Optimization of the spectral radius of a product for nonnegative matrices

Let A be an n × n irreducible nonnegative matrix. We show that over the set Ωnof all n × n doubly stochastic matrices S, the multiplicative spectral radius ρ (SA) attains a minimum and a maximum at a permutation matrix. For the case when A is a symmetric nonnegative matrix, a by-product of our technique of proof yields a result allowing us to show that ρ (S1A) ≥ ρ (S2A), when S1and S2are two symmetric matrices such that both S1A and S2A are nonnegative matrices and S1- S2is a positive semidefinite matrix. This result has several corollaries. One corollary is that ρ (S1A) ≥ ρ (S2A), when S1= (1 / n) J and S2= (1 / (n - 1)) (J - I), where J is the matrix of all 1's. A second corollary is a comparison theorem for weak regular splittings of two monotone matrices.