On decomposing any matrix as a linear combination of three idempotents

TL;DR

This paper proves that any square matrix over an arbitrary field can be expressed as a linear combination of three idempotent matrices, extending previous results limited to characteristic zero fields.

Contribution

It generalizes the decomposition of matrices into three idempotents to any field, building on prior characterizations of two-idempotent decompositions.

Findings

Every square matrix is a linear combination of three idempotents over any field.

Improves upon Rabanovich's theorem by removing the characteristic zero restriction.

Provides a full characterization of matrices decomposable into two idempotents with given coefficients.

Abstract

In a recent article, we gave a full characterization of matrices that can be decomposed as a linear combination of two idempotents with prescribed coefficients. In this one, we use those results to improve on a recent theorem of V. Rabanovich: we establish that every square matrix is a linear combination of three idempotents (for an arbitrary coefficient field rather than just a field of characteristic 0).