  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.
