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 ⋅ ⋅ ⋅ 1 1 with factors from two sets of matrices, ∈ A and ∈ B, due to a suitable choice of matrices { }. It is assumed that for any sequence of matrices { } there is a sequence of matrices { } such that the corresponding matrix products ⋅ ⋅ ⋅ 1 1 converge to zero. We show that, in this case, the convergence of the matrix products under consideration is uniformly exponential; that is, ‖ ⋅ ⋅ ⋅ 1 1 ‖ ≤ , where the constants > 0 and ∈ (0, 1) do not depend on the sequence { } and the corresponding sequence { }. Other problems of this kind are discussed and open questions are formulated.

Minimax joint spectral radius and stabilizability of discrete-time linear switching control systems

“…. Lemma 7.3 (see [28,Theorem 2]). Let A and B be finite sets of matrices of dimension N × M and M ×N , respectively, and · be a norm on the space of matrices of dimension N ×N .…”