I am working with a partitioned matrix $A =\begin{bmatrix}E\\D\end{bmatrix}$. E and D have the same properties. $E\in R^{n\times n}$, $E$ is full column rank and nonsingular. $E = I\cdot f(\bar{x}) + \bar{x}\otimes\frac{\partial}{\partial \bar{x}}f$, $D = I\cdot g(\bar{x}) + \bar{x}\otimes\frac{\partial}{\partial \bar{x}}g(\bar{x})$.
In this case, I have two blocks but in general $A \in R^{n^2\times n}$.
I want to know if there is a way to find $A^+= \begin{bmatrix}U&V\end{bmatrix}$ by using $E^{-1}$ and $D^{-1}$. I have read Cline's and Baksalary's papers and I believe that it can be done. When applying Backsalary's theorems I run into the issue that the orthogonal projector $P = EE^+$ always becomes the identity and thus the other projector $Q$ is always zero.
Thanks to everyone in advance!