Continuous products of matrices

Abstract

We answer the question if the continuous product of square matrices $M(t)$ over $t\in [0,1]$ can be correctly defined. The case where all $M(t)$ are taken from a finite set $\Sigma$ is studied. We find necessary and sufficient conditions on $\Sigma$ that ensure the convergence of products $M(t_{0}=0)M(t_{1})\dots M(t_{N}=1)$ as the partition $0<t_{1}<\dots<1$ refines. These conditions are properties LCP (left convergent product) and RCP (right convergent product) of the set $\Sigma$. That is, it suffices to require the convergence of all finite products $M_{1}M_{2}\dots M_{K}$ and $M_{K}\dots M_{2}M_{1}$ as $K\to\infty$, where $M_{i}\in\Sigma$. The theory of joint spectral radius is heavily used.