Generalized Inverses of a Partitioned Matrix

Let A{i,j,…,k} be the set of matrices X=A~(i,j,…k) which satisfy equations, , …, from among <1> AXA=A, <2> XAX=X,<3> (AX)~*=AX, <4>(XA)~*=XA.G~* denotes the conjugate transpose of G, A~+=A~(,2,3,4).(A) and (A~*) denote the range of A and the null space of A~*, respectively. In this paper, the formula for computing inverse A(A_0B_1)]~+ is presented, i.e.,