Matrices with Sign Consistency of a Given Order

In this paper, the matrices whose minors of a given order have the same sign are characterized in several ways. In particular, given an $n\times d$ matrix $A$ (with $n>d$), a criterion involving $(n-d)d+1$ minors to determine if all $d\times d$ minors of $A$ have the same strict sign is obtained. A test of $O(m^3)$ elementary operations (with $m=\max \{n-d,d\}$) to check if a given matrix satisfies these properties is also provided. Finally, these results are applied to improve the characterizations of alternating polytopes.