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) variation diminishing property, (b) sign non-reversal property, and (c) Linear Complementarity Problem. (The last of these provides a novel connection between total negativity and optimization/game theory.) More strongly, each of these three characterizations uses a single test vector whose coordinates alternate in sign. 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^±(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 and provide a finite set of test matrices to detect the total non-positivity property of an interval hull. These novel characterizations may be considered similar in spirit to fundamental results characterizing totally positive matrices by Brown et al. (1981) [5] (see also Gantmacher–Krein, 1950), Choudhury et al. (2021) [9] and Choudhury (2022) [8]. Finally, 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 variation diminishing property or the sign non-reversal property.