The linear preservers of non-singularity in a large space of matrices

Abstract

Let K be an arbitrary (commutative) field, and V be a linear subspace of M_n(K) such that codim V<n-1. Using a recent generalization of a theorem of Atkinson and Lloyd, we show that every linear embedding of V into M_n(K) which strongly preserves non-singularity must be M->PMQ or M->PM^TQ for some pair (P,Q) of non-singular matrices of M_n(K), unless n=3, codim V=1 and K is isomorphic to F_2. This generalizes a classical theorem of Dieudonn\'e with a similar strategy of proof. Weak linear preservers are also discussed, as well as the exceptional case of a hyperplane of M_3(F_2).