论文标题
签署完全非负和完全积极矩阵的非反转特性，并测试其间隔船体的总阳性
Sign non-reversal property for totally non-negative and totally positive matrices, and testing total positivity of their interval hull
论文作者
论文摘要
如果所有大小$ \ leq k $的未成年人均为正（或非负），则矩阵$ a $是$ k $（或$ tn_k $）的订单$ k $（或$ tn_k $）的完全正（或非负）。众所周知，此类矩阵的特征是变化属性下降以及符号非反转属性。我们取消了前者，并证明$ a $是$ tp_k $，并且仅当每个由$ k $ conteceent Rows and Coments and Coments组成的每个子元素都具有符号非反转属性。实际上，这可以得到加强，以仅考虑带有交替标志的$ \ mathbb {r}^k $中的测试向量。我们还显示了所有$ tn_k $矩阵的类似表征 - 更强烈的是，这两个特征都使用一个向量（带有交替的符号）对每个Square Submatrix。这些特征是新颖的，在精神上类似于Gantmacher-Krein [Compos。数学。 [1937年]和$ P $ - MATRICES撰写的Gale-Nikaido [Math。安。 1965]。 作为一个应用程序，我们研究了两个$ m \ times n $矩阵$ a =（a_ {ij}）$和$ b =（b_ {ij}）$的两个$ m \ times n $矩阵$ \ times n $矩阵的$ \ mathbb {i}（a，b）$。这是$ c \ in \ mathbb {r}^{m \ times n} $的集合，使每个$ c_ {ij} $在$ a_ {ij} $和$ b_ {ij} $之间。使用符号非逆转属性，我们确定$ \ mathbb {i}（a，b）$的两元素子集检测所有$ \ mathbb {i}（a，b）的$ tp_k $属性的$ tp_k $属性。特别是，这为整个矩形矩阵同时提供了总阳性（任何顺序）的测试。同时，我们还提供了一个有限的集合来测试间隔船体$ \ mathbb {i}（a，b）$的总非负（任何顺序）。
A matrix $A$ is totally positive (or non-negative) of order $k$, denoted $TP_k$ (or $TN_k$), if all minors of size $\leq k$ are positive (or non-negative). It is well-known that such matrices are characterized by the variation diminishing property together with the sign non-reversal property. We do away with the former, and show that $A$ is $TP_k$ if and only if every submatrix formed from at most $k$ consecutive rows and columns has the sign non-reversal property. In fact this can be strengthened to only consider test vectors in $\mathbb{R}^k$ with alternating signs. We also show a similar characterization for all $TN_k$ matrices - more strongly, both of these characterizations use a single vector (with alternating signs) for each square submatrix. These characterizations are novel, and similar in spirit to the fundamental results characterizing $TP$ matrices by Gantmacher-Krein [Compos. Math. 1937] and $P$-matrices by Gale-Nikaido [Math. Ann. 1965]. As an application, we study the interval hull $\mathbb{I}(A,B)$ of two $m \times n$ matrices $A=(a_{ij})$ and $B = (b_{ij})$. This is the collection of $C \in \mathbb{R}^{m \times n}$ such that each $c_{ij}$ is between $a_{ij}$ and $b_{ij}$. Using the sign non-reversal property, we identify a two-element subset of $\mathbb{I}(A,B)$ that detects the $TP_k$ property for all of $\mathbb{I}(A,B)$ for arbitrary $k \geq 1$. In particular, this provides a test for total positivity (of any order), simultaneously for an entire class of rectangular matrices. In parallel, we also provide a finite set to test the total non-negativity (of any order) of an interval hull $\mathbb{I}(A,B)$.
