Question about rotation $2\times 2$ rotation matrices

Question

How do I prove that, if for a $2 \times 2$ matrix $A$ and a fixed integer $n> 0$ we have that $$A^n= \begin{bmatrix} \cos{x}& -\sin{x}\\ \sin{x} & \cos{x}\end{bmatrix} ,$$ for some real $x$, then $A$ is also a rotation matrix? I know that it is very obvious, especially if you think about the isomorphism with complex numbers, but I can't seem to come up with a simple and rigorous proof.

Answer 1

Even for the case $A^2=R$ there are many possible roots.

Look for instance at:

https://www.maa.org/sites/default/files/pdf/cms_upload/Square_Roots-Sullivan13884.pdf

With the additional constraint $\det(R)=1$ here you get $$\operatorname{tr}(A)A=R\pm I$$

and have to discuss according values of $x$, whether the trace is zero or not and so on.

• e.g. $R=I$ and null trace, gives $\begin{pmatrix}a&b\\c&-a\end{pmatrix}$ with $a^2+bc=1$

[ $a=0,b=1,c=1$ is the case presented by David C. Ulrich in his answer ]

For $A^n=R$ you get even more solutions.

Starting with $n=2p$ even, you have to solve $A^p=B$ for every $B^2=R$ with $B$ not necessarily a rotation already...

Answer 2

"Very obvious" or not, it's not true. If $A=\begin{bmatrix}0&1\\1&0\end{bmatrix}$ then $A^2=I$ but $A$ is not a rotation.

Answer 3

$\DeclareMathOperator{\Tr}{Tr}$For the case $A^2=R$ where $R$ is a 2x2 rotation different from identity, we can apply Sullivan's article quoted by zwim, and find:

If $R=\begin{pmatrix}-1\\&-1\end{pmatrix}$, then $A=\begin{pmatrix}\alpha&\beta\\\gamma&-\alpha\end{pmatrix}$ with $\alpha^2+\beta\gamma=-1$.

If $R$ is any other rotation over an angle $\phi$, then $\displaystyle A=\pm\frac{R+I}{\sqrt{2\cos\phi+2}}$. Working through the possibilities, it means that $A$ is a rotation matrix over either $\frac 12\phi$ or $\frac 12\phi + \pi$.