Pseudoinverse of the pseudoinverse of a matrix

Question

If $ A^+ $ is the psuedoinverse of matrix $ A $, then is it always true that $ (A^+)^+ = A$?

Answer 1

Yes, https://en.wikipedia.org/wiki/Moore%E2%80%93Penrose_inverse#Basic_properties

We have that $(A^+)^+ = A$.

To see this, let $B = A^+$ and recall that the pseudoinverse is unique.

Then, for the four properties which define pseudoinverses, we have

$$A B A = A A^+ A = A\\ B A B = A^+ A A^+ = A^+ = B\\ (BA)^* = (A^+A)^* =A^+A = BA\\ (AB)^* = (AA^+)^* = AA^+ = AB$$

Thus $A$ is a pseudoinverse of $B$.