Contractive inhomogeneous products of non-negative matrices
Joel E. Cohen · Mathematical Proceedings of the Cambridge Philosophical Society · 1979
1. Introduction. Hajnal (5) showed that under wide conditions a sequence of products H(1, q) = Mq…M1q = 1,2,…, of square non-negative matrices Mq approaches a sequence of positive matrices of rank 1. We call a product H(1,q) inhomogeneous if its factors M1,…, Mq are not necessarily all equal to one another. When the matrices Mq are members of an ‘ergodic set’, and x and y are positive vectors, the projective distance d(H(1,q)x, H(1,q)y) decays at least exponentially fast as q increases. An important condition on an ergodic set is that any product of g members from the set be a matrix in which all elements are (strictly) positive, where g is some fixed positive integer.