Totally Positive Matrices and the Highest-Order Coefficients of the Characteristic Polynomial
2026-07-20 • Machine Learning
Machine Learning
AI summaryⓘ
The authors studied special positive matrices called totally positive matrices and tried to tell them apart from other matrices using just a few numbers from their characteristic polynomials. They used neural networks to find which numbers are most helpful and found that three top coefficients are already very good at distinguishing these matrices in various sizes. They also discovered that these coefficients form shapes (ellipsoids) that nicely group the totally positive matrices separately from others. Different types of these matrices make different shapes, and these differences become clearer as matrix size grows.
Totally Positive MatrixCharacteristic PolynomialNeural Network ClassifierFeature AttributionBidiagonal MatrixVandermonde MatrixCauchy MatrixMahalanobis EllipsoidNonlinear SeparationMatrix Dimension
Authors
Tiago Closs, Leandro Farina
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.