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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Large affine spaces of non-singular matrices

Author: Clément de Seguins Pazzis
Journal: Trans. Amer. Math. Soc. 365 (2013), 2569-2596
MSC (2010): Primary 15A03, 15A30
Published electronically: December 12, 2012
MathSciNet review: 3020109
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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}$.

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Clément de Seguins Pazzis
Affiliation: Lycée Privé Sainte-Geneviève, 2, rue de l’École des Postes, 78029 Versailles Cedex, France

Keywords: Affine subspaces, non-zero eigenvalues, alternate matrices, simultaneous triangularization, non-isotropic quadratic forms, Gerstenhaber theorem
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
Article copyright: © Copyright 2012 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.