Construction of nonnegative symmetric matrices with given spectrum

Let σ = (λ1, … , λn) be the spectrum of a nonnegative symmetric matrix A with the Perron eigenvalue λ1, a diagonal entry c and let τ = (μ1, … , μm) be the spectrum of a nonnegative symmetric matrix B with the Perron eigenvalue μ1. We show how to construct a nonnegative symmetric matrix C with the spectrum(λ1+max{0,μ1-c},λ2,…,λn,μ2,…,μm).(λ1+max{0,μ1-c},λ2,…,λn,μ2,…,μm).