Log-majorization of the moduli of the eigenvalues of a matrix polynomial by tropical roots

TL;DR

This paper establishes a log-majorization relationship between the eigenvalues of matrix polynomials and tropical roots, extending classical bounds and providing new estimates especially for sparse or well-separated roots.

Contribution

It introduces a novel log-majorization bound linking eigenvalues of matrix polynomials to tropical roots, generalizing classical scalar polynomial bounds.

Findings

Eigenvalues' moduli are log-majorized by tropical roots up to universal constants.

New bounds for scalar polynomials are accurate for fewnomials and well-separated roots.

Extension of classical bounds by Hadamard, Ostrowski, and Pólya to matrix polynomials.

Abstract

We show that the sequence of moduli of the eigenvalues of a matrix polynomial is log-majorized, up to universal constants, by a sequence of "tropical roots" depending only on the norms of the matrix coefficients. These tropical roots are the non-differentiability points of an auxiliary tropical polynomial, or equivalently, the opposites of the slopes of its Newton polygon. This extends to the case of matrix polynomials some bounds obtained by Hadamard, Ostrowski and P\'olya for the roots of scalar polynomials. We also obtain new bounds in the scalar case, which are accurate for "fewnomials" or when the tropical roots are well separated.