Eigenvalue Invariance of Inhomogeneous Matrix Products in Distributed Algorithms

Abstract

This letter establishes a general theorem concerning the eigenvalue invariance of certain inhomogeneous matrix products with respect to changes of individual multiplicands’ orderings. Instead of detailed entries, it is the zero-nonzero structure that matters in determining such eigenvalue invariance. The theorem is then applied in analyzing the convergence rate of a distributed algorithm for solving linear equations over networks modeled by undirected graphs.