Polar Decomposition: derivative of each factor wrt. original matrix

Question

If the arbitrary square real matrix $F$ is decomposed into $F=RU$ with orthogonal $R$ and positive semi-definite symmetric $U$, is there any way to express $$\frac{\partial R}{\partial F}$$ or $$\frac{\partial U}{\partial F}$$ analytically? I know one could try several numerical approaches, but atm only analytic expressions are of interest. I also know how to differentiate eigenvectors (and eigenvalues) wrt. their matrix. But that does not seem to help since eigenvectors/-values of $F$ and $U$ don't necessarily have anything in common, not even existence, because $R$ may have none.

If it helps, the answer can be specialized for $3\times3$ matrices.

Answer 1

I feel like I'm tired for nothing because you don't read the answers to your questions...

"quadratic", what do you mean?

The decomposition $F=RU$ must be unique; then, necessarily $F\in GL_n(\mathbb{R})$. Moreover $U=\sqrt{F^TF},R=FU^{-1}$. After, it's not difficult.

Answer 2

The polar decomposition $F=RM$ can be obtained from the SVD (with $R=UV^T$ and $M = VSV^T$) which you can differentiate (see e.g. https://j-towns.github.io/papers/svd-derivative.pdf) to obtain the partial derivatives you're looking for.