Perturbations on constructible lists preserving realizability in the NIEP and questions of Monov

• Published in: Linear algebra and its applications Vol. 445; pp. 206 - 222
• Main Authors: Cronin, Anthony G., Laffey, Thomas J.
• Format: Journal Article
• Language: English
• Published: Elsevier Inc 15-03-2014
• Subjects: Inverse eigenvalue problem; Nonnegative matrices; Inverse eigenvalue problem; Nonnegative matrices; 15A18; 15A29
• ISSN: 0024-3795; 1873-1856
• DOI: 10.1016/j.laa.2013.12.003

In [5] the authors showed that if σ=(λ1,λ2,λ2¯,λ4,…,λn) is realizable where λ1 is the Perron eigenvalue and λ2 is non-real, then so too is σ=(λ1+4t,λ2+t,λ2¯+t,λ4,…,λn). They asked if 4 can be replaced by 1 or 2 or what is the least possible multiple c⩾1 of t in order for this perturbation to be realizable. In [2] the authors showed that for n=4 one can find certain spectra for which the result holds when c=1 provided t is “small”. In this paper we show that c=2 is best possible and we construct a realizing matrix for c=2 when t is sufficiently small. We also address some questions of Monov concerning the realizability of the derivative of a realizable polynomial and if such a polynomial must have positive power sums.