In this paper, we present necessary and sufficient conditions for X to be idempotent and orthogonal idempotent, where X ∈ {A † , A D , A D, † , A †,D , A w }. Several characteristics when X is idempotent and orthogonal idempotent are derived by core-EP decomposition. Additionally, we give some equivalent conditions when matrix A is orthogonal idempotent, using the properties of some generalized inverses of A