The minimum rank problem: A counterexample |
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Authors: | Swastik Kopparty |
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Institution: | a Computer Science and Artificial Intelligence Laboratory, MIT, Cambridge, MA, USA b Department of Mathematics and Computer Science, Indiana State University, Terre Haute, IN 47809, USA |
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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. |
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Keywords: | 15A03 15A33 05C50 |
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