There is a 2d rotation matrix around point $(0, 0)$ with angle $\theta$.
$$
\left[ \begin{array}{ccc}
\cos(\theta) & -\sin(\theta) \\
\sin(\theta) & \cos(\theta) \end{array} \right]
$$
Next, there is a 3d rotation matrix around point $(0, 0, 0)$ and unit axis $(u_x, u_y, u_z)$ with angle $\theta$ (Rodrigues' Rotation Formula).
\begin{bmatrix} \cos \theta +u_x^2 \left(1-\cos \theta\right) & u_x u_y \left(1-\cos \theta\right) – u_z \sin \theta & u_x u_z \left(1-\cos \theta\right) + u_y \sin \theta \\ u_y u_x \left(1-\cos \theta\right) + u_z \sin \theta & \cos \theta + u_y^2\left(1-\cos \theta\right) & u_y u_z \left(1-\cos \theta\right) – u_x \sin \theta \\ u_z u_x \left(1-\cos \theta\right) – u_y \sin \theta & u_z u_y \left(1-\cos \theta\right) + u_x \sin \theta & \cos \theta + u_z^2\left(1-\cos \theta\right)
\end{bmatrix}
How it is possible to generalize rotation matrix on $N$ dimension around zero point and $N-2$ dimensional unit axis with angle $\theta$?
Best Answer
The definition is that $A\in M_{n}(\mathbb{R})$ is called a rotation matrix if there exist a unitary matrix $P$ s.t $P^{-1}AP$ is of the form $$\begin{pmatrix}\cos(\theta) &-\sin(\theta)\\ \sin(\theta) & \cos(\theta)\\ & & 1\\ & & & 1\\ & & & & 1\\ & & & & & .\\ & & & & & & .\\ & & & & & & & .\\ & & & & & & & & 1 \end{pmatrix}$$
If we consider $A:\mathbb{R}^{n}\to\mathbb{R}^{n}$ then the meaning is that there exist an orthonormal basis where we rotate the $2-$dimensional space spanned by the first two vectors by angle $\theta$ and we fix the other $n-2$ dimensions