# General form of matrices in $\mathrm{O}_{3}(\mathbb{R})$

It can be shown that every matrix in $\mathrm{O}_{2}(\mathbb{R})$ is of the form

$$\begin{pmatrix} \cos \varphi & -\sin \varphi \\ \sin \varphi & \cos \varphi \end{pmatrix}$$

or

$$\begin{pmatrix} \cos \varphi & \sin \varphi \\ \sin \varphi & -\cos \varphi \end{pmatrix}$$

for some angle $\varphi$ where the former is a counter-clockwise rotation by $\varphi$ and the latter is a reflection about the line which passes through the origin and $(\cos \varphi/2, \sin \varphi/2)$.

My question is: can a similar "general" form can be found for matrices in $\mathrm{O}_{3}(\mathbb{R})$?

So far, I attempted this by showing that the first column corresponds to a point on the unit sphere in $\mathbb{R}^3$. Therefore, it must be of the form $(\sin \theta \cos \varphi, \sin \theta \sin \varphi, \cos \theta)$ for some choice of angles $\theta$ and $\varphi$. However, I am unable to determine the other two columns from this.

• The nicest form I know gives three matrices, each dependent on a choice of angle (rotations about the three axes) that can be shown to generate $SO(3)$. I'm sure someone here knows more. – Randall Aug 16 '17 at 16:45
• That could be useful. After all, if $H = \mathrm{SO}_{3}(\mathbb{R})$ and $K = \Bigl\{ \Bigl( \begin{smallmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{smallmatrix} \Bigr), \Bigl( \begin{smallmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & -1 \end{smallmatrix} \Bigr) \Bigr\}$, is it not true that $\mathrm{O}_{3}(\mathbb{R}) = HK$? – Peace - A Theme Aug 16 '17 at 17:25
• Yes, you can use the matrix exponential for rotations and then multiply by any reflection for reflections. – Qiaochu Yuan Aug 16 '17 at 17:53

I'm a little rusty, but here goes. Given a rotation angle $\theta$, we have rotations by $\theta$ radians $$R_z(\theta) = \begin{pmatrix} \cos{\theta} & -\sin{\theta} & 0 \\ \sin{\theta} & \cos{\theta} & 0 \\ 0 & 0 & 1 \\\end{pmatrix}$$
$$R_y(\theta) = \begin{pmatrix} \cos{\theta} & 0 & -\sin{\theta} \\ 0 & 1 & 0 \\ \sin{\theta} & 0 & \cos{\theta} \\\end{pmatrix}$$
$$R_x(\theta) = \begin{pmatrix} 1 & 0 & 0 \\ 0 & \cos{\theta} & -\sin{\theta} \\ 0 & \sin{\theta} & \cos{\theta} \\\end{pmatrix}$$ in the $xy$-, $xz$-, and $yz$-planes respectively (subscript denotes the axis of rotation). You can always use these to generate any rotation in $SO(3)$ by moving the axis first (then moving it back).
Let $A$ be a non-identity element of $SO(3)$. Then $A$ has a 1-diml eigenspace (the axis of rotation), about which we rotate by an angle $\theta$. This eigenspace intersects the unit sphere somewhere, so write this point of intersection as $$\mathbf{n} = \langle \cos{\alpha}\sin{\beta},\sin{\alpha}\sin{\beta},\cos{\beta}\rangle^T$$ with spherical coordinates (and as a column vector).
Note that $R_y(\beta)R_z(\alpha)^{T}\mathbf{n} = \langle 0,0,1\rangle^T$ so we have moved the axis so that this rotation occurs in the $xy$-plane. We now rotate by our original angle $\theta$ and then rotate our axis back to its original position. You can do the math and check that $$A = R_z(\alpha)R_y(\beta)^TR_z(\theta)R_y(\beta)R_z(\alpha)^{T}$$ accomplishes this.