Viewing $\begin{bmatrix}\cos \theta \\ \sin \theta\end{bmatrix}$ as an eigenvector of $\frac{\partial}{\partial \theta}$

Question

It's easy to see that

$$\frac{\partial}{\partial \theta}\begin{bmatrix}\cos \theta \\ \sin \theta\end{bmatrix} = \begin{bmatrix}0 & -1 \\ 1 & 0\end{bmatrix}\begin{bmatrix}\cos \theta \\ \sin \theta\end{bmatrix}.$$

So in some sense, the function $\begin{bmatrix}\cos \theta \\ \sin \theta\end{bmatrix}$ is an eigenvector of $\dfrac{\partial}{\partial \theta}$.

Question. Is there an accepted definition of the term "eigenvector" in ring theory or representation theory, generalizing its usual meaning, such that this is actually true?

Answer 1

This is not true since $$\begin{bmatrix} 0 & -1 \\ 1 & 0 \end{bmatrix} $$ is itself a linear operator. Vector $\begin{bmatrix} \cos(\theta) & \sin(\theta) \\ \end{bmatrix}^{T} $ is an eigenvector of the operator $$\partial_{\theta}-\begin{bmatrix} 0 & -1 \\ 1 & 0 \end{bmatrix}$$ With the corresponding eigenvalue zero.