### Abstract

We provide a counterexample to a recent conjecture that the minimum rank over the reals 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 of the graph over the reals is strictly smaller than the minimum rank of the graph over the rationals. We also make some comments on the minimum rank of sign pattern matrices over different subfields of R.

Original language | English (US) |
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Pages (from-to) | 1761-1765 |

Number of pages | 5 |

Journal | Linear Algebra and Its Applications |

Volume | 428 |

Issue number | 7 |

DOIs | |

State | Published - Apr 1 2008 |

Externally published | Yes |

### All Science Journal Classification (ASJC) codes

- Algebra and Number Theory
- Numerical Analysis
- Geometry and Topology
- Discrete Mathematics and Combinatorics

### Keywords

- Minimum rank
- Minimum rank of a graph
- Sign pattern matrix
- Zero nonzero pattern

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## Cite this

Kopparty, S., & Bhaskara Rao, K. P. S. (2008). The minimum rank problem: A counterexample.

*Linear Algebra and Its Applications*,*428*(7), 1761-1765. https://doi.org/10.1016/j.laa.2007.10.016