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

摘要

Let $M= left[ egin{array}{cc} A& B C& O end{array} ight]$ be a complex square matrix where A is square. When BCB^{Omega} =0, rank(BC) = rank(B) and the group inverse of $left[ egin{array}{cc} B^{Omega} A B^{Omega} & 0 CB^{Omega} & 0 ight]$ exists, the group inverse of M exists if and only if rank(BC + A)B^{Omega}AB^{Omega})^{pi}B^{Omega}A)= 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 S^2GI, if the group inverse of each matrix ar{A} in Q(A) exists and its sign pattern is independent of e A. By using the group inverse representation, a necessary and sufficient condition for a real block matrix to be an S^2GI-matrix is given.