A sign pattern (matrix) is a matrix whose entries are from the set {+, −, 0}. The minimum rank of a sign pattern matrix A is the smallest possible rank of a real matrix whose entries have signs indicated by A.
I establish a direct connection between an m × n sign pattern with minimum rank r ≥ 2 and an m point–n hyperplane configuration in Rr−1. I
give a possibly smallest example of a sign pattern (with minimum rank 3) whose minimum rank cannot be realized rationally. For every sign pattern with at most 2 zero
entries in each column, the minimum rank can be realized rationally.
Using a new approach involving sign vectors of subspaces and oriented matroid duality, I show that for every m × n sign pattern with minimum rank ≥ n − 2,
rational realization of the minimum rank is possible. Also, for every integer n ≥ 9, there is a positive integer m, such that there exists an m × n sign pattern
with minimum rank n − 3 for which rational realization is not possible.
I give a characterization of m × n sign patterns A with minimum rank n − 1, along with a more general description of sign patterns with minimum rank r, in
terms of sign vectors of certain subspaces.
I discuss a number of results on the maximum and minimum numbers of sign vectors of k-dimensional subspaces of Rn; this maximum number is equal to the
total number of cells of a generic central hyperplane arrangement in Rk. For example, the maximum number of sign vectors of a 2-dimensional subspace of
Rn is 4n + 1 and the maximum number of sign vectors of a 3-dimensional subspace of Rn is 4n(n − 1) + 3.
Along the way I state related results and open problems.
To the Combinatorics Seminar Web page.