Before revision: We establish a novel characterization of totally positive matrices via a sign non-reversal property.
After revision: 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.