1 research outputs found
On some properties of contracting matrices
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 sup-norm 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 ||.||_2, but not pseudocontractive for
||.||_2 or sup-norm. 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.Comment: 17 page