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arXiv · 1807.04874

The sepr-sets of sign patterns

Leslie Hogben·Jephian C. -H. Lin·D. D. Olesky·P. van den Driessche
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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.

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Leslie Hogben, Jephian C. -H. Lin, D. D. Olesky, P. van den Driessche. 2018-07-13. The sepr-sets of sign patterns. https://arxiv.org/abs/1807.04874

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