Minimum Rank of Matrices Described by a Graph or Pattern over the Rational, Real and Complex Numbers

  • Avi Berman
  • Shmuel Friedland
  • Leslie Hogben
  • Uriel G. Rothblum
  • Bryan Shader

Abstract

We use a technique based on matroids to construct two nonzero patterns $Z_1$ and $Z_2$ such that the minimum rank of matrices described by $Z_1$ is less over the complex numbers than over the real numbers, and the minimum rank of matrices described by $Z_2$ is less over the real numbers than over the rational numbers. The latter example provides a counterexample to a conjecture by Arav, Hall, Koyucu, Li and Rao about rational realization of minimum rank of sign patterns. Using $Z_1$ and $Z_2$, we construct symmetric patterns, equivalent to graphs $G_1$ and $G_2$, with the analogous minimum rank properties. We also discuss issues of computational complexity related to minimum rank.

Published
2008-02-04
Article Number
R25