Commutators from a hyperplane of matrices

TL;DR

This paper extends a classical result by showing that for matrices over a field with more than 3 elements, any trace-zero matrix can be expressed as a commutator with both matrices lying in any chosen hyperplane of the matrix algebra, given that n>2.

Contribution

It proves that for n>2 and fields with more than 3 elements, matrices A and B can be chosen in any specified hyperplane to form a commutator equal to a given trace-zero matrix.

Findings

Any trace-zero matrix can be expressed as a commutator within any hyperplane.

The result holds for matrices over fields with more than 3 elements.

The theorem generalizes the classical Albert-Muckenhoupt result.

Abstract

Denote by $M_n(K)$ the algebra of $n$ by $n$ matrices with entries in the field $K$. A theorem of Albert and Muckenhoupt states that every trace zero matrix of $M_n(K)$ can be expressed as $AB-BA$ for some pair $(A,B)$ of matrices of $M_n(K)$. Assuming that $n>2$ and that $K$ has more than 3 elements, we prove that the matrices $A$ and $B$ can be required to belong to an arbitrary given hyperplane of $M_n(K)$.