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

Verifying whether a matrix is Totally Positive (TP)—meaning all its minors are positive—is computationally expensive, as the number of minors grows combinatorially with the matrix dimension. This paper investigates whether the highest-order coefficients of the characteristic polynomial, which are related to the sums of principal minors, can serve as a compact, informative proxy for identifying TP matrices. The authors use neural network classifiers and feature-attribution methods (Integrated Gradients and SHAP) to rank the importance of various spectral features. They then analyze the geometric distribution of TP matrices in the three-dimensional space defined by the coefficients (a_{n-1}, a_{n-2}, a_{n-3}).

The researchers found that the three highest-order coefficients contain sufficient information to separate TP from non-TP matrices with test accuracy exceeding 0.998. In the three-dimensional coefficient space, TP matrices form distinct, localized clusters that can be effectively bounded by Mahalanobis ellipsoids. These ellipsoids not only separate TP from non-TP samples but also exhibit unique geometric signatures for different structured TP families (e.g., Vandermonde vs. Cauchy). As the matrix dimension increases, the separation between these ellipsoidal signatures becomes more pronounced, suggesting a robust geometric structure inherent to total positivity.

This work offers a significant reduction in complexity for characterizing TP matrices. By mapping high-dimensional matrix properties into a three-dimensional coefficient space, researchers can bypass the need for exhaustive minor verification. The identification of these ellipsoidal signatures provides a new, intuitive way to classify structured matrices and suggests that the highest-order characteristic coefficients capture the essential spectral constraints imposed by the total positivity condition.