Sets of Matrices All Infinite Products of Which Converge

An infinite product IIT= lMi of matrices converges (on the right) if limi __ M, . . . Mi exists. A set Z = (Ai: i > l} of n X n matrices is called an RCP set (rightconvergent product set) if all infinite products with each element drawn from Z 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 X to be an RCP set. These are conditions on the eigenvalues and left eigenspaces of matrices in 2 and finite products of these matrices. Necessary and sufficient conditions are given for a finite set Z to be an RCP set having a limit function M,(d) = rIT= lAd,, where d = (d,, . , d,, . .>, 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 X is an RCP set.