Does polar decomposition work when the 'orthogonal' matrices are not orthogonal wrt to the identity ($O^TO=Id$), but wrt to some other symmetric matrix $K$ (i.e. $O^TKO=K$)?

Specifically, $GL(3,R)$ can be decomposed as $SO(3)\times Diag(3)\times SO(3)$, but can it also be decomposed as $SO(3)\times Diag(3)\times SO_K(3)$? (SO_K are the K-orthogonal matrices mentioned above)

Thanks for any help.

EDIT:

Sorry to resurrect this, but I've had a further thought. First of all thank you for you answers (over my head as they were, they're still appreciated). From what I gather the $SO_k \times Diag^+ \times SO_k$ decomposition works, but $SO \times Diag^+ \times SO_k$ does not?

What if we were to use the standard polar decomposition on $Q\in GL(3)$ so that $Q=RS$ for orthogonal $R$ and symmetric $S$, and then further decompose this symmetric matrix into a $SO_k \times Diag^+ \times SO_k$ product? This should be valid, since $S\in GL(3)$, right? So you'd end up with a matrix that looks like $SO\times [something]\times SO_k$. This is really what I was after when I first asked the question (I realise I could have been clearer about that). The something in the middle doesn't really matter to me.

Does that make any sense?

Thank you again.

4more comments