Totally Positive Matrices and the Highest-Order Coefficients of the Characteristic Polynomial

Abstract: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.

26 pages, 12 figures, 3 tables. Published open access in Linear Algebra and its Applications

Machine Learning (cs.LG); Rings and Algebras (math.RA)

arXiv:2607.18148 [cs.LG]

(or arXiv:2607.18148v1 [cs.LG] for this version)

https://doi.org/10.48550/arXiv.2607.18148 Focus to learn more arXiv-issued DOI via DataCite (pending registration)

https://doi.org/10.1016/j.laa.2026.07.003 Focus to learn more DOI(s) linking to related resources

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