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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
Published electronically: June 13, 2018
MathSciNet review: 3825826
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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.

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Additional Information

Yaroslav Shitov
Affiliation: 129346 Russia, Moscow, Izumrudnaya ulitsa, dom 65, kvartira 4
MR Author ID: 864960

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