Let $A$ be a $2\times 2$ real matrix without eigenvalues, and the roots of its characteristic polynomial be $\alpha+i \beta$ and $\alpha - i \beta$. Show that there exists a basis of $\mathbb{R^2}$ such that $A=\begin{pmatrix} \alpha & \beta \\ - \beta & \alpha \end{pmatrix}$.
I've tried to use the fact that those are its eigenvalues, when considering it a complex matrix, and try to construct a basis to $\mathbb{R^2}$ from the eigenvectors somehow, but I didn't achieve anything.