Sign patterns which require or allow the strong multiplicity property

TL;DR

This paper investigates sign patterns in matrices that necessitate or permit the strong multiplicity property, providing characterizations for various classes of patterns and their eigenvalue properties.

Contribution

It introduces new classifications of sign patterns based on the strong multiplicity property and characterizes patterns that require, allow, or do not allow this property.

Findings

Cycle patterns require the nSMP regardless of diagonal entries.

Certain Hessenberg patterns require the nSMP.

Patterns that require distinct eigenvalues also require the nSMP.

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.