I am working through a multi-part proof of how orthogonal projection matrices give specific results from their properties. I read through the Gauss-Markov model theory to get a start. This is only a part of the proof, but if I can get help on this part, then I will be better equipped to try and tackle the rest.
Part (a) Prove $X'XA = X'XB$ iff $XA = XB$
Part (b) Use result of (a) to prove $X(X'X)^-X'X = X$ for any generalized inverse of $X'X$.
Part (c) Prove if $A$ is symmetric and $G$ is a generalized inverse of $A$, then it must be true that $G'$ is also a generalized inverse of $A$.
Part (d) Use these to show $X'X(X'X)^-X' = X'.$
(There's a lot more, but this is the stuff I'm stuck on, and I honestly am struggling with how these matrices prove these types of properties.)
Introductory (a) Proof
$XA=XB$ therefore $X'XA = X'XB$ holds only trivially
Must prove that $X'XA = X'XB$ therefore $XA =XB.$
Rewrite $X'XA=X'XB$ as $X'XA-X'XB = 0.$
Factor $X'X(A-B) = 0$
Multiply both sides $(A'-B') X'X(A-B) = (A'-B') 0$
I know I should Somehow show $A'A = 0$, then use this.