On convergence of infinite matrix products with alternating factors from two sets of matrices
Victor Kozyakin
TL;DR
The paper addresses whether, for finite matrix sets \mathscr{A} and \mathscr{B}, one can choose the alternating factors $B_n$ so that the products $A_n B_n \cdots A_1 B_1$ converge to zero for all sequences $A_n\in\mathscr{A}$. Under path-dependent stabilizability, it proves a strong result: the convergence is uniformly exponential, i.e., $\|A_n B_n \cdots A_1 B_1\|\le C\lambda^{n}$ for some $C>0$ and $\lambda\in(0,1)$ independent of the sequences. The proof hinges on a uniform block-contraction lemma that guarantees a bounded block length $k_{*}$ producing a contraction $\|A_{k_{*}}B_{k_{*}}\cdots A_1 B_1\|\le\mu<1$, which in turn yields the global exponential bound. The paper also shows that path-independent periodic stabilizability implies the same exponential rate by reducing to a finite family of products over one period and applying exponential convergence there; these results extend to complex matrices and to compact sets, and the authors discuss related notions and open questions about weaker stabilizability conditions and spectral-radius concepts.
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, $\|A_{n}B_{n}\cdots A_{1}B_{1}\|\le Cλ^{n}$, where the constants $C>0$ and $λ\in(0,1)$ do not depend on the sequence $\{A_{i}\}$ and the corresponding sequence $\{B_{i}\}$.