Given a diagonal matrix $D$, we symmetrize matrix $W$ by the following transformation: $S \equiv D^{1/2} W D^{-1/2} $
Now, $S$ is a symmetric matrix which is diagonalizable as $S = X \Lambda X^T$
The left eigenvectors of $W$ are \begin{align} u_i = x_i D^{1/2} \end{align}
and the right eigenvectors of $W$ are \begin{align} v_i = x_i D^{-1/2} \end{align}
I tried $ S = X \Lambda X^T = (D^{1/2}U) \Lambda (VD^{-1/2}) $, because matrices $S$ and $W$ have the same eigenvalues. But I do not get how to derive the expressions for left and right eigenvectors as shown in previous equations. Can someone please explain how that works?