Characterizing total negativity and testing their interval hulls
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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^±(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., in press] and Choudhury [2021 preprint]. 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.
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