Rotation Dilation Matrices

A rotation and dilation can be done like so:

$$\begin{array}{r}\left[\begin{array}{cc}r& 0\\ 0& r\end{array}\right]\left[\begin{array}{cc}\cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{array}\right]\end{array}$$

More generally, any such matrix is a rotation dilation matrix:

$$\left[\begin{array}{cc}a& -b\\ b& a\end{array}\right]$$

For this the eigenvalues will work out to be $a\pm ib$ or in polar form $r(\cos \theta \pm \sin \theta )$. Where $r^2=a^2+b^2$ and $\theta$ is the angle of rotation ($\tan \theta =\frac{b}{a}$).