Spectral Radii of Fixed Frobenius Norm Perturbations of Nonnegative Matrices

Abstract

Let A be an n X n nonnegative matrix. In this paper we consider the problems of maximizing the spectral radii of (i) A + ,X. and (ii) A + D, where X is a real n X n matrix whose Frobenius norm is restricted to be 1 and where D is as X but is further constrained to be a diagonal matrix. For both problems the maximums occur at nonnegative X and D, and we use tools of nonnegative matrices, most notably due to Levinger and Fiedler, as well as the Kuhn--Tucker criterion for constrained optimization, to find upper and lower bounds on the maximums, and, when A is additionally assumed to be irreducible, to characterize cases of equalities in these bounds. In the case of the first problem, when A is irreducible, we characterize a matrix which gives the global maximum. A matrix which yields a global maximum to the second problem is more complicated to characterize as, depending on A, the problem admits local maximums within the nonnegative diagonal matrices of Frobenius norm 1.