# Pseudoinverse of block diagonal matrix

Suppose I have some block diagonal matrix $$A$$, defined as:

$$A = \begin{bmatrix} A_1 & 0 & ... & 0 \\ 0 & A_2 & ... & 0 \\ \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & ... & A_n \end{bmatrix}$$

Where $$\{A_i\}_{i \in \{1,...n\}}$$ are the blocks of $$A$$. Is it true that the Pseudo-inverse of $$A$$, $$A^+$$, is given by:

$$A^+ = \begin{bmatrix} A_1^+ & 0 & ... & 0 \\ 0 & A_2^+ & ... & 0 \\ \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & ... & A_n^+ \end{bmatrix}$$

If so, why?/why not?