The singular linear preservers of non-singular matrices

TL;DR

This paper characterizes singular linear maps on matrix spaces over a field that preserve invertible matrices, linking the problem to classifying division algebras, and provides a new proof for the classical case.

Contribution

It reduces the problem of classifying singular endomorphisms stabilizing invertible matrices to the classification of division algebras over the field.

Findings

Reduction of the problem to division algebra classification

New proof for the classical non-singular case

Application of Dieudonné's theorem to singular subspaces

Abstract

Given an arbitrary field K, we reduce the determination of the singular endomorphisms $f$ of M_n(K) that stabilize GL_n(K) to the classification of n-dimensional division algebras over K. Our method, which is based upon Dieudonn\'e's theorem on singular subspaces of M_n(K), also yields a proof for the classical non-singular case.