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Zhongshan Li, Georgia State University, Atlanta, GA
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| Title: Sign vectors of subspaces of R^n and minimum ranks of sign patterns |
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Abstract: 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. A direct connection between an m by n sign pattern with minimum rank r>=2 and an m point–n hyperplane configuration in R^{r-1} is established. A possibly smallest example of a sign pattern (with minimum rank 3) whose minimum rank cannot be realized rationally is given. It is shown that 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, it is shown that for every m by n sign pattern with minimum rank >= n-2, rational realization of the minimum rank is possible. It is also shown that for every integer n>=9, there is a positive integer m, such that there exists an m by n sign pattern with minimum rank n-3 for which rational realization is not possible. A characterization of m by n sign patterns A with minimum rank n-1 is given, along with a more general description of sign patterns with minimum rank r, in terms of sign vectors of certain subspaces. A number of results on the maximum and minimum numbers of sign vectors of k-dimensional subspaces of R^n are discussed; this maximum number is equal to the total number of cells of a generic central hyperplane arrangement in R^k. In particular, it is shown that the maximum number of sign vectors of a 2-dimensional subspace of R^n is 4n+1 and the maximum number of sign vectors of a 3-dimensional subspace of R^n is 4n(n – 1) + 3. Related results and open problems are stated along the way.
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