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

A matrix $A$ is totally positive (or non-negative) of order $k$, denoted $TP_k$ $TN_k$), if all minors size $\leq k$ are non-negative). It well-known that such matrices characterized by the variation diminishing property together with sign non-reversal property. We do away former, and show only every submatrix formed from at most $k$ consecutive rows columns has In fact this can be strengthened to consider test vectors in $\mathbb{R}^k$ alternating signs. also a similar characterization for $TN_k$ - more strongly, both these characterizations use single vector (with signs) each square submatrix. These novel, spirit fundamental results characterizing $TP$ Gantmacher-Krein [Compos. Math. 1937] $P$-matrices Gale-Nikaido [Math. Ann. 1965]. As an application, we study interval hull $\mathbb{I}(A,B)$ two $m \times n$ $A=(a_{ij})$ $B = (b_{ij})$. This collection $C \in \mathbb{R}^{m n}$ $c_{ij}$ between $a_{ij}$ $b_{ij}$. Using property, identify two-element subset detects arbitrary $k \geq 1$. particular, provides total positivity (of any order), simultaneously entire class rectangular matrices. parallel, provide finite set non-negativity order) $\mathbb{I}(A,B)$.