Total negativity: Characterizations and single-vector tests

TL;DR

This paper introduces new characterizations of totally negative and non-positive matrices using single-vector tests, with applications to interval hulls and connections to classical matrix theory results.

Contribution

It provides novel characterizations of total negativity via sign non-reversal, variation diminishing, and LCP properties, each using a single test vector, extending classical matrix theory.

Findings

Characterizations of total negativity using single test vectors.

Identification of matrices testing total negativity in interval hulls.

Limitations on test vectors for detecting total negativity/non-positivity.

Abstract

A matrix is called totally negative (totally non-positive) of order $k$, if all its minors of size at most $k$ are negative (non-positive). The objective of this article is to provide several novel characterizations of total negativity via the (a) sign non-reversal property, (b) variation diminishing property, and (c) Linear Complementarity Problem. More strongly, each of these three characterizations uses a single test vector. As an application of the sign non-reversal property, we study the interval hull of two rectangular matrices. In particular, we identify two matrices $C^\pm(A,B)$ in the interval hull of matrices $A$ and $B$ that test total negativity of order $k$, simultaneously for the entire interval hull. We also show analogous characterizations for totally non-positive matrices. These novel characterizations may be considered similar in spirit to fundamental results characterizing totally positive matrices by Brown--Johnstone--MacGibbon [J. Amer. Statist. Assoc. 1981] (see also Gantmacher--Krein, 1950), Choudhury--Kannan--Khare [Bull. London Math. Soc., 2021] and Choudhury [Bull. London Math. Soc., 2022]. Finally using a 1950 result of Gantmacher--Krein, we show that totally negative/non-positive matrices can not be detected by (single) test vectors from orthants other than the open bi-orthant that have coordinates with alternating signs, via the sign non-reversal property or the variation diminishing property.