 Solution

Assume the known covariance matrix $\Sigma$ is positive definite. The Gaussian likelihood, up to constants, isIts gradient is $X^T{\Sigma }^{-1}(Y-X\beta )$. The negative Hessian matrix $X^T{\Sigma }^{-1}X$ is positive definite because $X$ has full column rank. Thus the unique maximum-likelihood estimator is the generalized least squares estimatorIt is unbiased, and multiplication of its linear coefficient matrix by $\Sigma$ givesIn fact ${\hat{\beta }}_{\mathrm{GLS}}\sim N_p(\beta ,(X^T{\Sigma }^{-1}X{)}^{-1})$. Equivalently, a whitening transformation premultiplies $Y$ and $X$ by ${\Sigma }^{-1/2}$, after which ordinary least squares applies. There is no extra ${\sigma }^2$ factor when $\Sigma$ is the complete known covariance; such a factor appears if only a shape matrix $\Omega$ is known and $\Sigma ={\sigma }^2\Omega$.