Characterizing total positivity: Single vector tests via linear complementarity, sign non‐reversal and variation diminution

Abstract

A matrix A $A$ is called totally positive (or totally non-negative) of order k $k$ , denoted by T P k $TP_k$ (or T N k $TN_k$ ), if all minors of size at most k $k$ are positive (or non-negative). These matrices have featured in diverse areas in mathematics, including algebra, analysis, combinatorics, differential equations 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 totally positive 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 $A$ is T P k $TP_k$ if and only if for every submatrix A r $A_r$ of A $A$ formed from r $r$ consecutive rows and r $r$ consecutive columns (with r ⩽ k $r\leqslant k$ ), LCP ( A r , q ) $\mathrm{LCP}(A_r,q)$ has a unique solution for each vector q < 0 $q<0$ . In fact this can be strengthened to check the solution set of the LCP at a single vector for each such square submatrix. These novel characterizations are in the spirit of classical results characterizing T P $TP$ matrices by Gantmacher–Krein [Compos. Math. 1937] and P $P$ -matrices by Ingleton [Proc. London Math. Soc. 1966]. Our work contains two other contributions, both of which characterize total positivity using single test vectors whose coordinates have alternating signs — that is, lie in a certain open bi-orthant. First, we improve on one of the main results in recent joint work [Bull. London Math. Soc., 2021], which provided a novel characterization of T P k $TP_k$ matrices using sign non-reversal phenomena. We further improve on a classical characterization of total positivity by Brown–Johnstone–MacGibbon [J. Amer. Statist. Assoc. 1981] (following Gantmacher–Krein, 1950) involving the variation diminishing property. Finally, we use a Pólya frequency function of Karlin [Trans. Amer. Math. Soc. 1964] to show that our aforementioned characterizations of total positivity, involving (single) test-vectors drawn from the ‘alternating’ bi-orthant, do not work if these vectors are drawn from any other open orthant.