Group Inverse Of Block Matrix

Let C^m×n be the set of all m×n matrices over the field of complexuumbers. Suppose A∈C^m×n, let A^* be the conjugate transpose of A. Inthis paper, we study the group inverses of partitioned matrices of the formwhereA, Band C are square matrices of order n.In this paper, on one hand, making use of the definition and theexistential condition of the group inverses, we prove the existence of thegroup inverses for the five matrices of complex numbers of the formwhere. A is idempotent matrix; on the other hand, using the operatingformula of the group inverses M^#=M(M³)⁽²⁾M and a series of the ele-mentary transformation of block matrices,we cbtain the representationsof the group inverses for block matrices above.