Generalized inverses − idempotents and projectors

Abstract

In this paper, we present necessary and sufficient conditions for $\widetilde{X}$ to be idempotent and orthogonal idempotent, where $\widetilde{X}\in\mathbb \{A^{\textcircled{\dag}},A^{D},A^{D,\dag}, A^{\dag,D},A^{\textcircled{w}}\}$. Several characteristics that $\widetilde{X}$ is idempotent and orthogonal idempotent are derived by core-EP decomposition. Additionally, we give some equivalent conditions for a matrix $A$ which is orthogonal idempotent by making use of the properties of some generalized inverses of the matrix $A$.