$2 \times 2$ matrix transformations on the unit circle I have been given the following problems to solve:


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*Find all $2 \times 2$ matrices that transform each point on the circle $x^2 + y^2 = 1$ into a point on the same circle. What happens to points on other circles?


*Find all $2 \times 2$ matrices that transform each point on the hyperbola $x^2 - y^2 = 1$ into a point on the same hyperbola. What happens to points on other hyperbolas?


*How would you extend your methods to other sets of points in the plane?

I am really struggling to prove that multiplying the rotation by any angle theta matrix with any point on the unit circle results in a point on the same circle, and likewise with the hyperbola question.

To prove that rotation of a point on the unit circle by any angle $\theta$ results in a point on the same circle,
note the following.
If $x^2+y^2=1$ and $\begin{pmatrix}x' \\ y'\end{pmatrix}=\begin{pmatrix}\cos\theta &-\sin\theta\\ \sin\theta & \cos\theta \end{pmatrix}\begin{pmatrix}x \\y\end{pmatrix}=\begin{pmatrix}x\cos\theta-y\sin\theta\\ x\sin\theta+y\cos\theta\end{pmatrix}$,
then $x'^2+y'^2=(x\cos\theta-y\sin\theta)^2+(x\sin\theta+y\cos\theta)^2$
$=x^2\cos^2\theta-2xy\cos\theta\sin\theta+y^2\sin^2\theta+x^2\sin^2\theta+2xy\sin\theta\cos\theta+y^2\cos^2\theta$.
Can you now combine terms and simplify using $\cos^2\theta+\sin^2\theta=1$?
