduces a method to derive eigenvalue bounds for matrix polynomials using companion forms called $\,ell$-ifications, enhancing the ability to estimate eigenvalues effectively.

Contribution

It demonstrates how $\,ell$-ifications can be combined with recent techniques to produce improved eigenvalue bounds for matrix polynomials.

Findings

$\,ell$-ifications facilitate eigenvalue bounding.

The method improves existing eigenvalue estimation techniques.

The approach is applicable to complex matrix polynomials.

Abstract

We show how $\ell$-ifications, which are companion forms of matrix polynomials, namely, lower order matrix polynomials with the same eigenvalues as a given complex square matrix polynomial, can be used in combination with other recent results to produce eigenvalue bounds.