Total negativity: Characterizations and single-vector tests

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.