On convergence of infinite matrix products with alternating factors from two sets of matrices

TL;DR

This paper investigates conditions under which infinite products of matrices from two sets, with alternating factors, converge to zero uniformly exponentially, highlighting a strong form of convergence for such matrix sequences.

Contribution

It proves that if for any sequence of matrices from one set, there exists a sequence from the other set making the product converge to zero, then this convergence is uniformly exponential.

Findings

Convergence to zero implies uniform exponential convergence.

Constants for exponential decay do not depend on specific sequences.

Results apply to matrix products with alternating factors from two sets.

Abstract

We consider the problem of convergence to zero of matrix products $A_{n}B_{n}\cdots A_{1}B_{1}$ with factors from two sets of matrices, $A_{i}\in\mathscr{A}$ and $B_{i}\in\mathscr{B}$, due to a suitable choice of matrices $\{B_{i}\}$. It is assumed that for any sequence of matrices $\{A_{i}\}$ there is a sequence of matrices $\{B_{i}\}$ such that the corresponding matrix product $A_{n}B_{n}\cdots A_{1}B_{1}$ converges to zero. We show that in this case the convergence of the matrix products under consideration is uniformly exponential, that is, $\|A_{n}B_{n}\cdots A_{1}B_{1}\|\le C\lambda^{n}$, where the constants $C>0$ and $\lambda\in(0,1)$ do not depend on the sequence $\{A_{i}\}$ and the corresponding sequence $\{B_{i}\}$.