# « Generalized simultaneous diagonalization » of a pair of symmetric, non-commuting, positive semi-definite matrices

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.

• The problem becomes easy enough when one of the matrices is positive definite. But I'm also unsure on how to approach the semi-definite case. Can you maybe give some context on where you encountered the problem? Apr 3, 2018 at 12:04
• @MaikPickl Yes, should $A$ or $B$ be positive definite, I would not ask the question, the problem would reduce to the generalized eigenproblem. The underlying context is probabilistic but it would be too long to describe it here. Apr 3, 2018 at 12:44
• Fair enough. Here seems to be something: maths.manchester.ac.uk/~higham/narep/narep460.pdf Apr 3, 2018 at 14:59
• The interesting part starts in chapter 10 but they heavily use Kronecker canonical forms of matrix pencils. And there seems to be an additional requirement to $A$ and $B$, but see for yourself. Apr 3, 2018 at 15:00
• @MaikPickl Thanks for the paper. Checking................ Apr 3, 2018 at 20:38