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 $0<t_1<\cdots <1$ 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.

9 pages

References in corpus (1)

• An explicit Lipschitz constant for the joint spectral radius