Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

Counterexamples on spectra of sign patterns


Author: Yaroslav Shitov
Journal: Proc. Amer. Math. Soc. 146 (2018), 3709-3713
MSC (2010): Primary 15A18, 15B35
DOI: https://doi.org/10.1090/proc/14041
Published electronically: June 13, 2018
MathSciNet review: 3825826
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: An $n\times n$ sign pattern $S$, which is a matrix with entries $0,+,-$, is called spectrally arbitrary if any monic real polynomial of degree $n$ can be realized as a characteristic polynomial of a matrix obtained by replacing the nonzero elements of $S$ by numbers of the corresponding signs. A sign pattern $S$ is said to be a superpattern of those matrices that can be obtained from $S$ by replacing some of the nonzero entries by zeros. We develop a new technique that allows us to prove spectral arbitrariness of sign patterns for which the previously known Nilpotent Jacobian method does not work. Our approach leads us to solutions of numerous open problems known in the literature. In particular, we provide an example of a sign pattern $S$ and its superpattern $S’$ such that $S$ is spectrally arbitrary but $S’$ is not, disproving a conjecture proposed in 2000 by Drew, Johnson, Olesky, and van den Driessche.


References [Enhancements On Off] (What's this?)

References

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2010): 15A18, 15B35

Retrieve articles in all journals with MSC (2010): 15A18, 15B35


Additional Information

Yaroslav Shitov
Affiliation: 129346 Russia, Moscow, Izumrudnaya ulitsa, dom 65, kvartira 4
MR Author ID: 864960
Email: yaroslav-shitov@yandex.ru

Keywords: Matrix theory, eigenvalues, sign pattern
Received by editor(s): December 22, 2016
Received by editor(s) in revised form: October 31, 2017, and November 14, 2017
Published electronically: June 13, 2018
Communicated by: Patricia L.Β Hersh
Article copyright: © Copyright 2018 American Mathematical Society