Construction of nonnegative symmetric matrices with given spectrum

Abstract

Let σ = ( λ 1, … , λ n ) be the spectrum of a nonnegative symmetric matrix A with the Perron eigenvalue λ 1, a diagonal entry c and let τ = ( μ 1, … , μ m) be the spectrum of a nonnegative symmetric matrix B with the Perron eigenvalue μ 1. We show how to construct a nonnegative symmetric matrix C with the spectrum ( λ 1 + max { 0 , μ 1 - c } , λ 2 , … , λ n , μ 2 , … , μ m ) .