Existence and construction of nonnegative matrices with complex spectrum

The following inverse spectrum problem for nonnegative matrices is considered: given a set of complex numbers σ={λ1,λ2,…,λn}, find necessary and sufficient conditions for the existence of an n×n nonnegative matrix A with spectrum σ. Our work is motivated by a relevant theoretical result of Guo Wuwen [Linear Algebra Appl. 266 (1997) 261, Theorem 2.1]: there exists a real parameter λ0 max2 j nλj such that the problem has a solution if and only if λ1 λ0. In particular, we discuss how to compute λ0 and the solution matrix A for certain class of matrices. A sufficient condition for the problem to have a solution is also derived.