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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).
Bulletin of the London Mathematical Society – Wiley
Published: Aug 1, 2021
Keywords: ; ;
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