# Pseudoinverse of $I-MM^+$

Are there any identities for the Moore-Penrose pseudoinverse that would let me simplify and analize the expression $(I-MM^+)^+$ where $I$ is the identity matrix and $M$ is a rectangular matrix of rank $r$? ($M^+$ denotes the Moore-Penrose pseudoinverse of $M$)

Ultimately I want to simplify a more complex expression, that is

$$I-(I-MM^+)^+(I-MM^+)$$

EDIT: If it is of any use, we can additionally assume that $M$ is $m\times n$ with $m\leq n$ and that its rank is either $m$ or $m-1$ (those two cases may be treated separately)

$I-MM^+$ is an orthogonal projection, so its Moore-Penrose pseudoinverse is equal to itself and $$I-(I-MM^+)^+(I-MM^+)=I-(I-MM^+)^2=I-(I-MM^+)=MM^+.$$