How come you don't just solve the eigenvalues for the matrix that you have denoted by [tex]S_{z'}[/tex]? We are not interested in the eigenvalues of [tex]S_{z'}^2[/tex], but just on the eigenvalues of the spin matrix you have constructed in the fourth line of your attachment. I think you have got a bit messed up with the eigenvalues related to the total spin angular momentum [tex]S^2=S_x^2+S_y^2+S_z^2[/tex] and the component of spin angular momentum related to an arbitrary direction.
I hope this made any sense to you, it's my first post!
