Standard set-up:
$$
Y=X\beta+e
$$
where $e\sim N(0,\Sigma)$
We know that the GLS estimate of $\beta$ is $\hat{\beta}=(X'\Sigma^{-1}X)^{-1}X'\Sigma^{-1}Y$
The (generalized) residual sum of square is then:
\begin{align}
SS(\hat{\beta})&=(Y-X\hat{\beta})'\Sigma^{-1}(Y-X\hat{\beta})\\
&=(Y-X(X'\Sigma^{-1}X)^{-1}X'\Sigma^{-1}Y)'\Sigma^{-1}(Y-X(X'\Sigma^{-1}X)^{-1}X'\Sigma^{-1}Y)\\
&=Y'\Sigma^{-1}Y-Y'\Sigma^{-1}X(X'\Sigma^{-1}X)^{-1}X'\Sigma^{-1}Y
\end{align}
The second equality is by substituting $\hat{\beta}$ into the LHS, the third equality is by brute force expanding the brackets but I think it is just standard algebra stuff one do with least square problems. In fact, if $\Sigma$ is the Identity matrix it becomes $Y'(I-H)Y$ where $H$ is the 'hat- matrix'
The thing I got quite confused now is that, if I am correct,
$$
\Sigma^{-1}X(X'\Sigma^{-1}X)^{-1}X'\Sigma^{-1}=\Sigma^{-1}\quad(*)
$$
which would make $SS(\hat{\beta})=0$.
Which does not make sense, as $SS(\hat{\beta})$ is the objective function we try to minimize. Can someone tell me what I did wrong?
The way I got $(*)$ is by a bunch of factorization (inspired by this post):
We first use cholesky to get $\Sigma=LL'$
And QR to get $L^{-1}X=QR$
Therefore
\begin{align}
&\Sigma^{-1}X(X'\Sigma^{-1}X)^{-1}X'\Sigma^{-1}\\
=&(LL')^{-1}X(X'(LL')^{-1}X)^{-1}X'(LL')^{-1}\\
=&(LL')^{-1}X(R'Q'QR)^{-1}X'(LL')^{-1}\\
=&L'^{-1}QR(R'R)^{-1}R'Q'L^{-1}\\
=&L'^{-1}L^{-1}\\
=&\Sigma^{-1}
\end{align}
Best Answer
I found by problem by generating random matrices and checking every equality. The problem is in the second last equality. Turns out, $$ L'^{-1}QR(R'R)^{-1}R'Q'L^{-1}\neq L'^{-1}L^{-1} $$ because $$ R(R'R)^{-1}R' =\begin{bmatrix} I_p&0\\ 0&0 \end{bmatrix} \neq I $$ where $I_p$ is the identity matrix of dimension $p<N$, where $N$ is the dimension of the full matrix.