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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.

rehctawrats
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3 Answers3

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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.

  • Sorry you are tired, but I had a great weekend ;-) Replaced "quadratic" by "square". As soon as I understand your answer to https://math.stackexchange.com/questions/2397169/ I will mark this answer as accepted. My $F$ is always invertible (deformation gradient), thus the polar decomposition is unique. Thanks in advance :-) – rehctawrats Aug 21 '17 at 08:40
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If you add the constraint that $F$ is full rank and $U $ is positie definite, then there is indeed an analytical expression in terms of the singular values and vectors of $F$, which you can find page in Proposition 2.23, p74 of this thesis

https://www.researchgate.net/publication/319481099_Riemannian_Geometry_of_Matrix_Manifolds_for_Lagrangian_Uncertainty_Quantification_of_Stochastic_Fluid_Flows

Anonymous
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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.

max
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