发表机构
Institute of Informatics, Federal University of Rio Grande do Sul; Institute of Mathematics and Statistics, Federal University of Rio Grande do Sul（南里奥格兰德联邦大学信息学院; 南里奥格兰德联邦大学数学与统计学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
研究全正矩阵能否通过特征多项式最高阶系数区分，采用神经网络分类器等方法，利用多族矩阵构建数据集，发现特定系数有判别力，分离呈非线性，不同族有不同椭球特征并据此提出猜想。
AI 中文摘要
我们研究了全正矩阵能在多大程度上通过其特征多项式的最高阶系数来区分。为了确定最具信息的系数，我们还使用了神经网络分类器和特征归因方法。利用从几个结构化全正族构建的数据集，包括正双对角矩阵、范德蒙德矩阵和柯西矩阵的乘积，我们发现系数\((a_{n - 1}, a_{n - 2}, a_{n - 3})\)在维度5、10和30中已经包含了区分全正矩阵和非全正矩阵的强大判别信息。由此产生的分离明显是非线性的，并且在相应的三维系数空间中通过马氏椭球体有自然的几何描述。这些椭球体包围全正样本，同时排除大多数非全正样本。此外，不同的结构化全正族表现出不同的椭球特征，并且这些特征之间的分离随着维度增加。这些观察结果使我们对由三个最高阶特征系数确定的空间中结构化全正族的几何分离提出了一个猜想。
英文摘要
We investigate the extent to which totally positive matrices can be distinguished through the highest-order coefficients of their characteristic polynomials. To identify the most informative coefficients, we also employed neural-network classifiers together with feature-attribution methods. Using datasets built from several structured totally positive families, including products of positive bidiagonal matrices, Vandermonde matrices, and Cauchy matrices, we find that the coefficients (a_{n-1}, a_{n-2}, a_{n-3}) already contain strong discriminatory information for separating totally positive from non-totally positive matrices in dimensions 5, 10, and 30. The resulting separation is markedly nonlinear and admits a natural geometric description in the corresponding three-dimensional coefficient space by means of Mahalanobis ellipsoids. These ellipsoids enclose the totally positive samples while excluding most non-totally positive ones. Moreover, different structured totally positive families exhibit distinct ellipsoidal signatures, and the separation between these signatures increases with the dimension. These observations lead us to formulate a conjecture on the geometric separation of structured totally positive families in the space determined by the three highest-order characteristic coefficients.
Comments26 pages, 12 figures, 3 tables. Published open access in Linear Algebra and its Applications
Journal refLinear Algebra and its Applications 750 (2026) 24-52
