Hello,


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?



Eigendecomposition after multiplying by diagonal matrix
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](http://en.wikipedia.org/wiki/Singular_value_decomposition#Singular_values.2C_singular_vectors.2C_and_their_relation_to_the_SVD) 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](http://www.jstor.org/pss/2695237), particularly Section 11, for a gentle introduction.


