Geodesic
Minimum rank of matrices described by a graph or pattern over the rational, real and complex numbers
The electronic journal of combinatorics, Tome 15 (2008)
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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 $Z_2$, 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.
DOI : 10.37236/749
Classification : 05C50, 15A03
Mots-clés : nonzero patterns, minimum rank of matrices 
@article{10_37236_749,
     author = {Avi Berman and Shmuel Friedland and Leslie Hogben and Uriel G. Rothblum and Bryan Shader},
     title = {Minimum rank of matrices described by a graph or pattern over the rational, real and complex numbers},
     journal = {The electronic journal of combinatorics},
     year = {2008},
     volume = {15},
     doi = {10.37236/749},
     zbl = {1179.05070},
     url = {http://geodesic.mathdoc.fr/articles/10.37236/749/}
}
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Avi Berman; Shmuel Friedland; Leslie Hogben; Uriel G. Rothblum; Bryan Shader. Minimum rank of matrices described by a graph or pattern over the rational, real and complex numbers. The electronic journal of combinatorics, Tome 15 (2008). doi: 10.37236/749

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