Decomposition of Singular Matrices into Idempotents

Adel Alahmadi (1) , Surender Jain (2) , André Leroy (3)

1 department of Mathematics
2 Department of Mathematics, Ohio University, Athens
3 LML - Laboratoire de Mathématiques de Lens

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 Bézout domains as to when such a matrix is a product of idempotent matrices.

Mots clés: Idempotent; Singular Matrix; Local Ring; Principal Right; Ideal Domain; B\'{e}zout Domain; Injective Module; Regular Ring; Stable Range 1

Domaines: Anneaux et algèbres [math.RA]

Adel Alahmadi, Surender Jain, André Leroy. Decomposition of Singular Matrices into Idempotents. Linear and Multilinear Algebra, 2013, Decomposition of singular matrices. ⟨10.1080/03081087.2012.754439⟩. ⟨hal-00784158⟩