Eigenvectors of complex matrix I'm working on a problem where I am trying to find the eigenvectors of a pretty complicated matrix, and I am in need of some assistance. The matrix in question is:
$$A =\begin{bmatrix}
        \sin(x) & \cos(x)\cos(y) - i\cos(x)\sin(y)\\
        \cos(x)\cos(y) + i\cos(x)\sin(y) & -\sin(x)\\
        \end{bmatrix}$$
I know that the matrix is Hermitian, so that it is equal to its own conjugate transpose. Moreover, the eigenvalues are $\lambda = \pm 1$, as $A^2 = I$. However, I'm not sure how to use these properties to find the possible eigenvectors (if that would even help), and I would like to avoid doing it by brute force if possible, as it seem unruly. 
Thus far, I have tried to separate the matrix into real and imaginary parts, but that didn't seem to help. I also had the thought to assume diagonalization in an attempt to find the diagonalizing unitary matrix (and, in turn, the eigenvectors), but I don't see that making things much nicer either. Any help would be greatly appreciated.

Note that
$$
A=DQD^{-1},\ D=\pmatrix{1\\ &e^{iy}},\ Q=\pmatrix{\sin x&\cos x\\ \cos x&-\sin x}.
$$
It follows that if $v$ is an eigenvector of $Q$, then $Dv$ is an eigenvector of $A$.
