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Handelman (J. Operator Theory, 1981) proved that if the spectral radius of amatrix $A$ is a simple root of the characteristic polynomial and is strictlygreater 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 explicitconjugate matrix $Z$, and prove that the spectral radius of $A$ is a simple anddominant 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 bedeclined into checking that each entry of some power is strictly larger thanthe average of the entries of the same column minus $\frac{1}{n}$. We apply thecriterion to elements of irreducible infinite nonaffine Coxeter groups toprovide evidences for the dominance of the spectral radius, which is stillunknown.

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