Rotation Matrix

linear-algebra

Definition

Rotation Matrix

A rotation matrix is an orthogonal matrix with determinant $1$.

It represents a linear map that preserves lengths, angles, and orientation.

Examples

Rotation in ${\mathbb{R}}^2$

A rotation in ${\mathbb{R}}^2$ turns every vector about the origin by a fixed angle $\phi$.
It does not change the length of the vector; it only changes its direction.

Derivation

A rotation is determined once we know what it does to the standard basis vectors $e_1$ and $e_2$.

Since $R$ is a rotation by $\phi$, the image of $e_1$ is the unit vector making angle $\phi$ with the positive $x_1$-axis. Hence

$$R{e}_1=\left[\begin{array}{c}\cos \phi \\ \sin \phi \end{array}\right].$$

The vector $e_2$ is orthogonal to $e_1$. A rotation preserves angles and orientation, so $R{e}_2$ must be the unit vector obtained by rotating $R{e}_1$ by another $\frac{\pi }{2}$. Therefore

$$R{e}_2=\left[\begin{array}{c}-\sin \phi \\ \cos \phi \end{array}\right].$$

Now write an arbitrary vector as

$$v=x{e}_1+y{e}_2.$$

By linearity,

$$Rv=xR{e}_1+yR{e}_2.$$

Substituting the two basis images gives

$$R(\phi )v=x\left[\begin{array}{c}\cos \phi \\ \sin \phi \end{array}\right]+y\left[\begin{array}{c}-\sin \phi \\ \cos \phi \end{array}\right]=\left[\begin{array}{c}x\cos \phi -y\sin \phi \\ x\sin \phi +y\cos \phi \end{array}\right].$$

The minus sign is therefore forced: it is the $x_1$-component of $R{e}_2$, and $R{e}_2$ points to the left.

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

$$R(\phi )=\left[\begin{array}{cc}\cos \phi & -\sin \phi \\ \sin \phi & \cos \phi \end{array}\right].$$

If $v=\left[\begin{array}{c}x\\ y\end{array}\right]$, then

$$R(\phi )v=\left[\begin{array}{c}x\cos \phi -y\sin \phi \\ x\sin \phi +y\cos \phi \end{array}\right].$$

The columns of $R(\phi )$ are the images of the standard basis vectors $e_1$ and $e_2$ under the rotation.

Special case: For $\phi =\frac{\pi }{2}$,

$$R\left(\frac{\pi }{2}\right)=\left[\begin{array}{cc}0& -1\\ 1& 0\end{array}\right].$$

Mirroring on the $x_1$-Axis

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

Projection onto the First Median

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