The Stability of the Products of a Finite Set of Matrices

Let \(F = \{ {{A}_{t}}, \ldots ,{{A}_{N}}\}\) be a set of n × n matrices. Given a sequence \(S = \{ {{A}_{{{{i}_{k}}}}}\} _{{k = 1}}^{\infty }\), with \({{A}_{{{{i}_{k}}}}} \in F\), we consider products of the form \({{B}_{{M,S}}} = \prod\limits_{{k = 1}}^{M} {{{A}_{{{{i}_{k}}}}}}\). We are interested in questions of the following type: 1. Is the set {B M,S: M = 1, 2, ... } bounded for all sequences S? (We will then say that F is stable.) Does B M,S converge to zero, as M → ∞ for all S? 2. What happens if we impose some restrictions on the set of allowed sequences S? 3. What are some simple classes of matrices for which the answers to 1 and 2 become simpler?