SPECTRAL RADII OF CERTAIN ITERATION MATRICES AND CYCLE MEANS OF DIGRAPHS

Motivated by questions arising in the study of asynchronous iterative methods for solving linear systems, we consider the spectral radius of products of certain one cycle matrices. The spectral radius of a matr ix in our class is a monotonic increasing function of the length of th e cycle of the matrix, but this is known to be false for products of s uch matrices. The thrust of our investigation is to determine sufficie nt conditions under which the spectral radius of the product increases (decreases) when the lengths of the cycles of the factors increase (d ecrease). We also find sufficient conditions for the spectral radius o f the product to be independent of the order of the factors. Our chief tool is an auxiliary directed weighted graph whose cycle means determ ine the eigenvalues of the matrix product, and our main results are st ated in terms of the maximal cycle mean of this graph.