Invertibly convergent infinite products of matrices, with applications to difference equations

The standard definition of convergence of an infinite product of scalars is extended to the infinite product P = П∞m=1 Bm of k × k matrices; that is, P is convergent according to the definition given here if and only if there is an integer N such that Bm is invertible for m ≥ N and P = limn→∞ Пnm=N (I + Am) is invertible. Sufficient conditions for this kind of convergence are given. Some of the results seem to be new even for infinite products of scalars. The results are derived by considering related systems of difference equations, and have implications concerning the asymptotic behavior of solutions of these systems.