I hope my question is trivial for some of you but for the time being I’m lost somewhere between the generalized eigenproblem, simultaneous diagonalization of quadratic forms, simultaneous SVD, generalized SVD, etc.
Let $A$ and $B$ be two symmetric, positive semi-definite (but not positive-definite) matrices in ${\mathbb{R}^{n \times n}}$. $\left[ {A,B} \right] \ne 0$ . Both of them are diagonalizable, one of them is diagonal. Find a pair of non-singular matrices $P$ and $Q$ such as
$\left\{ \begin{gathered} PAQ = {D_1} \hfill \\ PBQ = {D_2} \hfill \\ \end{gathered} \right.$
${D_1}$ and ${D_2}$ diagonal. No other property of $P$ and $Q$ is required.