I'm trying to prove the following:
If $A$ is an $n \times m$ matrix with $n \geq m$ then
$$rank(I -A(A^TA)^GA^T) = n - rank(A)$$
Note: $G$ here means the generalized inverse matrix ie. $A^G$ is the generalized inverse of $A$.
How would one go about proving this? Should I be assuming that $rank(A - B) = rank(A) - rank(B)$ or is this not true?