Sets of Matrices All Infinite Products of Which Converge.

An infinite product prod (i=1) to inf M sub i of matrices converges (on the right) if lim (i->inf) M sub 1 ...M sub i exists. A set sum = {A sub i :i>=1} of n x n matrices is called an RCP-set (right-convergent product set) if all infinite products with each element drawn from sum converge. Such sets of matrices arise in constructing self-similar objects like vonKoch's snowflake curve, in various interpolation schemes, in constructing wavelets of compact support, and in studying non-homogeneous Markov chains. This paper gives necessary conditions and also some sufficient conditions for a set sum to be an RCP-set. These are conditions on the eigenvalues and left-eigenspaces of matrices in sum and finite products of these matrices. Necessary and sufficient conditions are given for a finite set sum to be an RCP-set having a limit function M sub sum (d) = prod i=1 to inf A sub (di), where d = (d sub 1 , ... , d sub 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 sum is an RCP-set.