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 R^k with alternating signs. We also show a similar characterization for all TN_k matrices - more strongly, using at most one 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. 1937] and P-matrices by Gale-Nikaido [Math. Ann. 1965]. As an application, we study the interval hull I(A,B) of two m \times n matrices A=(a_{ij}) and B = (b_{ij}). This is the collection of C \in 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 I(A,B) that detects the TP_k property for all of 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  .
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 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 R^k with alternating signs. We also show a similar characterization for all TN_k 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. 1937] and P-matrices by Gale-Nikaido [Math. Ann. 1965]. As an application, we study the interval hull I(A,B) of two m \times n matrices A=(a_{ij}) and B = (b_{ij}). This is the collection of C \in 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 I(A,B) that detects the TP_k property for all of 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 . In parallel, we also provide a finite set to test the total non-negativity (of any order) of an interval hull \mathbb{I .
R
contiguous submatrix of size \leq k
submatrix formed from at most k consecutive rows and columns
R
using at most one
both of these characterizations use a single
A
. In parallel, we also provide a finite set to test the total non-negativity (of any order) of an interval hull \mathbb{I