If we possess the eigendecomposition of a positive definite matrix: $X = U \Sigma U^T$, is there an efficient way to compute the eigendecomposition of $D X D$ where $D$ is a diagonal matrix?
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Write $\Sigma$ as $T^2$, for positive definite $T$. Set $Y = U T$.
So the eigenvalues of $X$ are the squares of the singular values of $Y$, and what you want to compute are the singular values of $DY$.
There is no formula which gives the singular values of $DY$ in terms of those of $Y$ and $D$. However, there is a famous set of inequalities relating the three sets of singular values, called the Horn inequalities. See Bhatia's article Linear Algebra to Quantum Cohomology, particularly Section 11, for a gentle introduction.