Representations and sign pattern of the group inverse for some block matrices

Let M= ( A B C 0 ) be a complex square matrix where A is square. When BCB = 0, rank(BC) = rank(B) and the group inverse of ( BAB 0 CB 0 ) exists, the group inverse of M exists if and only if rank(BC + A ( BAB ) π BA) = rank(B). In this case, a representation of M in terms of the group inverse and Moore-Penrose inverse of its subblocks is given. Let A be a real matrix. The sign pattern of A is a (0,+,−)-matrix obtained from A by replacing each entry by its sign. The qualitative class of A is the set of the matrices with the same sign pattern as A, denoted by Q(A). The matrix A is called SGI, if the group inverse of each matrix à ∈ Q(A) exists and its sign pattern is independent of Ã. By using the group inverse representation, a necessary and sufficient condition for a real block matrix   A ∆1 Y1 ∆2 0 0 Y2 0 0   to be an SGI-matrix is given, where A is square, ∆1 and ∆2 are invertible, Y1 and Y2 are sign orthogonal.