The affine preservers of non-singular matrices

TL;DR

This paper characterizes affine automorphisms of matrix spaces that preserve invertibility over arbitrary fields, providing new proofs and explicit examples, especially highlighting differences in small fields like F_2.

Contribution

It extends the classification of affine preservers of invertible matrices to arbitrary fields and includes a new proof of Flanders' theorem for bounded rank affine subspaces.

Findings

Affine automorphisms stabilizing GL_n(K) are characterized for arbitrary fields.

A new proof of Flanders' theorem for affine subspaces with bounded rank is provided.

Explicit isomorphisms are constructed between certain affine transformation groups and classical groups over F_2.

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.