On the asymptotic regularity of a family of matrices.

In this paper we consider bounded families ${\cal F}$ of complex $n\times n$ matrices. We give sufficient conditions under which the sequence $\{\bar{\rho}_k({\cal F})^{1/k}\}_{k\ge 1}$, where $\bar{\rho}_k({\cal F})$ is the supremum of the spectral radii of all possible products of $k$ matrices chosen in ${\cal F}$, is convergent to its supremum $\rho({\cal F})$, the so-called {\em (generalized) spectral radius} of ${\cal F}$. We also illustrate a possible practical application.