通过特征多项式高阶系数，可有效区分正定矩阵与非正定矩阵。
Totally Positive Matrices and the Highest-Order Coefficients of the Characteristic Polynomial
- 利用前三个高阶系数识别正定矩阵特征
- 5到30维下分离效果显著，且呈非线性分布
- 不同结构矩阵在系数空间中呈现独特椭球形态
我们研究了完全正定矩阵能否通过其特征多项式的最高阶系数进行区分。基于由正双对角矩阵乘积、范德蒙德矩阵和柯西矩阵构成的多个结构化完全正定族数据集，结合神经网络分类器与特征归因方法，发现系数 (a_{n-1}, a_{n-2}, a_{n-3}) 在维度为5、10和30时已包含强区分信息，能有效分离完全正定与非完全正定矩阵。该分离具有显著非线性特征，在三维系数空间中可通过马哈拉诺比斯椭球自然描述：完全正定样本被椭球包围，而多数非完全正定样本被排除在外。此外，不同结构的完全正定族在系数空间中表现出独特的椭球特征，且随维度增加，各特征间的分离度增强。这些发现促使我们提出关于三类最高阶系数空间中结构化完全正定族几何分离的猜想。
原文摘要 · Abstract (English)
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.
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