Sign patterns with minimum rank 3 and point-line configurations

الأصل · EN

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. In this paper, a new direct connection between condensed m × n sign patterns with minimum rank r and m point--n hyperplane configurations in Rʳ⁻¹ is established. In particular, condensed sign patterns with minimum rank 3 are closed related to point--line configurations on the plane. It is proved that for any sign 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. Furthermore, we construct the smallest known sign pattern whose minimum rank is 3 but whose rational minimum rank is greater than 3.