Let $ Y = A\beta + \epsilon $ be a linear model with a design matrix $ A \in \mathbb{R}^{n \times p} $, parameter vector $ \beta \in \mathbb{R}^{p} $, and error terms $ \epsilon $ such that $ \operatorname{Cov}(\epsilon) = \sigma^{2} \Sigma $. How can we transform this linear model into the 'classical' form (with $ \operatorname{Cov}(\tilde{\epsilon}) = \sigma^{2} I $)? Write the new model $ \tilde{Y} $ and calculate its expected value and covariance.
Attempt/Idea: (Is the following the correct answer?) Sometimes, linear models are also considered with $ \operatorname{Cov}(\varepsilon)=\sigma^{2} \Sigma $, where $ \Sigma $ is a positive definite matrix. However, such linear models can be transformed into the form described above by a simple transformation.
To do this, consider a symmetric and positive definite matrix $\Sigma^{1 / 2} $ such that $ \Sigma^{1 / 2} \cdot \Sigma^{1 / 2}=\Sigma $. Then, the covariance matrix of $ \Sigma^{-1 / 2} Y=\Sigma^{-1 / 2} \Lambda \beta+\Sigma^{-1 / 2} \varepsilon $ is given by:
$$ \operatorname{Cov}\left(\Sigma^{-1 / 2} Y\right)=\Sigma^{-1 / 2} \operatorname{Cov}(Y) \Sigma^{-1 / 2}=\sigma^{2} I_{n} $$