Continuous products of matrices

TL;DR

This paper investigates the conditions under which continuous products of matrices from a finite set converge, introducing properties LCP and RCP, and utilizing the joint spectral radius theory to establish necessary and sufficient criteria.

Contribution

It provides a complete characterization of convergence for continuous matrix products from finite sets using LCP and RCP properties, advancing the understanding of matrix product behavior.

Findings

Necessary and sufficient conditions for convergence are established.

Convergence depends on properties LCP and RCP of the matrix set.

Joint spectral radius theory is key to the analysis.

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.