On Linear spaces of of matrices bounded rank

TL;DR

This paper classifies all basic linear spaces of matrices with rank at most four, introducing two new non-classical examples and bridging algebraic geometry and linear algebra methods.

Contribution

It provides the first known non-classical examples of bounded rank four spaces and completes their classification up to isomorphism.

Findings

Two non-classical examples of bounded rank four spaces are constructed.

A full classification of basic spaces of bounded rank four is achieved.

Methods from algebraic geometry and linear algebra are integrated for the classification.

Abstract

Motivated by questions in theoretical computer science and quantum information theory, we study the classical problem of determining linear spaces of matrices of bounded rank. Spaces of bounded rank three were classified in 1983, and it has been a longstanding problem to classify spaces of bounded rank four. Before our study, no non-classical example of such a space was known. We exhibit two non-classical examples of such spaces and give the full classification of basic spaces of bounded rank four. There are exactly four such up to isomorphism. We also take steps to bring together the methods of the linear algebra community and the algebraic geometry community used to study spaces of bounded rank.