Sign patterns which require or allow the strong multiplicity property

TL;DR

This work develops a comprehensive program to classify sign patterns by their obligation or permission to have the non-symmetric strong multiplicity property ($nSMP$). By exploiting digraph structure, composite cycles, Hessenberg forms, and bifurcation arguments, the authors establish exact results for several natural pattern families: any pattern with all off-diagonal entries nonzero, cycle patterns, and a broad Hessenberg class must have the $nSMP$; star patterns are completely characterized in terms of loop/sign configurations, with clear criteria for when they require, allow, or do not allow the property. A central equivalence is proven: a pattern allows the $nSMP$ if and only if it allows distinct eigenvalues, with the latter tied to composite cycles of length $n-1$ or $n$, resolving open questions and providing a practical digraph-based criterion. The paper also constructs patterns that allow but do not require the $nSMP$, including spectrally arbitrary patterns, and provides complete classifications for small patterns, highlighting the nuanced interplay between combinatorial structure and spectral properties. Overall, the results offer a concrete, structure-driven framework to predict eigenvalue multiplicities from sign-pattern digraphs and lay groundwork for broader spectral-characterization problems.

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.

Figures (5)

• Figure 1: Digraphs of patterns that allow but do not require nSMP.
• Figure 2: Digraph of a pattern that allows but does not require the nSMP.
• Figure 3: Digraph constructions for patterns that allow but do not require the nSMP.
• Figure 4: Patterns whose fixed signings require the nSMP
• Figure 5: Patterns whose fixed signings allow but do not require the nSMP

Theorems & Definitions (42)

• Example 2.1
• Lemma 2.2
• Theorem 2.3
• Theorem 2.4
• Lemma 3.1
• proof
• Theorem 3.2
• proof
• Theorem 3.3
• proof