Choudhury, PN (2022) Characterizing total positivity: Single vector tests via linear complementarity, sign non-reversal and variation diminution. In: Bulletin of the London Mathematical Society, 54 (2). pp. 791-811.
Full text not available from this repository.Abstract
A matrix (Formula presented.) is called totally positive (or totally non-negative) of order (Formula presented.), denoted by (Formula presented.) (or (Formula presented.)), if all minors of size at most (Formula presented.) 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 (Formula presented.) is (Formula presented.) if and only if for every submatrix (Formula presented.) of (Formula presented.) formed from (Formula presented.) consecutive rows and (Formula presented.) consecutive columns (with (Formula presented.)), (Formula presented.) has a unique solution for each vector (Formula presented.). 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 (Formula presented.) matrices by Gantmacher–Krein [Compos. Math. 1937] and (Formula presented.) -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 (Formula presented.) 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.
| Item Type: | Journal Article |
|---|---|
| Publication: | Bulletin of the London Mathematical Society |
| Publisher: | John Wiley and Sons Ltd |
| Additional Information: | The copyright for this article belongs to the Author. |
| Department/Centre: | Division of Physical & Mathematical Sciences > Mathematics |
| Date Deposited: | 24 Jun 2022 11:52 |
| Last Modified: | 24 Jun 2022 11:52 |
| URI: | https://eprints.iisc.ac.in/id/eprint/73702 |
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