Large affine spaces of non-singular matrices
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Abstract:
Let $\mathbb {K}$ be an arbitrary (commutative) field with at least three elements. It was recently proven that an affine subspace of $\operatorname {M}_n(\mathbb {K})$ consisting only of non-singular matrices must have a dimension less than or equal to $\binom {n}{2}$. Here, we classify, up to equivalence, the subspaces whose dimension equals $\binom {n}{2}$. This is done by classifying, up to similarity, all the $\binom {n}{2}$-dimensional linear subspaces of $\operatorname {M}_n(\mathbb {K})$ consisting of matrices with no non-zero invariant vector, reinforcing a classical theorem of Gerstenhaber. Both classifications only involve the quadratic structure of the field $\mathbb {K}$.References
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Additional Information
- Clément de Seguins Pazzis
- Affiliation: Lycée Privé Sainte-Geneviève, 2, rue de l’École des Postes, 78029 Versailles Cedex, France
- Email: dsp.prof@gmail.com
- Received by editor(s): February 26, 2011
- Received by editor(s) in revised form: June 25, 2011, August 24, 2011, and September 14, 2011
- Published electronically: December 12, 2012
- © Copyright 2012
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Trans. Amer. Math. Soc. 365 (2013), 2569-2596
- MSC (2010): Primary 15A03, 15A30
- DOI: https://doi.org/10.1090/S0002-9947-2012-05705-9
- MathSciNet review: 3020109