Some Convergence Properties of Matrix Sets

A set A = \ Aⱼ : j ∈ J \ of n × n matrices is pointwise convergent provided each n-vector x can be steered to zero by iterated multiplication by matrices in A. The convergence is uniform if the sequence of multipliers may be chosen independently of x. This paper discusses conditions related to convergence for sets of diagonal, triangular, and general matrices, real and complex. It generalizes known conditions for convergence of a single matrix and characterizes convergence of a set of diagonal matrices in terms of semipositivity of a matrix derived from the set.