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.

翻译：若一个矩阵的所有阶数不超过$k$的子式均为负（非正），则称其为$k$阶全负（全非正）矩阵。本文旨在通过以下三个性质给出全负性的若干新刻划：(a) 符号非反转性质，(b) 变差缩减性质，以及(c) 线性互补问题。更关键的是，这三类刻划均使用单一测试向量。作为符号非反转性质的应用，我们研究了两矩形矩阵的区间包。特别地，我们在矩阵$A$和$B$的区间包中识别出两个矩阵$C^\pm(A,B)$，它们能同时检测整个区间包中$k$阶全负性。我们还给出了全非正矩阵的类似刻划。这些新刻划在思想上可类比于Brown-Johnstone-MacGibbon [J. Amer. Statist. Assoc. 1981]（另见Gantmacher-Krein, 1950）、Choudhury-Kannan-Khare [Bull. London Math. Soc., 2021]及Choudhury [Bull. London Math. Soc., 2022]关于全正矩阵的基础性结果。最后，利用Gantmacher-Krein 1950年的一个结果，我们证明全负/全非正矩阵无法通过来自（除具有交替符号坐标的开双正象限以外的）其他卦限的（单一）测试向量，经由符号非反转性质或变差缩减性质加以检测。