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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)

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

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It´s the same matrix consider de USV factorization of M I-MM+=I-(USVH)(V*S^-1*UH)
where UH is the transpose conjugate of U and VH is similar for V

Then I-MM+=I-U*UH Now consider W orthonormal basis of U thus [U W] has as inverse [UH;WH] denoted ; as column vector similar to matlab

Then I=[U W][UH;WH] = UUH+W*WH

Thus I-MM+=WWH where WH*W=I

I-MM+=WWH=WIWH=W*I^-1*WH=(WWH)+ = (I-MM+)+

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