# Blockwise Moore-Penrose pseudoinverse?

There exists a convenient formula for computing the inverse of a block matrix consisting of 4 matrices $\mathbf{A, B, C, D}$

$\begin{bmatrix}\mathbf{A} & \mathbf{B} \\ \mathbf{C} & \mathbf{D}\end{bmatrix} ^{-1}$

the inverse can be written as a function of $A^{-1}$ and $(A-B D^{-1}C)^{-1}$ (wikipedia)

$\begin{bmatrix} \mathbf{A}^{-1}+\mathbf{A}^{-1}\mathbf{B}(\mathbf{D}-\mathbf{CA}^{-1}\mathbf{B})^{-1}\mathbf{CA}^{-1} & -\mathbf{A}^{-1}\mathbf{B}(\mathbf{D}-\mathbf{CA}^{-1}\mathbf{B})^{-1} \\ -(\mathbf{D}-\mathbf{CA}^{-1}\mathbf{B})^{-1}\mathbf{CA}^{-1} & (\mathbf{D}-\mathbf{CA}^{-1}\mathbf{B})^{-1} \end{bmatrix}$

I wonder if a similar formula exists for the pseudo-inverse of non-invertible block matrices.

Using the fabulous search engine netted me quite a lot of references on the pseudoinverses of partitioned matrices. Miao’s paper gives a pretty general formula, Cline’s paper concentrates on column-partitioned matrices $\mathbf A=[\mathbf U\quad\mathbf V]$, Rohde’s paper handles the pseudoinverse of a partitioned Hermitian matrix, Hall and Hartwig’s paper give formulae for column-partitioned matrices and row-partitioned matrices $\mathbf A=\bigl[\begin{smallmatrix}\mathbf U\\\mathbf V\end{smallmatrix}\bigr]$, and this paper by Hartwig gives an expression for the pseudoinverse of a bordered matrix, $\mathbf A=\bigl[\begin{smallmatrix}\mathbf B&\mathbf u\\\mathbf v^T&c\end{smallmatrix}\bigr]$.