Partial matrices whose completions have ranks bounded below |
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Authors: | James McTigue |
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Affiliation: | School of Mathematics, Statistics and Applied Mathematics, National University of Ireland, Galway, Ireland |
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Abstract: | A partial matrix over a field F is a matrix whose entries are either elements of F or independent indeterminates. A completion of such a partial matrix is obtained by specifying values from F for the indeterminates. We determine the maximum possible number of indeterminates in a partial m×n matrix whose completions all have rank at least equal to a particular k, and we fully describe those examples in which this maximum is attained. Our main theoretical tool, which is developed in Section 2, is a duality relationship between affine spaces of matrices in which ranks are bounded below and affine spaces of matrices in which the (left or right) nullspaces of elements possess a certain covering property. |
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Keywords: | 15A03 15A83 |
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