Let $A$ be an orthogonal matrix with elements $a_{ij}$ so that $\sum_k a_{ik} a_{jk} = \delta_{ij}$. I'd like to know what is $\frac{ \partial a_{ij} }{ \partial a_{kl}}$. If $A$ was a generic matrix (with no constraints on its elements) then the answer would be $\delta_{ik} \delta_{jl}$. However, now we have the quadratic constraint on the matrix. What is the answer in this case?


Your question is non-sense but it is difficult to explain why. That follows is an example for $n=3$.

There is a one to one local diffeomorphism $f:K\in SK_3\rightarrow (I_3-K)(I_3+K)^{-1}\in SO_3$ where $SK_3$ is the set of skew symmetric matrices. In particular,

let $U=\begin{pmatrix}-18/23& 14/23& 3/23\\6/23& 3/23& 22/23\\13/23& 18/23& -6/23\end{pmatrix}\in O_3=f(\begin{pmatrix}0&-4&5\\4&0&-2\\-5&2&0\end{pmatrix})$.

Thus $U=[u_{i,j}]\in O_3$ depends on $3$ independent parameters. Locally, we can choose these parameters amongst the $(u_{i,j})$, but they cannot stand on the same row or column. We are interested by the derivative $\frac{ \partial u_{1,2} }{ \partial u_{1,1}}$. Then $u_{1,1}$ must be a parameter and $u_{1,2}$ must not (otherwise, the result is obvious).

We consider the $2$ following parametrizations

Choice 1. $u_{1,1},u_{2,2},u_{3,3}$. Then the derivative in our $U$ is $\frac{ \partial u_{1,2} }{ \partial u_{1,1}}\approx 1.46246$.

Choice 2. $u_{1,1},u_{2,3},u_{3,2}$. Then the derivative in our $U$ is $\frac{ \partial u_{1,2} }{ \partial u_{1,1}}\approx 0.666603$.

You can see that the result depends on the choice of the chosen local parametrization of $SO_3$.

EDIT. Answer to @Adam . Yes, in $SO(2)$, there is no problem because the parametrization contains only one parameter; for example, if $U(\theta)=\begin{pmatrix}\cos(\theta)&-\sin(\theta)\\\sin(\theta)&\cos(\theta)\end{pmatrix}$ with $u_{1,1}=f(u_{2,1})>0$, then $\dfrac{ \partial u_{1,1} }{ \partial u_{2,1}}=\dfrac{-\sin(\theta)}{\cos(\theta)}=\dfrac{-u_{2,1}}{\sqrt{1-u_{2,1}^2}}$. Yet, an element of $SO(3)$ depends on $3$ parameters and you must choose these parameters to calculate a partial derivative with respect to one of these parameters.

  • $\begingroup$ @Prahar , I get tired of writing an answer to your question that does not interest me -believe me-. So the least you could have done is to say that you read it... I have the impression that there is no politeness class in Princeton. $\endgroup$ – loup blanc Mar 19 '18 at 15:24
  • $\begingroup$ physicists like to defined derivatives like that for matrix differentials. For example, they define $\partial a_{ij}/\partial a_{kl}=\frac12(\delta_{ik}\delta_{jl}+\delta_{il}\delta_{jk}$ for a symmetric operator. Would you say that is meaningless too ? I am asking because I am interested exactly in the same question than Prahar. I think one can make sense of this in the case of SO(2), and I would like to find a similar answer in the case of SO(3) (or more generally in the case of non-linear constrained matrices). $\endgroup$ – Adam May 24 '18 at 14:29
  • $\begingroup$ Thanks for the edit. I've rephrased my question in a more general setting here : math.stackexchange.com/questions/2798047/… and I think it is possible to make sense of the SO(3) case, if you want to read it. $\endgroup$ – Adam May 28 '18 at 10:09

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