If $A$ and $B$ are real symmetric matrices, the matrix $BAB$ is also a real symmetric matrix. My question is : knowing the orthogonal diagonalization of $A$ and $B$, can we obtain the orthonormal diagonalization of $BAB$ ?
Starting from $A = UDU^{\top}$ and $B = VD'V^{\top}$ with $U,V$ real orthogonal matrices and $D,D'$ diagonal matrices, we get :
$$ BAB = VD'V^{\top}UDU^{\top}VD'V^{\top} $$
which I do not find very helpful. My intuition is that the answer to the question is no. It is not even clear what the eigenvalues of $BAB$ would be in terms of $A$ and $B$. Can an additional hypothesis, such as "$B$ is positive definite" make the problem easier?