Convergence of infinite products of matrices and inner-outer iteration schemes
L. Elsner, Rafael Bru, Michael Neumann · Publikationen an der Universität Bielefeld (Universität Bielefeld) · 1994
We develop conditions under which a product ∞ i=0 T i of matrices chosen from a possibly infinite set of matrices S = {T j |j ∈ J} converges.We obtain the following conditions which are sufficient for the convergence of the product: There exists a vector norm such that all matrices in S are nonexpansive with respect to this norm and there exists a subsequence {i k } ∞ k=0 of the sequence of the nonnegative integers such that the corresponding sequence of operators T i k ∞ k=0 converges to an operator which is paracontracting with respect to this norm.We deduce the continuity of the limit of the product of matrices as a function of the sequences {i k } ∞ k=0 .But more importantly, we apply our results to the question of the convergence of inner-outer iteration schemes for solving singular consistent linear systems of equations, where the outer splitting is regular and the inner splitting is weak regular.