The minimum rank problem: A counterexample
| Autor: | K. P. S. Bhaskara Rao, Swastik Kopparty |
|---|---|
| Rok vydání: | 2008 |
| Předmět: |
Numerical Analysis
Rational number Zero nonzero pattern Algebra and Number Theory Conjecture Minimum rank Minimum rank of a graph Graph theory Rational function Combinatorics Matrix (mathematics) Sign pattern matrix ComputingMethodologies_SYMBOLICANDALGEBRAICMANIPULATION Discrete Mathematics and Combinatorics Geometry and Topology Mathematics Counterexample Projective geometry |
| Zdroj: | Linear Algebra and its Applications. 428:1761-1765 |
| ISSN: | 0024-3795 |
| DOI: | 10.1016/j.laa.2007.10.016 |
| Popis: | 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. |
| Databáze: | OpenAIRE |
| Externí odkaz: |
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