We establish a novel characterization of totally positive matrices via a sign non-reversal property. In this we are inspired by the analogous results for ]  P-matrices (matrices with all principal minors positive) . Next, the interval hull ] of two m \times n matrices A=(a_{ij}) and B = (b_{ij}) , denoted by \mathbb{I is the collection of all matrices C such that c_{ij=t_{ij}a_{ij} +(1-t_{ij})b_{ij}for all i,j, where t_{ij} \in }%DIFDELCMD < [%%% 0,1^{m \times n} such that each c_{ij} is between a_{ij} and b_{ij}} . Using the sign non-reversal property, we identify a finite subset of \mathbb{I}(A,B) that detects the total positivity of all of \mathbb{I}(A,B) . This provides a test for  an entire class of matrices simultaneously to be totally positive. We also establish analogous results for other classes of matrices: almost P-matrices, N-matrices and (minimally) semipositive matrices.
A matrix A is totally positive (or non-negative) of order k, denoted TP_k (or TN_k), if all minors of size \leq 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 TP_k if and only if every contiguous submatrix of size \leq k has the sign non-reversal property. In fact this can be strengthened to only consider test vectors in \mathbb{R Compos. Math. 1937] and P-matrices by Gale-Nikaido Math. Ann. 1965]. As an application, we study the interval hull \mathbb{I of two m \times n matrices A=(a_{ij}) and B = (b_{ij}) . This is the collection of =t_{ij}a_{ij} +(1-t_{ij})b_{ij}for all i,j, where t_{ij} \in }%DIFDELCMD < [%%% C \in \mathbb{R^{m \times n} such that each c_{ij} is between a_{ij} and b_{ij}} . Using the sign non-reversal property, we identify a two-element subset of \mathbb{I}(A,B) that detects the TP_k property for all of \mathbb{I}(A,B) for arbitrary k \geq 1. In particular, this provides a test for total positivity (of any order), simultaneously for an entire class of rectangular matrices.
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A matrix A is totally positive (or non-negative) of order k, denoted TP_k (or TN_k), if all minors of size \leq k are positive (or non-negative). It is well-known that such matrices are characterized by the variation diminishing property together with the
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In this we are inspired by the analogous results for
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matrices simultaneously to be totally positive. We also establish analogous results for other classes of matrices: almost P-matrices, N-matrices and (minimally) semipositive
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