Matrices with Sign Consistency of a Given Order
Juan Manuel Peña · SIAM Journal on Matrix Analysis and Applications · 1995
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.