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Let $X\in\mathbb{R}^{3\times 3}$ be an orthogonal matrix. Then $\mathrm{vec}X\in\mathbb{R}^9$ is a 9 by 1 vector formed by stacking the columns of the matrix $X$ on top of one another. Given a matrix $A\in\mathbb{R}^{9\times 9}$, find the optimal orthogonal matrix $X$ minimizing the following objective function. $$J=\left(\mathrm{vec}X\right)^T A \mathrm{vec}X$$ I think Kronecker product may be useful for solving this problem. Does a closed-form solution exist? If not, is it possible to solve it iteratively? Thanks.

EDIT

a) Here $A$ doesn't have any special property. But it is also acceptable if solutions can be obtained by adding some properties on $A$.

b) In the original problem, $X$ is constrained as a rotation matrix. But I think that would be even harder, so I put $X$ as an orthogonal matrix herein. Of course, optimal rotation matrices are better.

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  • $\begingroup$ Does $A$ has any property, e.g. being positive definite? $\endgroup$ – B0rk4 Oct 9 '11 at 15:28
  • $\begingroup$ @Hauke Strasdat: Thanks, I've edited the question about this point. $\endgroup$ – Shiyu Oct 9 '11 at 15:36
  • $\begingroup$ Note sure whether there is a closed form solution. But you could solve it iteratively (e.g. using the Newton method or Gradient Descent) and the Lie algebra so3. Is X a proper rotation $\det(X)=1$ or can it also be a reflection? $\endgroup$ – B0rk4 Oct 9 '11 at 15:42
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    $\begingroup$ You can preserve orthogonality, if you do the update in the tangent space (Lie algerba). Thus, you calculate the gradient: $\nabla J =\frac{\partial }{\partial \epsilon} \text{vec} (\exp(\epsilon)X)^\top A \text{vec} (\exp(\epsilon)X)|_{\epsilon=0}$ with $\exp(\cdot)$ being the Rodriguez formula. Now you can iteratively update $X$ using gradient descent: $X' = \exp(-\alpha\nabla J) X$. $\endgroup$ – B0rk4 Oct 9 '11 at 16:23
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    $\begingroup$ On the issue of rotations vs. reflections, note since X is of odd dimension and can be replaced wlog in the objective function by -X if necessary, there is no problem restricting attention to det(X)=1. $\endgroup$ – hardmath Oct 9 '11 at 16:36
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This is a bit late, but I believe this might help.

Use the Cayley parametrization of orthogonal matrices. Example usage in optimization: A Feasible Method for Optimization with Orthogonality Constraints

Broken link: http://www.caam.rice.edu/~wy1/paperfiles/Rice_CAAM_TR10-26_OptManifold.PDF

You might want to start by just understanding the Wiki article on the Cayley transform and why a parametrization helps (you can do the optimisation without constraints, which should be easier)

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    $\begingroup$ Please explain how this "might help". A link-only Answer risks becoming obsolete as links break, and it is advantageous to the Reader to have a clear expectation of what they would find if the link were followed. $\endgroup$ – hardmath Jul 23 '14 at 13:08
  • $\begingroup$ @hardmath The link is now broken :( $\endgroup$ – Antoine Jun 27 at 13:52
  • $\begingroup$ @Patrick: The old "example usage" link was successfully crawled by the Wayback Machine in May of 2018. Here's a link at UCLA.edu associated with the second named author, so presumably of potential longevity. Of course an elaborate self-contained explanation would be welcome here. $\endgroup$ – hardmath Jun 27 at 20:13

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