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

Abstract

A matrix A is called totally positive (or totally non-negative) of order k, denoted by T P k (or T N 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, 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 is T P k if and only if for every submatrix Ar of A formed from r consecutive rows and r consecutive columns (with r ≤ k), LCP(Ar, 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 T P 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 whose co-ordinates have alternating signs -i.e., lie in a certain open bi-orthant. First, we improve on one of the main results in recent joint work [Bull. London Math. Soc., in press], which provided a novel characterization of T P 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.

Characterizing total negativity and testing their interval hulls

“…More precisely, we provide novel characterizations of such matrices using (a) sign non-reversal and (b) Linear Complementarity, to go along with Motzkin's classical characterization using (c) variation diminution. In addition, we strengthen all three tests (a), (b), and (c) to work with only a single test vector, in parallel to our recent works [8,9] for TP matrices.…”

“…(7) Let adj(A) denote the adjugate matrix of A and let e i ∈ R n denote the vector whose i-th component is 1, and other components are zero. (8) We say that an array X is ≥ 0 (respectively X > 0, X ≤ 0, X < 0) if all coordinates of X are ≥ 0 (respectively > 0, ≤ 0, < 0). (9) Define the vector d [n]…”

“…Remark 1.2. A careful look at the characterization of TP in [8,9] reveals the importance of the signs of the entries of A. Indeed, if A > 0 in Theorem A, the assertions (3), ( 4) are instead equivalent to T P k .…”

“…Then I(A, B) is totally negative of order k if and only if the matrices C ± (A, B) are totally negative of order k. We now turn our attention to one of the most well known and widely used properties of TP matrices: their variation diminution on the test set of all nonzero real vectors [5]. Recently in [8], we characterized total positivity via the variation diminishing property by providing a finite set of test vectors -in fact a single vector for each contiguous submatrix. Our next main result similarly characterizes total negativity using the variation diminishing property.…”

$k$-positivity of dual canonical basis elements from 1324- and 2143-avoiding Kazhdan-Lusztig immanants

“…In this situation, G S >0 is the set of k-positive matrices, matrices where all minors of size k and smaller are positive. Cluster algebra structures, topology, and variation diminishing properties of these matrices have been previously studied in [2,4,7,8].…”