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 R^r−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.