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

Sign non‐reversal property for totally non‐negative and totally positive matrices, and testing... A matrix A is totally positive (or non‐negative) of order k, denoted TPk (or TNk), if all minors of size ⩽k 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 A is TPk if and only if every submatrix formed from at most k consecutive rows and columns has the sign non‐reversal property. In fact, this can be strengthened to only consider test vectors in Rk with alternating signs. We also show a similar characterization for all TNk 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 TP matrices by Gantmacher–Krein (Compos. Math. 4 (1937) 445–476) and P‐matrices by Gale–Nikaido (Math. Ann. 159 (1965) 81–93). http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Bulletin of the London Mathematical Society Wiley

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

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References (32)

Publisher
Wiley
Copyright
© 2021 London Mathematical Society
ISSN
0024-6093
eISSN
1469-2120
DOI
10.1112/blms.12475
Publisher site
See Article on Publisher Site

Abstract

A matrix A is totally positive (or non‐negative) of order k, denoted TPk (or TNk), if all minors of size ⩽k 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 A is TPk if and only if every submatrix formed from at most k consecutive rows and columns has the sign non‐reversal property. In fact, this can be strengthened to only consider test vectors in Rk with alternating signs. We also show a similar characterization for all TNk 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 TP matrices by Gantmacher–Krein (Compos. Math. 4 (1937) 445–476) and P‐matrices by Gale–Nikaido (Math. Ann. 159 (1965) 81–93).

Journal

Bulletin of the London Mathematical SocietyWiley

Published: Aug 1, 2021

Keywords: ; ;

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