  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.

Group Theory;Combinatorics;Dynamical Systems