A Perron theorem for matrices with negative entries and applications to Coxeter groups

TL;DR

This paper extends Perron’s theorem to matrices with negative entries, providing explicit criteria for spectral dominance and applying these results to Coxeter groups to investigate spectral properties.

Contribution

It introduces an explicit conjugate matrix for Perron’s theorem and applies the criterion to Coxeter group elements to explore spectral dominance.

Findings

Explicit conjugate matrix construction for Perron’s theorem

Criterion for spectral radius dominance in matrices with row sum 1

Evidence for spectral radius dominance in Coxeter group elements

Abstract

Handelman (J. Operator Theory, 1981) proved that if the spectral radius of a matrix $A$ is a simple root of the characteristic polynomial and is strictly greater than the modulus of any other root, then $A$ is conjugate to a matrix $Z$ some power of which is positive. In this article, we provide an explicit conjugate matrix $Z$, and prove that the spectral radius of $A$ is a simple and dominant eigenvalue of $A$ if and only if $Z$ is eventually positive. For $n\times n$ real matrices with each row-sum equal to $1$, this criterion can be declined into checking that each entry of some power is strictly larger than the average of the entries of the same column minus $\frac{1}{n}$. We apply the criterion to elements of irreducible infinite nonaffine Coxeter groups to provide evidences for the dominance of the spectral radius, which is still unknown.