Spectra of principal submatrices of nonnegative matrices

Let AA be an n×nn×n nonnegative matrix with the spectrum (λ1,λ2,…,λn)(λ1,λ2,…,λn) and let A1A1 be an m×mm×m principal submatrix of AA with the spectrum (μ1,μ2,…,μm)(μ1,μ2,…,μm). In this paper we present some cases where the realizability of (μ1,μ2,…,μm,ν1,ν2,…,νs)(μ1,μ2,…,μm,ν1,ν2,…,νs) implies the realizability of (λ1,λ2,…,λn,ν1,ν2,…,νs)(λ1,λ2,…,λn,ν1,ν2,…,νs) and consider the question whether this holds in general. In particular, we show that the list(λ1,λ2,…,λn,-μ1,-μ2,…,-μm)(λ1,λ2,…,λn,-μ1,-μ2,…,-μm)is realizable.