I have a block matrix (which is overall symmetric)
$$ M = \begin{bmatrix} A & B \\ B^T & C \\ \end{bmatrix} $$
and it's inverse
$$ M^{-1} = \begin{bmatrix} \tilde{A} & \tilde{B} \\ \tilde{B}^T & \tilde{C} \\ \end{bmatrix} $$
Note that while $A$ and $C$ (along with their tilde counterparts) are both square, they are not, in general, the same size.
I'd like to find the eigenvalues of the product of the subblocks: $A\tilde{A}$.
However, I also have easy access to the diagonalization of $M = U D U^T$, which brings me to the actual question:
Can I make use of the diagonalization of $M$ to compute the eigenvalues of $A\tilde{A}$ (or at least speed up the computation somehow)? All of the matrices are real, if that helps at all.