Show that $PX=X$, where $P=X(X^{\prime}X)^{-}X^{\prime}$ denotes the projection matrix onto $C(x)$
My work:
Suppose $(X^{\prime}X)^{-}$ is g-inverse of $X^{\prime}X$.
Then, if we can show that
$(X^{\prime}X)^{-}X^{\prime}$ is g-inverse of $X$ The rest of the parts will follow.
Also, I have tried to use the following result here,
$GA^{\prime}$ is a g-inverse of $A$ s.t $AGA^{\prime}A=A$
but couldn't make it sense. I was wondering if you could give me some idea. I appreciate your time!
I don't know what is the g-inverse or $C(x)$; yet, that works when you consider the Moore Penrose inverse denoted by $U^+$; cf.
https://en.wikipedia.org/wiki/Moore%E2%80%93Penrose_inverse#Projectors
I assume that your $X'$ is the notation for $X^T$ (real matrices) or $X^*$ (complex matrices).
Firstly, $(X'X)^+X'=X^+$; secondly $PX=XX^+X=X$.