The sepr-sets of sign patterns
Abstract
Given a real symmetric matrix, the sepr-sequence 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.
Cite
@article{arxiv.1807.04874,
title = {The sepr-sets of sign patterns},
author = {Leslie Hogben and Jephian C. -H. Lin and D. D. Olesky and P. van den Driessche},
journal= {arXiv preprint arXiv:1807.04874},
year = {2018}
}