A matrix $A$ is totally positive (or non-negative) of order $k$, denoted $T{P}_k$ (or $T{N}_k$), if all minors of size $⩽k$ 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 $T{P}_k$ 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 $T{N}_k$ 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. 4 (1937) 445–476) and $P$-matrices by Gale–Nikaido (Math. Ann. 159 (1965) 81–93).

中文翻译：

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为完全非负和完全正矩阵签署非反转性质，并测试其区间包的总正性

一个矩阵 $一$ 是完全正（或非负）阶 $克$, 表示 $吨{磷}_{克}$ （或 $吨N_{克}$)，如果所有未成年人的大小 $⩽克$为正（或非负）。众所周知，这种矩阵的特点是变化减小特性和符号不反转特性。我们废除了前者，并表明$一$ 是 $吨{磷}_{克}$ 当且仅当每个子矩阵最多由 $克$连续的行和列具有符号非反转特性。事实上，这可以加强，只考虑测试向量${\mathbb{电阻}}^{克}$与交替的迹象。我们还展示了所有类似的特征$吨N_{克}$矩阵——更强烈的是，这两种特征对每个方形子矩阵都使用单个向量（具有交替符号）。这些表征是新颖的，在精神上与表征的基本结果相似。$吨磷$Gantmacher–Krein 的矩阵 ( Compos. Math . 4 (1937) 445–476) 和$磷$- Gale-Nikaido 的矩阵 ( Math. Ann . 159 (1965) 81-93)。