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

2008
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Electronic Journal of Combinatorics
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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 $Z_2$,

doi:10.37236/749
fatcat:cybj234mtzazjdp2fnrv5rjpne