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

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, $A_i\in \mathcal{A}$ and $B_i\in \mathcal{B}$, 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, $\Vert A_n{B}_n\cdots A_1{B}_1\Vert \le C{\lambda }^n$, where the constants $C>0$ and $\lambda \in (0,1)$ do not depend on the sequence $\{A_i\}$ and the corresponding sequence $\{B_i\}$.

7 pages, 13 bibliography references, expanded Introduction and Section 4 "Remarks and Open Questions", minor misprints correction

References in corpus (3)

• Entropy Games and Matrix Multiplication Games
• Minimax joint spectral radius and stabilizability of discrete-time linear switching control systems
• On convergence of infinite matrix products with alternating factors from two sets of matrices

Cited by in corpus (2)

• Minimax joint spectral radius and stabilizability of discrete-time linear switching control systems
• On convergence of infinite matrix products with alternating factors from two sets of matrices