$X$ is an $n×p$ matrix, with $Rank(X)=r$. Then what is the Rank of the matrix $P=X(X^TX)^{-1}X'$? Is it always true that $Rank(X)= Rank(P)$? Is it true when $X$ is of full column rank, i.e. $p=r$?
I know that $Rank(X)=Rank(X'X)$, but I am confused with the above questions. Any help will be appreciated. Thanks in advance.
Like Ted's comment says, $X^\top X$ is invertible only when $r = p$, so we can assume that $X$ has rank $p$ (otherwise $P$ does not exist). There is already a simple proof in this StackExchange answer here that the rank of $P$ is $p$, using the fact that $P$ is idempotent and that the trace of an idempotent matrix is equal to the rank of the matrix. If you want a proof of the fact that the trace of an idempotent matrix is equal to the rank of the matrix, please see this StackExchange post.
