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I want to rotate several vectors in $\mathbb{R}^6$ by the same rotation matrix $M$. $M$ should be random but uniformly sample the rotation space. I know that the $6\times6$ length-preserving rotation matrix $M$ must fulfill $M^T=M^{-1}$ and for preservation of orientation we would additionally require $\text{det}(M)=1$ (Link). How can I calculate these random matrices? How to achieve random matrices with uniform random sampling?

I found this answer to calculate uniformly sampled random vectors but I also need the rotation matrix as I want to apply the same transformation several times.

granular_bastard
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  • I think uniform in all directions and uniform on all matrices are the same, and thus the link you post shall solve your problem. – Vezen BU Aug 23 '22 at 06:03
  • @VezenBU I think the answer there is only about random vectors but I need also the rotation matrix. I clarified my post. – granular_bastard Aug 23 '22 at 13:33
  • Hi@granularbastard : combining the two answers in the post you link, should suggest that in the QR decomposition of a Gaussian Matrix, the Q matrix is uniformly distributed on the sphere – G Cab Aug 23 '22 at 13:53

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