Projections and Idempotents in star-reducing Rings Involving the Moore-Penrose Inverse

TL;DR

This paper generalizes previous characterizations of certain idempotent matrices involving the Moore-Penrose inverse from complex matrices to *-reducing rings, expanding the theoretical framework for projections and inverses.

Contribution

It extends known results about idempotent matrices and Moore-Penrose inverses from complex matrices to the broader setting of *-reducing rings, providing a more general algebraic context.

Findings

Generalization of matrix results to *-reducing rings

Characterization of projections in *-reducing rings

Broader applicability of Moore-Penrose inverse properties

Abstract

In [Comput. Math. Appl. 59 (2010) 764-778], Baksalary and Trenkler characterized some complex idempotent matrices of the form $(I-PQ)^{\dag}P(I-Q)$, $(I-QP)^{\dag}(I-Q)$ and $P(P+Q-QP)^{\dag}$ in terms of the column spaces and null spaces of $P$ and $Q$, where $P, Q\in \mathbb{C}^{\textsf{OP}}_n = \{L\in \mathbb{C}_{n,n}\ |\ L^2 = L = L^*\}$. We generalize these results from $\mathbb{C}_{n,n}$ to any *-reducing rings.