On sums of idempotent matrices over a field of positive characteristic

TL;DR

This paper investigates the decomposition of matrices into sums of idempotents over fields with positive characteristic, establishing bounds on the number of idempotents needed based on the field's properties.

Contribution

It proves that sufficiently large matrices over such fields can be expressed as sums of five idempotents, reducing to four in prime characteristic fields.

Findings

Any large enough matrix over a field of positive characteristic is a sum of five idempotents.

Over prime characteristic fields, matrices can be expressed as a sum of four idempotents.

The results specify bounds depending on the field's characteristic.

Abstract

We study which matrices are sums of idempotents over a field of non-zero characteristic; in particular, we prove that any such matrix, provided it is large enough, is actually a sum of five idempotents, and even of four when the field is a prime one.