The sepr-sets of sign patterns

TL;DR

This paper extends the concept of sepr-sequences from symmetric matrices to sign patterns, providing conditions for uniqueness, analyzing structure in semi-stable patterns, and characterizing sequences for small symmetric nonnegative matrices.

Contribution

It introduces the extension of sepr-sequences to sign patterns, offers a sufficient condition for uniqueness, and characterizes sequences for small symmetric nonnegative matrices.

Findings

Unique sepr-sequence condition for sign patterns

Structured sepr-sequences in sign semi-stable patterns

Characterization of sepr-sequences for symmetric nonnegative matrices of orders two and three

Abstract

Given a real symmetric $n\times n$ matrix, the sepr-sequence $t_1\cdots t_n$ records information about the existence of principal minors of each order that are positive, negative, or zero. This paper extends the notion of the sepr-sequence to matrices whose entries are of prescribed signs, that is, to sign patterns. A sufficient condition is given for a sign pattern to have a unique sepr-sequence, and it is conjectured to be necessary. The sepr-sequences of sign semi-stable patterns are shown to be well-structured; in some special circumstances, the sepr-sequence is enough to guarantee the sign pattern being sign semi-stable. In alignment with previous work on symmetric matrices, the sepr-sequences for sign patterns realized by symmetric nonnegative matrices of orders two and three are characterized.