On some properties of contracting matrices

Abstract

The concepts of paracontracting, pseudocontracting and nonexpanding operators have been shown to be useful in proving convergence of asynchronous or parallel iteration algorithms. The purpose of this paper is to give characterizations of these operators when they are linear and finite-dimensional. First we show that pseudocontractivity of stochastic matrices with respect to {norm of matrix} · {norm of matrix}∞ is equivalent to the scrambling property, a concept first introduced in the study of inhomogeneous Markov chains. This unifies results obtained independently using different approaches. Secondly, we generalize the concept of pseudocontractivity to set-contractivity which is a useful generalization with respect to the Euclidean norm. In particular, we demonstrate non-Hermitian matrices that are set-contractive for {norm of matrix} · {norm of matrix}2, but not pseudocontractive for {norm of matrix} · {norm of matrix}2 or {norm of matrix} · {norm of matrix}∞. For constant row sum matrices we characterize set-contractivity using matrix norms and matrix graphs. Furthermore, we prove convergence results in compositions of set-contractive operators and illustrate the differences between set-contractivity in different norms. Finally, we give an application to the global synchronization in coupled map lattices. © 2007 Elsevier Inc. All rights reserved.