Can an infinite left-product of nonnegative matrices be expressed in terms of infinite left-products of stochastic ones?

Abstract

If a left-product $M_n... M_1$ of square complex matrices converges to a nonnull limit when $n\to\infty$ and if the $M_n$ belong to a finite set, it is clear that there exists an integer $n_0$ such that the $M_n$, $n\ge n_0$, have a common right-eigenvector $V$ for the eigenvalue 1. Now suppose that the $M_n$ are nonnegative and that $V$ has positive entries. Denoting by $\Delta$ the diagonal matrix whose diagonal entries are the entries of $V$, the stochastic matrices $S_n=\Delta^{-1}M_n\Delta$ satisfy $M_n... M_{n_0}=\Delta S_n... S_{n_0}\Delta^{-1}$, so the problem of the convergence of $M_n... M_1$ reduces to the one of $S_n... S_{n_0}$. In this paper we still suppose that the $M_n$ are nonnegative but we do not suppose that $V$ has positive entries. The first section details the case of the $2\times2$ matrices, and the last gives a first approach in the case of $d\times d$ matrices.