I've found the eigenvectors, and plugged one of them in. I now need to find the eigenvectors. I understand that I need to find a vector that when I multiply the matrix by the vector, I'll get the zero vector.
$\begin{bmatrix}\omega_x+\sqrt{\omega_z^2+w_x^2} & \omega_x\\\omega_x & -\omega_z+\sqrt{\omega_z^2+w_x^2}\end{bmatrix}$
But it seems that each time I try to to solve this system of equations by substitution, I get that one of the items in the vector is zero. I'd appreciate some help