On invertibility and group inverse of combinations of two orthogonal projectors about a complex square matrix

Abstract

For any complex square matrix A, this paper characterizes the invertibility and group inverse of the combinations P = a1 PR(A) + a2 PR(A∗) +a3 PR(A) PR(A∗) +a4 PR(A∗) PR(A) by M-C-S decomposition of A. Necessary and sufficient conditions of the invertibility and its inverse are presented completely. Also, we characterize the group inverse and give an expression for P^# when P is group invertible.