Let $\theta \in [0,2\pi), \theta \neq 0,\pi$. What is $C_{\mathbb{R}^{2\times 2}}(R(\theta))$, where $R(\theta) := \begin{pmatrix} \cos(\theta)&-\sin(\theta)\\ \sin(\theta)&\cos(\theta)\end{pmatrix}$ and $C_{\mathbb{R}^{2\times 2}}(R(\theta))$ is the centralizer of $R(\theta)$ in $\mathbb{R}^{2\times 2}$?
I think one way to show that $C_{\mathbb{R}^{2\times 2}}(R(\theta)) = \mathbb{R}\cdot SO(2)$ is to take some $U \in U(2)$ with $R(\theta) = U \begin{pmatrix}e^{i\theta}& 0 \\ 0 & e^{-i\theta}\end{pmatrix}U^H$. Then $C_{\mathbb{R}^{2\times 2}}(R(\theta)) = \mathbb{R}^{2\times 2}\cap U\text{diag}_{\mathbb{C}}(2)U^H$ and one has to show that these are exactly the matrices in $\mathbb{R}\cdot SO(2)$.
Solving the equations for real numbers and complex diagonalizing is a bit annoying. Is there a quick elegant way to show this? And is there some general approach for solving such problem?
Remember how we turn $n\times n$ complex matrices into $2n\times2n$ real matrices? In particular, we can represent complex numbers as $2\times 2$ real matrices, $a+bi\leftrightarrow a[\begin{smallmatrix}1&0\\0&1\end{smallmatrix}]+b[\begin{smallmatrix}0&-1\\1&\,0\end{smallmatrix}]$. (In particular, the identity matrix $I$ and $90^\circ$ rotation matrix $J$ represent multiplication-by-$1$ and by-$i$ on $\mathbb{C}$, resp.)
A matrix commutes with $R(\theta)=\cos(\theta)I+\sin(\theta)J$ (assuming $R\ne\pm I$, equivalently $\sin\theta\ne0$) iff it commutes with $J$. Since $J$ is a complex structure, we're talking about linear transformations of $\mathbb{R}^2\cong\mathbb{C}$ as a 1D complex vector space. Well, $M_1(\mathbb{C})$ is basically just $\mathbb{C}$, so we're talking about the matrices $aI+bJ$, or in other words of the form $[\begin{smallmatrix}a&-b\\b&\,a\end{smallmatrix}]$. If we scale by $\sqrt{a^2+b^2}$, it's in $\mathrm{SO}(2)$.
Alternatively, just write out $[\begin{smallmatrix}w&x\\y&z\end{smallmatrix}][\begin{smallmatrix}0&-1\\1&\,0\end{smallmatrix}]=[\begin{smallmatrix}0&-1\\1&\,0\end{smallmatrix}][\begin{smallmatrix}w&x\\y&z\end{smallmatrix}]$ and see where it leads.