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On the Consistency of Combinatorially Symmetric Sign Patterns and the Class of 2-Consistent Sign Patterns

Partha Rana, Sriparna Bandopadhyay

TL;DR

The paper investigates when sign-pattern matrices are consistent, i.e., when every matrix in a pattern’s qualitative class has a fixed number of real eigenvalues. It disproves a prior sufficiency conjecture for irreducible, tridiagonal patterns with a $0$-diagonal, and provides a complete analysis for such patterns of order $n\le 5$, showing consistency is equivalent to the absence of repeated real eigenvalues. It also develops necessary conditions for irreducible, combinatorially symmetric sign patterns with cycles, and introduces the class $\Delta$ of $2$-consistent sign patterns, establishing foundational constraints and highlighting that these conditions are not generally sufficient. Together, these results advance understanding of inertia and eigenvalue structure in sign-pattern families and open avenues for characterizing $2$-consistency more broadly.

Abstract

A sign pattern is a matrix that has entries from the set $\{+,-,0\}$. An $n\times n$ sign pattern $\mathcal{P}$ is called consistent if every real matrix in its qualitative class has exactly $k$ real eigenvalues and $n-k$ nonreal eigenvalues for some integer $k$, with $1\leq k\leq n$. In the article \cite{1}, the authors established a necessary condition for irreducible, tridiagonal patterns with a $0$-diagonal to be consistent. Subsequently, they proposed that this condition is also sufficient for such patterns to be consistent. In this article, we first demonstrate that this proposition does not hold. We characterize all irreducible, tridiagonal sign patterns with a $0$-diagonal of order at most five that are consistent. Moreover, we establish useful, necessary conditions for irreducible, combinatorially symmetric sign patterns to be consistent. Finally, we introduce the class $Δ$ of all $2$-consistent sign patterns and provide several necessary conditions for sign patterns to belong to this class.

On the Consistency of Combinatorially Symmetric Sign Patterns and the Class of 2-Consistent Sign Patterns

TL;DR

The paper investigates when sign-pattern matrices are consistent, i.e., when every matrix in a pattern’s qualitative class has a fixed number of real eigenvalues. It disproves a prior sufficiency conjecture for irreducible, tridiagonal patterns with a -diagonal, and provides a complete analysis for such patterns of order , showing consistency is equivalent to the absence of repeated real eigenvalues. It also develops necessary conditions for irreducible, combinatorially symmetric sign patterns with cycles, and introduces the class of -consistent sign patterns, establishing foundational constraints and highlighting that these conditions are not generally sufficient. Together, these results advance understanding of inertia and eigenvalue structure in sign-pattern families and open avenues for characterizing -consistency more broadly.

Abstract

A sign pattern is a matrix that has entries from the set . An sign pattern is called consistent if every real matrix in its qualitative class has exactly real eigenvalues and nonreal eigenvalues for some integer , with . In the article \cite{1}, the authors established a necessary condition for irreducible, tridiagonal patterns with a -diagonal to be consistent. Subsequently, they proposed that this condition is also sufficient for such patterns to be consistent. In this article, we first demonstrate that this proposition does not hold. We characterize all irreducible, tridiagonal sign patterns with a -diagonal of order at most five that are consistent. Moreover, we establish useful, necessary conditions for irreducible, combinatorially symmetric sign patterns to be consistent. Finally, we introduce the class of all -consistent sign patterns and provide several necessary conditions for sign patterns to belong to this class.
Paper Structure (7 sections, 29 theorems, 52 equations, 12 figures)

This paper contains 7 sections, 29 theorems, 52 equations, 12 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Consistency of tree sign patterns with a $0$-diagonal and their relation to sign patterns requiring a unique inertia
  4. Consistency of combinatorially symmetric sign patterns with cycles in their underlying signed undirected graph
  5. $2$-consistent sign patterns
  6. Conclusion
  7. Acknowledgement

Key Result

Theorem 2.1

Let $\mathcal{A}$ be an irreducible, tridiagonal pattern with a $0$-diagonal. Then $\mathcal{A}$ is consistent only if the signed undirected graph of $\mathcal{A}$ has, at most, one maximal signed path with odd length.

Figures (12)

  • Figure 1: G
  • Figure 2: The signed undirected graph $G$ of $\mathcal{P}$.
  • Figure 3: The signed undirected graph $G_-$ of $\mathcal{P}_-$.
  • Figure 5: $G$
  • Figure 15: The signed undirected graph $G$ of $\mathcal{P}$.
  • ...and 7 more figures

Theorems & Definitions (74)

  • Theorem 2.1: Theorem 2.8, 1
  • Example 2.1
  • Example 2.1
  • Theorem 2.2: Theorem 2.8, 2025
  • Remark 2.1
  • Example 2.2
  • Remark 2.2
  • Lemma 3.1: Lemma 2.1, 2025
  • Theorem 3.1
  • proof
  • ...and 64 more