Negativity compensation in the nonnegative inverse eigenvalue problem

摘要

If a set Delta of complex numbers can be partitioned as Delta = Lambda(1) boolean OR...boolean OR Lambda(s) in such a way that each Lambda(i) is realized as the spectrum of a nonnegative matrix, say A(i) then Delta is trivially realized as the spectrum of the nonnegative matrix A = circle plus A(i). In [Linear Algebra Appl. 369 (2003) 169] it was shown that, in some cases, a real set Delta can be realized even if some of the Delta 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 Gamma a certain negativity N(Gamma), and to any realizable set A a certain positivity M(Lambda). Then, under appropriate conditions, if M(Lambda) greater than or equal to N(Gamma) we can conclude that Gamma boolean OR Lambda is the spectrum of a nonnegative matrix. Additionally, we prove a complex generalization of Suleimanova's theorem. (C) 2003 Elsevier Inc. All rights reserved.