I am looking at a paper where the covariance matrix needs to be scaled by another matrix where the scaling weights are on the diagonal elements. The formula for the scaled covariance matrix is written as:
$$ S = W \Sigma W^T $$
where $\Sigma$ is the original covariance matrix and $W$ is the weight matrix. I can verify this operation results in a valid covariance matrix but this operation is not intuitive to me. Is there an intuitive, geometric explanation of why we need to multiply by $W$ and $W^T$ rather than just having $W \Sigma$ as the scaled covariance matrix.