The affine preservers of non-singular matrices

Details The affine preservers of non-singular matrices The affine preservers of non-singular matrices

2010-03-29

arxiv.org/abs/1003.5675v2

Arch. Math. 95 (2010), 333-342

Mathematics

Rings and Algebras

13 pages, very minor corrections from the first version

Scientific paper

10.1007/s00013-010-0173-6

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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