Jones matrices of a mystery device When considering a Jones matrix
$$J=\ \left( \begin{array}{ccc}
\cos\phi & -\sin\phi \\
\sin\phi & \cos\phi \\ \end{array} \right) $$
I understand that the effect of a device described by this Jones matrix on a linearly polarized light is rotation by angle $\phi$. I identified the corresponding device as a Faraday rotator. 
I found eigenvalues to be 
$$\lambda_1=e^{i\phi} \quad \text{and} \quad \lambda_2=e^{-i\phi}$$
and corresponding eigenvectors as 
$$\vec{v}_1=\frac{1}{\sqrt{2}} \left( \begin{array}{ccc}
1 \\
i \\ \end{array} \right)
\quad \text{and} \quad 
\vec{v}_2=\frac{1}{\sqrt{2}}\ \left(\begin{array}{ccc}
1 \\
-i \\ \end{array} \right)\ $$
which have the form of left and right circular polarization states. 
I found Jones matrix in its own diagonal frame to be
$$J'=\ \left( \begin{array}{ccc}
e^{i\phi} & 0 \\
0 & e^{-i\phi} \\ \end{array} \right)\ $$
When asked to explain rotational effect of a device described by the first Jones matrix by considering an incident polarised wave to be a superposition of eigenpolarizations, how can I approach this?
 A: Since the matrix shown is a rotation matrix, which rotates an input vector by the angle $\phi $, it should be the Jones matrix for a half wave plate.
Click for a list of devices and their corresponding Jones matrix representations.
The half wave plate "flips" the normally incident linear polarization across the fast axis; this means that if the input polarization makes an angle $\theta $ with the fast axis, the total rotation will be $2\theta $.
A detailed analysis is found at Newport's polarization page.
A: The vectors are $v_1,v_2$ are circularly polarised light, in which the polarisation vector rotates over time. So the matrix $J'$ describes changing the phase relations between $v_1$ and $v_2$. One way to explain the relation would be just to work out the original vector as a superposition of $v_1$ and $v_2$ and then apply $J'$.
Another way to think about it is that $v_1$ and $v_2$ have a polarisation that rotates over time. Linear polarisation in some direction is like the rotation of $v_1$ and $v_2$ cancelling out at a particular angle. And the matrix $J'$ just makes the rotation cancel out at a different angle.
