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

TL;DR

This paper investigates the properties of sign patterns in matrices, disproves a previous conjecture about their consistency, and characterizes certain classes of consistent patterns, especially focusing on 2-consistent sign patterns.

Contribution

It disproves a previous sufficiency conjecture for tridiagonal sign patterns and characterizes all small irreducible, tridiagonal consistent patterns, introducing the class of 2-consistent sign patterns.

Findings

Disproved the sufficiency of a necessary condition for certain patterns.

Characterized all irreducible, tridiagonal sign patterns with zero diagonal of order up to five that are consistent.

Established necessary conditions for the class of 2-consistent sign patterns.

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 $\Delta$ of all $2$-consistent sign patterns and provide several necessary conditions for sign patterns to belong to this class.