Optimization of the spectral radius of a product for nonnegative matrices
For an \(n\times n\) irreducible nonnegative matrix \(A\), the authors show that over the set of all \(n\times n\) doubly stochastic matrices \(S\), the multiplicative spectral radius \(\rho (SA)\) attains a minimum and a maximum at a permutation matrix. When \(A\) is symmetric, they prove that \(\rho (S_1A)\geq \rho (S_2A)\), when \(S_1\) and \(S_2\) are two symmetric matrices such that both \(S_1A\) and \(S_2A\) are nonnegative matrices and \(S_1-S_2\) is a positive semidefinite matrix. Some consequences are considered.
scientific article; zbMATH DE number 5518761