The Minimum Rank Problem: a counterexample

TL;DR

This paper presents a counterexample disproving a conjecture about the minimum rank of sign pattern matrices, revealing differences between real and rational realizations and discussing implications for various subfields of real numbers.

Contribution

The paper provides the first known counterexample to the conjecture that minimum rank can always be realized rationally, highlighting differences between real and rational minimum ranks.

Findings

Counterexample disproves the conjecture

Minimum rank over reals can be smaller than over rationals

Implications for sign pattern matrices over different subfields

Abstract

We provide a counterexample to a recent conjecture that the minimum rank of every sign pattern matrix can be realized by a rational matrix. We use one of the equivalences of the conjecture and some results from projective geometry. As a consequence of the counterexample, we show that there is a graph for which the minimum rank over the reals is strictly smaller than the minimum rank over the rationals. We also make some comments on the minimum rank of sign pattern matrices over different subfields of $\mathbb R$.