全负性：刻画与单向量检验
Total negativity: Characterizations and single-vector tests
AI总结：
本文利用符号不反转、变差消减和线性互补问题给出全负矩阵的单检验向量刻画，并将其用于矩阵区间包络的全负性检验。
摘要：
若一个矩阵的所有阶数不超过 $k$ 的子式均为负（非正），则称该矩阵为 $k$ 阶全负（全非正）矩阵。本文旨在通过（a）符号不反转性质、（b）变差消减性质以及（c）线性互补问题，给出全负性的若干新颖刻画。更强的是，这三种刻画中的每一种都只使用一个检验向量。作为符号不反转性质的应用，我们研究两个矩形矩阵的区间包络。具体而言，我们在矩阵 $A$ 与 $B$ 的区间包络中识别出两个矩阵 $C^\pm(A,B)$，它们可同时检验整个区间包络的 $k$ 阶全负性。我们还证明了全非正矩阵的类似刻画。这些新颖刻画在精神上可被认为类似于 Brown–Johnstone–MacGibbon [J. Amer. Statist. Assoc. 1981]（另见 Gantmacher–Krein，1950）、Choudhury–Kannan–Khare [Bull. London Math. Soc., 2021] 与 Choudhury [Bull. London Math. Soc., 2022] 刻画全正矩阵的基础结果。最后，利用 Gantmacher–Krein 在1950年的一项结果，我们证明：通过符号不反转性质或变差消减性质，无法用来自除坐标具有交替符号的开双正交限之外其他正交限的（单个）检验向量来检测全负/全非正矩阵。
英文摘要：
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.
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