Given an orthonormal matrix $R$ and two diagonal matrices $D_1$ and $D_2$, is there an easy way to obtain the eigenvalues of:
$$ D_2 R D_1 R^T D_2 $$
Edit: I found Singular value decomposition of product of matrices and and Most efficient method for computing Singular Value Decomposition of a triangular matrix?
If we define the QR decomposition $QU=qr(D_2R)$
We get $QUD_1U^TQ^T$
As the outer rotation would not change the eigenvalues, we need the eigenvalues of $UD_1U^T$ which looks similar to a Cholesky decomposition except that it uses upper triangular matrices and the triangular matrices are not normalized yet.
I'm not quite sure if this is the right direction. Is there any advantage of going the way of
- QR
- Bringing it to normalized lower triangular form
- Calculating the Eigenvalues using Cholesky
in comparison to just performing PCA on the whole matrix? Is there any more efficient way of doing QR decomposition knowing the properties of $D_2R$?
