How would Moore-Penrose pseudoinverse look like for a simple general example of a block matrix?
$$ A_1=\begin{pmatrix} X & 0 \\ 0 & 0 \\ \end{pmatrix} $$ $$ A_2=\begin{pmatrix} 0 & X \\ 0 & 0 \\ \end{pmatrix} $$
Where X is a block matrix.
I tried some online calculator and got:
$$ A^+_1=\begin{pmatrix} X^+ & 0 \\ 0 & 0 \\ \end{pmatrix} $$ $$ A^+_2=\begin{pmatrix} 0 & 0 \\ X^+ & 0 \\ \end{pmatrix} $$
Where $A_i^+$ and $X^+$ denote the pseudoinverses.
But why there is the flip for $A_2$? Because of the transposition?
Is there any general formula how to deal with the block matrices like this?
On Wikipedia I just found the block matrices written in the form of $A = (X | Y)$ or the general expression for the inverse of the block matrices but I wasnt able to find out why that flip actually happened in these formulae.
And what if $X$ isn't regular or square?