SETS OF MATRICES ALL INFINITE PRODUCTS OF WHICH CONVERGE

An infinite product II(i = 1)infinity M(i) of matrices converges (on the right) if lim(i--> infinity) M1 ... M(i) exists. A set SIGMA = {A(i):i greater-than-or-equal-to 1} of n x n matrices is called an RCP set (right-convergent product set) if all infinite products with each element drawn from SIGMA converge. Such sets of matrices arise in constructing self-similar objects like von Koch's snowflake curve, in various interpolation schemes, in constructing wavelets of compact support, and in studying nonhomogeneous Markov chains. This paper gives necessary conditions and also some sufficient conditions for a set SIGMA to be an RCP set. These are conditions on the eigenvalues and left eigenspaces of matrices in SIGMA and finite products of these matrices. Necessary and sufficient conditions are given for a finite set SIGMA to be an RCP set having a limit function M-SIGMA(d) = II(i = 1)infinity A(d)(i), where d = (d1,...,d(n),...), which is a continuous function on the space of all sequences d with the sequence topology. Finite RCP sets of column-stochastic matrices are completely characterized. Some results are given on the problem of algorithmically deciding if a given set SIGMA is an RCP set.