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...
