Assuming that I already know the eigen-decomposition of a real symmetric matrix $X$, is there any way to use it to retrieve efficiently the eigen-decomposition of $DXD$, where $D$ is a diagonal matrix? (I mean, more efficiently than computing $DXD$ and decomposing it).
Modulo a base change, this is the same as asking for an efficient way to eigen-decompose $A\Sigma A$ with $\Sigma$ diagonal and $A$ symmetric with known eigen-decomposition.
Maybe some iterative algorithm can use the known eigenvectors as a starting point and converge really fast?
In fact, I strongly doubt there is anything, but I will feel better if someone can provide an definitive argument to close the question.