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

• URL: http://arxiv.org/abs/2607.18148v1
• Date: Mon, 20 Jul 2026 16:48:24 GMT
• Title: Totally Positive Matrices and the Highest-Order Coefficients of the Characteristic Polynomial
• Abstract summary: 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.<n>The resulting separation is markedly nonlinear and admits a natural geometric description in the corresponding three-dimensional space by means of Mahalanobis ellipsoids.<n>These observations lead us to a conjecture on the geometric separation of structured totally positive families in the space determined by the three highest-order characteristic coefficients.
• Score: 0.0
• License: http://creativecommons.org/licenses/by/4.0/
• 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.

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