Dimensionality reduction of a large covariance matrix

I have a large covariance matrix $Sigma$ and I am reducing its dimensionality by using a truncated eigendecomposition. $Sigma approx VDV^T$. I remember somewhere that you could also decompose it as $VDV^T + Delta$, where $Delta$ is a diagonal matrix with positive entries and dimensionality similar to $Sigma$. Does anybody know this technique?

Can I just assume that diag($Sigma – VDV^T$) always has positive entries?