Suppose I have the following decomposition of a symmetric matrix $S$, $S = U \Lambda U^T$, where we have that $\Lambda$ is a diagonal matrix and $U U^T = D$ where $D$ is another diagonal matrix.
In other words, what I have is "almost" the eigenvalue decomposition of $S$, only $U$ is not orthogonal. This means that $U$ and $\Lambda$ solve the generalized eigenvalue problem $Su=\lambda Du$.
Now given $\Lambda$ and $U$ from the generalized eigenvalue problem, how can I find the eigenvalues and eigenvectors of $S$ itself?
An additional observation is that if we set $Z = D^{-1/2}U$, then we have $Z Z^T=D^{-1/2}UU^TD^{-1/2}=D^{-1/2}DD^{-1/2}=I$, so we have that Z is orthogonal.
