Negativity compensation in the nonnegative inverse eigenvalue problem Original Research Article

If a set Δ of complex numbers can be partitioned as Δ=Λ1union or logical sumcdots, three dots, centeredunion or logical sumΛs in such a way that each Λi is realized as the spectrum of a nonnegative matrix, say Ai, then Δ is trivially realized as the spectrum of the nonnegative matrix A=circled plusAi. In [Linear Algebra Appl. 369 (2003) 169] it was shown that, in some cases, a real set Δ can be realized even if some of the Λi are not realizable themselves. Here we systematize and extend these results, in particular allowing the sets to be complex. The leading idea is that one can associate to any nonrealizable set Γ a certain negativity image, and to any realizable set Λ a certain positivity image. Then, under appropriate conditions, if image we can conclude that Γunion or logical sumΛ is the spectrum of a nonnegative matrix. Additionally, we prove a complex generalization of Suleimanova’s theorem.