I was thinking about $$A =\pmatrix{ \cos \theta & -\lambda \sin \theta \\ \sin \theta & \lambda \cos \theta}$$ and its Eigenvectors and Eigenvalues.
Calculating with the characteristic polynomial gave me $$x_{1,2}= \frac{1}{2}(\lambda \cos \theta + \cos \theta \pm \sqrt{(- \lambda \cos \theta - \cos \theta)^2-4\lambda}).$$
Which is fine, I think. However, I failed calculating the eigenvectors with $(A - \lambda I)v = 0$, and I assume there must be a better solution here.
For $\lambda=1$, we have to counterclockwise rotational $2\times 2$ matrix. And in general, $A$ can be considered as the composition of $\pmatrix{ \cos \theta & - \sin \theta \\ \sin \theta & \cos \theta}$ and $\pmatrix{ 1 & 0 \\ 0 & \lambda}$
The eigenvectors of these matrices are known. Does this tell me anything about the eigenvectors of A?
Any help is much appreciated!