Existence and construction of nonnegative matrices with prescribed spectrum

We consider the following inverse spectrum problem for nonnegative matrices: given a set of real numbers σ={λ1,λ2,…,λn}, find necessary and sufficient conditions for the existence of an n×n nonnegative matrix A with spectrum σ. In particular, by the use of a relevant theorem of Brauer we obtain new simple sufficient conditions for the problem to have a solution. Moreover, we can always construct a solution matrix, which is nonnegative generalized stochastic.