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 Ωn of 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 S1 and S2 are two symmetric matrices such that both S1A and S2A are nonnegative matrices and S1-S2 is 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.