Joint bounds for the Perron roots of nonnegative matrices with applications

Abstract

Given a finite set {Ax}x∈X of nonnegative matrices, we derive joint upper and lower bounds for the row sums of the matrices D-1 A(x) D, x ∈ X, where D is a specially chosen nonsingular diagonal matrix. These bounds, depending only on the sparsity patterns of the matrices A(x) and their row sums, are used to obtain joint two-sided bounds for the Perron roots of given nonnegative matrices, joint upper bounds for the spectral radii of given complex matrices, bounds for the joint and lower spectral radii of a matrix set, and conditions sufficient for all convex combinations of given matrices to be Schur stable. Bibliography: 20 titles. © 2007 Springer Science+Business Media, Inc.