Sign patterns which require or allow the strong multiplicity property
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.
Cite
@article{arxiv.2505.08967,
title = {Sign patterns which require or allow the strong multiplicity property},
author = {Abhilash Saha and Leona Tilis and Kevin N. Vander Meulen and Adam Van Tuyl},
journal= {arXiv preprint arXiv:2505.08967},
year = {2025}
}
Comments
17 pages; comments welcomed