A note on sums of three square-zero matrices

TL;DR

This paper proves that over any field, trace-zero matrices can be extended to sums of three square-zero matrices, with a special case where all such matrices over fields of characteristic 2 are sums of three square-zero matrices.

Contribution

It establishes that every trace-zero matrix can be extended by a zero block to be expressed as a sum of three square-zero matrices, generalizing previous results.

Findings

Over any field, trace-zero matrices can be extended to sums of three square-zero matrices.

If the field has characteristic 2, every trace-zero matrix is itself a sum of three square-zero matrices.

The paper discusses a related result for sums of idempotent matrices.

Abstract

It is known that every complex trace-zero matrix is the sum of four square-zero matrices, but not necessarily of three such matrices. In this note, we prove that for every trace-zero matrix $A$ over an arbitrary field, there is a non-negative integer $p$ such that the extended matrix $A \oplus 0_p$ is the sum of three square-zero matrices (more precisely, one can simply take $p$ as the number of rows of $A$). Moreover, we demonstrate that if the underlying field has characteristic $2$ then every trace-zero matrix is the sum of three square-zero matrices. We also discuss a counterpart of the latter result for sums of idempotents.