On Convergence of Infinite Matrix Products with Alternating Factors from Two Sets of Matrices

Abstract EN

We consider the problem of convergence to zero of matrix products AnBn⋯A1B1 with factors from two sets of matrices, Ai∈A and Bi∈B, due to a suitable choice of matrices {Bi}.

It is assumed that for any sequence of matrices {Ai} there is a sequence of matrices {Bi} such that the corresponding matrix products AnBn⋯A1B1 converge to zero.

We show that, in this case, the convergence of the matrix products under consideration is uniformly exponential; that is, AnBn⋯A1B1≤Cλn, where the constants C>0 and λ∈(0,1) do not depend on the sequence {Ai} and the corresponding sequence {Bi}.

Other problems of this kind are discussed and open questions are formulated.