The sepr-sets of sign patterns

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 that the sign pattern is 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.

COMMUNICATED BY:

• Bryan Shader

Keywords:

• Signed enhanced principal rank characteristic sequence
• sign pattern
• sign semi-stable pattern
• principal minor
• digraph

AMS subject classifications:

• 15B35
• 15A15
• 15B48
• 05C50

Acknowledgments

We thank PIMS for supporting a visit to the University of Victoria by L.H. where this research was initiated. We thank the anonymous referee for a careful reading and helpful comments that improved the exposition.

Disclosure statement

No potential conflict of interest was reported by the author(s).

ORCID

Leslie Hogben  https://orcid.org/0000-0003-1673-3789

Jephian C.-H. Lin  http://orcid.org/0000-0003-0119-9376

Additional information

Funding

Research supported in part by an NSERC (Natural Sciences and Engineering Research Council of Canada) Discovery grant.