Counterexamples on spectra of sign patterns

Shitov, Yaroslav

doi:10.1090/proc/14041

Counterexamples on spectra of sign patterns
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by Yaroslav Shitov PDF

Proc. Amer. Math. Soc. 146 (2018), 3709-3713 Request permission

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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