On Convergence of Infinite Matrix Products with Alternating Factors from Two Sets of Matrices - Victor Kozyakin (Discrete Dynamics in Nature and Society)
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.