MLE of regression model

Question

Consider the regression model:

$Y = X.\beta+\epsilon$ where $\epsilon \sim N(0,\Omega)$.

$\Omega$ is a nxn symmetric positive matrix.

Show that, the MLE of $\beta$ is:

$\hat{\beta}=(X^T\Omega X^{-1})^{-1}X^T\Omega^{-1}Y$

Answer 1

Note that as $\Omega$ is symmetric positive definite one can find a non singular matrix $B$ such that $\Omega=BB^t$. Then make the transformation of variables $Z=B^{-1}X$, then $Z\sim N(0,I)$ where $I$ is the identity matrix. Try to modify the regression equation, so that it is $B^{-1}Y$ regressed on $Z$, and complete the rest.