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

TL;DR

This paper characterizes linear maps that strongly preserve non-singularity in large subspaces of matrices over any field, extending classical results and identifying exceptional cases.

Contribution

It generalizes Dieudonné's theorem to broader matrix subspaces and describes the form of linear embeddings that preserve non-singularity.

Findings

Linear embeddings must be of the form M->PMQ or M->PM^TQ.

The result holds for subspaces with codimension less than n-1.

An exceptional case occurs when n=3, codim V=1, and K is F_2.

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).