Decomposition of Singular Matrices into Idempotents

TL;DR

This paper develops methods to decompose singular matrices into products of idempotents over various rings, extending previous work to noncommutative cases and Bézout domains, with concrete constructions and proofs.

Contribution

It generalizes existing results on matrix factorizations into idempotents to noncommutative rings and Bézout domains, providing explicit constructions and filling gaps in prior proofs.

Findings

Concrete constructions of idempotents for singular matrices.

Extension of previous results to noncommutative rings.

Analysis of singular matrices over Bézout domains.

Abstract

In this paper we provide concrete constructions of idempotents to represent typical singular matrices over a given ring as a product of idempotents and apply these factorizations for proving our main results. We generalize works due to Laffey (Products of idempotent matrices. Linear Multilinear A. 1983) and Rao (Products of idempotent matrices. Linear Algebra Appl. 2009) to noncommutative setting and fill in the gaps in the original proof of Rao's main theorems. We also consider singular matrices over B\'ezout domains as to when such a matrix is a product of idempotent matrices.