Optimization of the spectral radius of a product for nonnegative matrices

Abstract

Let A be an n × n irreducible nonnegative matrix. We show that over the set Ω n 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 ( S 1 A ) ⩾ ρ ( S 2 A ) , when S 1 and S 2 are two symmetric matrices such that both S 1 A and S 2 A are nonnegative matrices and S 1 - S 2 is a positive semidefinite matrix. This result has several corollaries. One corollary is that ρ ( S 1 A ) ⩾ ρ ( S 2 A ) , when S 1 = ( 1 / n ) J and S 2 = ( 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.