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

Abstract

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