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arXiv · 2505.08967

Sign patterns which require or allow the strong multiplicity property

Abhilash Saha·Leona Tilis·Kevin N. Vander Meulen·Adam Van Tuyl
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Abstract

We initiate a study of sign patterns that require or allow the non-symmetric strong multiplicity property (nSMP). We show that all cycle patterns require the nSMP, regardless of the number of nonzero diagonal entries. We present a class of Hessenberg patterns that require the nSMP. We characterize which star sign patterns require, which allow, and which do not allow the nSMP. We show that if a pattern requires distinct eigenvalues, then it requires the nSMP. Further, we characterize the patterns that allow the nSMP as being precisely the set of patterns that allow distinct eigenvalues, a property that corresponds to a simple feature of the associated digraph. We also characterize the sign patterns of order at most three according to whether they require, allow, or do not allow the nSMP.

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Abhilash Saha, Leona Tilis, Kevin N. Vander Meulen, Adam Van Tuyl. 2025-05-13. Sign patterns which require or allow the strong multiplicity property. https://arxiv.org/abs/2505.08967

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