Existence and Construction of Nonnegative Matrices with Pure Image Spectrum

The nonnegative inverse eigenvalue problem (NIEP) is the problem of determining necessary and sufficient conditions for a list of complex numbers σ to be the spectrum of a nonnegative matrix. In this paper the problem is completely solved in the case when all numbers in the given list except for one (the Perron eigenvalue) are pure image numbers. Lets. Let σ = (ρ,b₁i, b₁i,···,b_ki, b_ki)be a list of complex numbers with ρ,bj >; 0 for j =1,2,···,k . A simple necessary and sufficient conditions for the existence of an entry wise nonnegative 2k +1 order matrix A with spectrum σ are presented, and the proof is elementary.