Characterizing total negativity

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 in the interval hull that test total negativity of order $k$, simultaneously for the entire interval hull. We also prove 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.}, in press] and Choudhury [2021 preprint].