Matroids and the Minimum Rank Problem for Matrix Patterns

Abstract

Suppose the only thing we know about a matrix is which of its entries are zero. What can we say about its rank? We develop a framework for applying matroid theory to this question. One consequence is a generalization of the question to the setting of matroids; over an infinite field, we recover the original question by considering only matroids representable over that field. This framework also lets us revisit prior work on the problem for matrices and clarify how previous results rest on facts about matroids. In the process, we unify and simplify the proofs of these results, and sometimes improve on them a bit. We also derive a few new results, and consider further directions for exploration.