Minimum ranks of sign patterns and zero-nonzero patterns and point-hyperplane configurations

A sign pattern (matrix) is a matrix whose entries are from the set {+,−,0}. The minimum rank (respectively, rational minimum rank) of a sign pattern matrix A is the minimum of the ranks of the real (respectively, rational) matrices whose entries have signs equal to the corresponding entries of A. A sign pattern A is said to be condensed if A has no zero row or column and no two rows or columns are identical or negatives of each other. A zero-nonzero pattern (matrix) is a matrix whose entries are from the set {0,⋆}, where ⋆ indicates a nonzero entry. Many of the sign pattern notions carry over to zero-nonzero patterns, assuming that the ground field is R. In this paper, a direct connection between condensed m×n sign patterns and zero-nonzero patterns with minimum rank r and m point-n hyperplane configurations in Rr−1 is established. In particular, condensed sign patterns (and zero-nonzero patterns) with minimum rank 3 are closely related to point-line configurations on the plane. Using this connection, we construct the smallest known sign pattern whose minimum rank is 3 but whose rational minimum rank is greater than 3. It is proved that for any sign pattern or zero-nonzero pattern A, if the number of zero entries on each column of A is at most 2, then the rational and real minimum ranks of A are equal. Further, it is shown that for any zero-nonzero pattern A with minimum rank r≥3, if the number of zero entries on each column of A is at most r−1, then the rational minimum rank of A is also r. A few related conjectures and open problems are raised.