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Sign non-reversal property for totally non-negative and totally positive matrices, and testing total positivity of their interval hull

Choudhury, PN and Kannan, MR and Khare, A (2021) Sign non-reversal property for totally non-negative and totally positive matrices, and testing total positivity of their interval hull. In: Bulletin of the London Mathematical Society, 53 (4). pp. 981-990.

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Official URL: https://doi.org/10.1112/blms.12475


A matrix (Formula presented.) is totally positive (or non-negative) of order (Formula presented.), denoted (Formula presented.) (or (Formula presented.)), if all minors of size (Formula presented.) are positive (or non-negative). It is well known that such matrices are characterized by the variation diminishing property together with the sign non-reversal property. We do away with the former, and show that (Formula presented.) is (Formula presented.) if and only if every submatrix formed from at most (Formula presented.) consecutive rows and columns has the sign non-reversal property. In fact, this can be strengthened to only consider test vectors in (Formula presented.) with alternating signs. We also show a similar characterization for all (Formula presented.) matrices � more strongly, both of these characterizations use a single vector (with alternating signs) for each square submatrix. These characterizations are novel, and similar in spirit to the fundamental results characterizing (Formula presented.) matrices by Gantmacher�Krein (Compos. Math. 4 (1937) 445�476) and (Formula presented.) -matrices by Gale�Nikaido (Math. Ann. 159 (1965) 81�93). As an application, we study the interval hull (Formula presented.) of two (Formula presented.) matrices (Formula presented.) and (Formula presented.). This is the collection of (Formula presented.) such that each (Formula presented.) is between (Formula presented.) and (Formula presented.). Using the sign non-reversal property, we identify a two-element subset of (Formula presented.) that detects the (Formula presented.) property for all of (Formula presented.) for arbitrary (Formula presented.). In particular, this provides a test for total positivity (of any order), simultaneously for an entire class of rectangular matrices. In parallel, we also provide a finite set to test the total non-negativity (of any order) of an interval hull (Formula presented.). © 2021 The Authors. The publishing rights in this article are licensed to the London Mathematical Society under an exclusive licence.

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 John Wiley and Sons Ltd.
Keywords: 15A24; 15B48 (primary); 65G30 (secondary)
Department/Centre: Division of Physical & Mathematical Sciences > Mathematics
Date Deposited: 14 Jul 2023 06:17
Last Modified: 14 Jul 2023 06:17
URI: https://eprints.iisc.ac.in/id/eprint/81782

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