Rotation Matrix

linear-algebra

Definition

Rotation Matrix

A rotation matrix is a square matrix that performs linear transformation preserving the euclidean norm and orientation of the vectors in a given vector space.

In ${\mathbb{R}}^2$, the rotation matrix for an angle $\phi$ is given by:

$$\left[\begin{array}{cc}\cos \phi & -\sin \phi \\ \sin \phi & \cos \phi \end{array}\right]$$

Examples

Mirroring on the $x_1$-Axis

$$S=\left[\begin{array}{cc}1& 0\\ 0& -1\end{array}\right]$$

Projection onto the First Median

$$P=\left[\begin{array}{cc}1/2& 1/2\\ 1/2& 1/2\end{array}\right]$$