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Suppose that $U_{\theta }=\begin{bmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{bmatrix}$,in which $\theta$ is a real parameter,it is easy to prove that a given matrix $ U\in M_{2}(\mathbb{R})$ is real orthogonal matrix if and only if either $ U=U_{\theta}$ or $U=\begin{bmatrix} 0 &1 \\ 1 & 0 \end{bmatrix}U_{\theta}$. But for 3-by-3 matrices, how to construct a parametric presentation of 3-by-3 real orthogonal group?

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  • $\begingroup$ Have you tried Euler angles? $\endgroup$ – Eric 2 days ago
  • $\begingroup$ @Eric I don’t know how to use. Would you mind giving me some tips? thank you very much. $\endgroup$ – Stephen Wong 2 days ago
  • $\begingroup$ I doubt you can get a neat formula but you could process in the same way as in the 2-dimensional case. That is, since U is orthogonal its rows (and columns) have length 1 and they are orthogonal to each other. Then solve the equations depending on two parameters. $\endgroup$ – Salcio 2 days ago
  • $\begingroup$ @Salcio Thanks for your tips! $\endgroup$ – Stephen Wong 2 days ago
  • $\begingroup$ Euler angles are effectively rotating around x axis, then z -axis, then x-axis again. You can compose those matrices. en.m.wikipedia.org/wiki/Euler_angles $\endgroup$ – Eric yesterday

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