Negativity compensation in the nonnegative inverse eigenvalue problem (original) (raw)

2004, Linear Algebra and its Applications

If a set of complex numbers can be partitioned as = 1 ∪ · · · ∪ s in such a way that each i is realized as the spectrum of a nonnegative matrix, say A i , then is trivially realized as the spectrum of the nonnegative matrix A = A i . In [Linear Algebra Appl. 369 (2003) 169] it was shown that, in some cases, a real set can be realized even if some of the i 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 a certain negativity N( ), and to any realizable set a certain positivity M( ). Then, under appropriate conditions, if M( ) N( ) we can conclude that ∪ is the spectrum of a nonnegative matrix. Additionally, we prove a complex generalization of Suleimanova's theorem.