Characterizing total positivity: single vector tests via Linear Complementarity, sign non-reversal, and variation diminution

TL;DR

This paper establishes new characterizations of totally positive matrices using Linear Complementarity Problems, sign non-reversal, and variation diminution, connecting total positivity with optimization and game theory.

Contribution

It introduces novel single-vector tests for total positivity, linking TP matrices to LCP solutions and improving classical characterizations with sign and variation properties.

Findings

TP matrices characterized by unique LCP solutions for submatrices.

Enhanced sign non-reversal criteria for TP_k matrices.

Demonstrated limitations of test vectors outside the bi-orthant.

Abstract

A matrix $A$ is called totally positive (or totally non-negative) of order $k$, denoted by TP_k (or TN_k), if all minors of size at most $k$ are positive (or non-negative). These matrices have featured in diverse areas in mathematics, including algebra, analysis, combinatorics, and probability theory. The goal of this article is to provide a novel connection between total positivity and optimization/game theory. Specifically, we draw a relationship between TP matrices and the Linear Complementarity Problem (LCP), which generalizes and unifies linear and quadratic programming problems and bimatrix games - this connection is unexplored, to the best of our knowledge. We show that $A$ is $TP_k$ if and only if for every contiguous square submatrix $A_r$ of $A$, $LCP(A_r,q)$ has a unique solution for each vector $q<0$. In fact this can be strengthened to check the solution set of LCP at a single vector for each such square submatrix. These novel characterizations are in the spirit of classical results characterizing $TP$ matrices by Gantmacher-Krein [Compos. Math. 1937] and P-matrices by Ingleton [Proc. London Math. Soc. 1966]. Our work contains two other contributions, both of which characterize TP using single test vectors. First, we improve on one of the main results in recent joint work [Bull. London Math. Soc., 2021], which provided a novel characterization of TP_k matrices using sign non-reversal phenomena. We further improve on a classical characterization of TP by Brown-Johnstone-MacGibbon [J. Amer. Statist. Assoc. 1981] (following Gantmacher-Krein, 1950) involving the variation diminishing property. Finally, we use a P\'olya frequency function of Karlin [Trans. Amer. Math. Soc. 1964] to show that our aforementioned characterizations of TP, involving test-vectors drawn from the `alternating' bi-orthant, do not work if these vectors are drawn from any other open orthant.