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If a left-product $M_n... M_1$ of square complex matrices converges to anonnull limit when $n\to\infty$ and if the $M_n$ belong to a finite set, it isclear that there exists an integer $n_0$ such that the $M_n$, $n\ge n_0$, havea common right-eigenvector $V$ for the eigenvalue 1. Now suppose that the $M_n$are nonnegative and that $V$ has positive entries. Denoting by $\Delta$ thediagonal matrix whose diagonal entries are the entries of $V$, the stochasticmatrices $S_n=\Delta^{-1}M_n\Delta$ satisfy $M_n... M_{n_0}=\Delta S_n...S_{n_0}\Delta^{-1}$, so the problem of the convergence of $M_n... M_1$ reducesto 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 firstsection details the case of the $2\times2$ matrices, and the last gives a firstapproach in the case of $d\times d$ matrices.

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