Colloquium: Matroids and the minimum rank of zero-nonzero matrix patterns
Matroids and the minimum rank of zero-nonzero matrix patterns
Louis Deaett, Ph.D., Assistant Professor of Mathematics, Quinnipiac University
A problem of interest in combinatorial matrix theory is the following. Suppose we know of a matrix only which entries are zero and which are nonzero. What can we say about the rank of the matrix based only on that information? The goal of this work is to generalize this problem to the setting of matroids. We investigate the extent to which combinatorial bounds in the matrix setting extend to the generalized setting of matroids. We also use insights from the matroid setting to shed light on the original matrix-theoretic problem. In some ways, this program parallels work on the minimum rank of matrix sign patterns (where the information we have is the sign of every entry) that draws on results from oriented matroid theory. In particular, results on the non-representability of certain low rank matroids allow us to construct zero-nonzero patterns for which the minimum rank depends on the field from which the matrix entries are chosen.