Subspaces of matrices with special rank properties |
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Authors: | Jean-Guillaume Dumas Gary McGuire |
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Institution: | a Laboratoire Jean Kuntzmann, Université de Grenoble, France b School of Mathematical Sciences, University College Dublin, Ireland |
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Abstract: | Let K be a field and let Mm×n(K) denote the space of m×n matrices over K. We investigate properties of a subspace M of Mm×n(K) of dimension n(m-r+1) in which each non-zero element of M has rank at least r and enumerate the number of elements of a given rank in M when K is finite. We also provide an upper bound for the dimension of a constant rank r subspace of Mm×n(K) when K is finite and give non-trivial examples to show that our bound is optimal in some cases. We include a similar a bound for the maximum dimension of a constant rank subspace of skew-symmetric matrices over a finite field. |
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Keywords: | 15A03 15A33 |
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