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

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