A connection between Schur and Dieudonn\'e's theorems on spaces of bounded rank matrices

TL;DR

This paper presents a novel proof of Dieudonne9's theorem on singular matrices by employing a double-duality approach, linking it to the structure of low-rank operator spaces, especially over algebraically closed fields.

Contribution

It introduces a new proof technique connecting Schur and Dieudonne9's theorems, emphasizing the role of double-duality and low-rank operator spaces.

Findings

New proof of Dieudonne9's theorem using double-duality.

Connection established between Schur and Dieudonne9's theorems.

Method works best over algebraically closed fields.

Abstract

We use a double-duality argument to give a new proof of Dieudonn\'e's theorem on spaces of singular matrices. The argument connects the situation to the structure of spaces of operators with rank at most $1$, and works best over algebraically closed fields.