Result: The weighted generalized inverses of a partitioned matrix

Given a matrix \(A\in \mathbb{C}^{m \times n}\), its weighted Moore-Penrose inverse \(X\) (denoted by \(A_{M,N}^{\dagger }\)) satisfies the following four conditions: (1) \(AXA=A\); (2) \(XAX=X\); (3) \((MAX)^{\ast }=MAX\); (4) \((NXA)^{\ast }=NXA\) where \(M,N\) are \(m \times m\) and \(n \times n\) positive definite matrices, respectively, and where \( ()^{\ast }\) denotes the conjugate transpose of a matrix. If it only satisfies the first and third condition it is denoted as \(A^{(1,3M)}\), and is referred as the \(\{1,3M\}\)-inverse of \(A\), and if it only satisfies the first and fourth condition it is denoted as \(A^{(1,4N)}\), and is referred as the \(\{1,4N\}\)-inverse of \(A\). The authors give a constructive proof by minimizing norm of the obtained following result by \textit{J. Miao} [J. Comput. Math. 7, No. 4, 321--323 (1989; Zbl 0711.15003)] of a recursive scheme for the computation of the weighted Moore-Penrose inverse \(A_{M,N}^{\dagger }\) of the matrix \( A=(U,V)\) when an additional block \(V\) of columns is added to \(U\). Let \(A=(U,V)\in \mathbb{C}^{m \times (r+p)}\) whose last \(p\) columns are denoted by \(V\) and \(N=\left( \begin{smallmatrix} N_{r} & L \\ L^{\ast } & N_{p} \end{smallmatrix} \right) \) a \((r+p) \times (r+p)\) positive definite matrix, then \( A_{M,N}^{\dagger }=\left( \begin{smallmatrix} U_{M,N_{r}}^{\dagger }(I-VH)-(I-U_{M,N_{r}}^{\dagger }U)N_{r}^{-1}LH \\ H \end{smallmatrix} \right) \) where \(C=(I-UU_{M,N_{r}}^{\dagger })V\), \(D=U_{M,N_{r}}^{\dagger }V\) , \(H=C_{M,K_{1}}^{\dagger }+(I-C_{M,N_{r}}^{\dagger }C)K_{1}^{-1}(D^{\ast }N_{r}-L^{\ast })U_{M,N_{r}}^{\dagger }\) and \(K_{1}=N_{p}+D^{\ast }N_{r}D-(D^{\ast }L+L^{\ast }D)-L^{\ast }(I-U_{M,N_{r}}^{\dagger }U)N_{r}^{-1}L\). Similar results are obtained for \(A^{(1,3M)}\) and \(A^{(1,4N)}\), using the same kind of reasoning. Moreover, all above weighted generalized inverses are expressed by a unified formula.