This question is from Principles of Quantum Mechanics by R. Shankar.
Given the operator (matrix) $\Omega$ with eigenvalues $e^{i\theta}$ and $e^{-i\theta}$ , I am told to find the corresponding eigenvectors.
I think these equations are relevant:
$$\begin{align}\Omega &= \begin{bmatrix}\cos{\theta} & \sin{\theta} \\ -\sin{\theta} & \cos{\theta} \end{bmatrix} \\ \Omega \begin{bmatrix} x_1 \\ x_2 \end{bmatrix} &= \begin{bmatrix} x_1 \cos{\theta} + x_2 \sin{\theta} \\ -x_1 \sin{\theta} + x_2 \cos{\theta} \end{bmatrix} \\ e^{i\theta} \begin{bmatrix} x_1 \\ x_2 \end{bmatrix} &= \begin{bmatrix} x_1 \cos{\theta} + x_1 i\sin{\theta} \\ x_2 \cos{\theta} + x_2 i\sin{\theta} \end{bmatrix}\end{align}$$
I let the matrix operate on the generic vector $(x_1, x_2)^T$ and demand that the resulting vector is equal to $(e^{i\theta}x_1, e^{i\theta}x_2)^T$ . From this i get the condition that $x_2 = ix_2$ and $x_1 = -ix_2$ , which implies that $x_1 = x_2 = 0$ . Am i missing something crucial?