Rotation matrix

Example

Let $\theta \in \R$. The rotation matrix of angle $\theta$ is $$R(\theta) = \begin{pmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{pmatrix}.$$ The map $x \mapsto R(\theta) x$ rotates the plane $\R^2$ about the origin by the angle $\theta$, counterclockwise. It is an orthogonal matrix with determinant $\cos^2\theta + \sin^2\theta = 1$, so it is an element of $SO(2)$.

Remarks

Rotations add their angles: $R(\alpha) R(\beta) = R(\alpha + \beta)$. So $R(\theta)^{-1} = R(-\theta) = R(\theta)^T$. For $\theta$ not a multiple of $\pi$, $R(\theta)$ has no real eigenvalue; its complex eigenvalues are $e^{\pm i\theta}$. Every element of $SO(2)$ is $R(\theta)$ for some $\theta$.

Explanations

L1 human

The rotation matrix sends $(1, 0)$ to $(\cos\theta, \sin\theta)$ and $(0, 1)$ to $(-\sin\theta, \cos\theta)$. These are the first and the second column. For $\theta = 90$ degrees, it is $\begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix}$.