I'm trying to solve the linear regression problem but I'm stuck and can't solve the question. imagine you have below form $$ Y = WX-\epsilon $$ and $\epsilon $ is from Gaussian distribution $\epsilon \sim normal(0,\Sigma)$. we define errors like below $$ error = \sum \epsilon^2 = (WX-Y)^T(WX-Y) $$ we know we have $W* = (X^TX)^{-1}X^TY$ $$$$ now we want to prove this
if we know E($\epsilon$) = 0 and var($\epsilon$) = $\Sigma$ then we can have new estimator $W*_{new} = (X^T\Sigma^-1X)^{-1} X^T\Sigma^{-1}Y$ equal to previous estimator iff there exist a nonsingular F that below condition holds $$ \Sigma X = XF $$ but I can not prove statement mentioned above. help!!
If $\Sigma X = XF$, then $X = \Sigma X F^{-1}$. If you plug $\Sigma X F^{-1}$ in for $X$ in $W^*_{new} = (X^\top \Sigma^{-1} X)^{-1} X^\top \Sigma^{-1} Y$ you will arrive at the expression for $W^*$ after a lot of cancellations. This proves one direction of the "if and only if" statement.