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

arXiv: https://arxiv.org/abs/2607.18148 · HTML: https://mlsanity.com/paper/2607.18148/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 7 of its 17 references that are in the map.
- First posted on arXiv 20 Jul 2026.

## Builds on (most cited first, 7 of 7)

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- [Solving the Conjugacy Decision Problem via Machine Learning](https://mlsanity.com/paper/1705.10417/) (2017)
- [Explaining Deep Network Classification of Matrices: A Case Study on Monotonicity](https://mlsanity.com/paper/2507.22570/) (2025)

## Abstract (arXiv)

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.