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

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.