Spectral Radii of Fixed Frobenius Norm Perturbations of Nonnegative Matrices

Let A be an n × 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 × 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.