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 ={ λ 1 , λ 2 ,…, λ n }, 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 λ 0 ⩾max 2⩽ j ⩽ n | λ j | such that the problem has a solution if and only if λ 1 ⩾ λ 0 . In particular, we discuss how to compute λ 0 and the solution matrix A for certain class of matrices. A sufficient condition for the problem to have a solution is also derived.