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arXiv · 1003.5675

The affine preservers of non-singular matrices

Clément de Seguins Pazzis
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Abstract

When K is an arbitrary field, we study the affine automorphisms of M_n(K) that stabilize GL_n(K). Using a theorem of Dieudonn\'e on maximal affine subspaces of singular matrices, this is easily reduced to the known case of linear preservers when n>2 or #K>2. We include a short new proof of the more general Flanders' theorem for affine subspaces of M_{p,q}(K) with bounded rank. We also find that the group of affine transformations of M_2(F_2) that stabilize GL_2(F_2) does not consist solely of linear maps. Using the theory of quadratic forms over F_2, we construct explicit isomorphisms between it, the symplectic group Sp_4(F_2) and the symmetric group S_6.

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Clément de Seguins Pazzis. 2010-03-29. The affine preservers of non-singular matrices. https://doi.org/10.1007/s00013-010-0173-6

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