### 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) |
---|---|

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 |

### Fingerprint

### All Science Journal Classification (ASJC) codes

- Algebra and Number Theory
- Numerical Analysis

### Keywords

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

### Cite this

*Linear Algebra and Its Applications*,

*428*(7), 1761-1765. https://doi.org/10.1016/j.laa.2007.10.016

}

*Linear Algebra and Its Applications*, vol. 428, no. 7, pp. 1761-1765. https://doi.org/10.1016/j.laa.2007.10.016

**The minimum rank problem : A counterexample.** / Kopparty, Swastik; Bhaskara Rao, K. P S.

Research output: Contribution to journal › Article

TY - JOUR

T1 - The minimum rank problem

T2 - A counterexample

AU - Kopparty, Swastik

AU - Bhaskara Rao, K. P S

PY - 2008/4/1

Y1 - 2008/4/1

N2 - 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.

AB - 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.

KW - Minimum rank

KW - Minimum rank of a graph

KW - Sign pattern matrix

KW - Zero nonzero pattern

UR - http://www.scopus.com/inward/record.url?scp=38849117737&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=38849117737&partnerID=8YFLogxK

U2 - 10.1016/j.laa.2007.10.016

DO - 10.1016/j.laa.2007.10.016

M3 - Article

AN - SCOPUS:38849117737

VL - 428

SP - 1761

EP - 1765

JO - Linear Algebra and Its Applications

JF - Linear Algebra and Its Applications

SN - 0024-3795

IS - 7

ER -