I have the following matrix
$$
\begin{pmatrix}
cos\phi & -sin\phi \\
sin\phi & cos\phi &
\end{pmatrix}
\in \mathbb C^{2 \times 2}, \phi\in \mathbb R$$
and I am trying to find eigenvalues and eigenvectors of this function. In case $\phi$ is a multiple of $\pi$, it is trivial, since I end up with an identity function (or the negative version). But in the other case, I am having troubles. I know that I can compute the roots of the characteristic polynomial to find eigenvalues. In this case, I end up with: $$x_{1,2} = cos\phi \pm \sqrt{cos^2\phi-1}$$
So far so good. Then I went on to find the subspace of eigenvectors, by finding $ker(f-x*id_V)$. Chosing one of the eigenvectors I end up with a system of linear equations:
$$\sqrt{cos^2\phi-1} * x_1 – sin\phi * x_2 = 0$$
$$sin\phi*x_1+\sqrt{cos^2\phi-1} * x_2 = 0$$
Is this the right way of doing it? If so, how to I correctly solve this homogenous system? I cant seem to solve it via row operations.
Best Answer
First, you can rewrite that eigenvalue as
$$ \newcommand{\ii}{\mathbf i} \cos (\varphi) \pm \ii \sin \varphi $$ The corresponding eigenvector (for the positive choice) is $\begin{bmatrix} \ii \\ 1\end{bmatrix}$.