TITLE: Eigenvalues involving the product of diagonal and Hermitian matrices
QUESTION [5 upvotes]: I have a question with regards to the eigenvalues involving the product of diagonal and Hermitian matrices.
Let $A$ be a Hermitian matrix and $B$ be a diagonal matrix. I am seeking to express the eigenvalues of the matrices $ABA$ and $BAB$ in terms of the eigenvalues of $A$ and $B$.
While I am able to find the eigenvalues of the two matrices under some very special conditions, I would like to know if there is any way of relating the eigenvalues of the matrices $ABA$ and $BAB$ in terms of the eigenvalues of $A$ and $B$ in general.

REPLY [2 votes]: This is not possible. The eigenvalues of the product of matrices do not depend solely on the eigenvalues of each multiplicand. For instance, consider
$$
A=\pmatrix{2&1\\ 1&1},\ \widetilde{A}=\pmatrix{1&1\\ 1&2},\ B=\pmatrix{1&0\\ 0&2}.
$$
$A$ and $\widetilde{A}$ have identical spectra, but $ABA$ and $\widetilde{A}BA$ do not share a common eigenvalue and similarly for $BAB$ and $B\widetilde{A}B$.