I am analyzing the polarization state of a monochromatic, coherent light source, for which I know the Jones vector of the polarization,
$$
\mathbf E
=\begin{pmatrix}E_x\\E_y\end{pmatrix}
=\begin{pmatrix}|E_x|e^{i\varphi_x}\\|E_y|e^{i\varphi_y}\end{pmatrix},
$$
and I would like to expand it in terms of a major and a minor axis of ellipticity, i.e. in the form
$$
\mathbf E=
e^{i\varphi}\left(
A \hat{\mathbf u} + i B\hat{\mathbf v}
\right)
=
e^{i\varphi}\left(
A
\begin{pmatrix}\cos(\theta)\\ \sin(\theta)\end{pmatrix}
+ i B
\begin{pmatrix}-\sin(\theta)\\ \cos(\theta)\end{pmatrix}
\right),
$$
or as shown graphically as follows:
Wikipedia provides a multi-step procedure going through the Stokes parameters, but I'm thinking there is surely a cleaner and more direct way to get $A$, $B$, $\hat{\mathbf u}$, $\hat{\mathbf v}$, $\theta$, and the components $A \hat{\mathbf u}$ and $B\hat{\mathbf v}$, from $E_x$ and $E_y$, and it's not particularly obvious from the search results I can find. What's the cleanest way to do this?
To be clear: what I think is lacking from the existing resources, and what the question is directly asking for, is an explicit set of connections, as simple as possible, for the named parameters (all of $A$, $B$, $\hat{\mathbf u}$, $\hat{\mathbf v}$, $\theta$, and the components $A \hat{\mathbf u}$ and $B\hat{\mathbf v}$), in terms of the Cartesian components $E_x$ and $E_y$. Schemes that simply send to some other set of complex manipulations are already available from Wikipedia and are not what the question is asking for.
Best Answer
Let me try a second time. I use https://math.stackexchange.com/questions/1204131/converting-a-rotated-ellipse-in-parametric-form-to-cartesian-form as a resource.
Prior Manipulations
The physical electric field is
$$\mathbf{E}_{phys} = Re\left[\mathbf{E} e^{i\omega t}\right] = Re\left[\begin{pmatrix}|E_x|e^{i\varphi_x}\\|E_y|e^{i\varphi_y}\end{pmatrix}e^{i\omega t}\right] = \begin{pmatrix}|E_x|\cos(\omega t + \varphi_x)\\|E_y| \cos(\omega t + \varphi_y)\end{pmatrix} = \begin{pmatrix}|E_x|\left[\cos(\omega t)\cos(\varphi_x) - \sin(\omega t) \sin(\varphi_x) \right]\\|E_y| \left[\cos(\omega t)\cos(\varphi_y) - \sin(\omega t) \sin(\varphi_y) \right]\end{pmatrix}$$
This is a parametric equation for an ellipse, which is traced by the electric field.
Major axes angle
Let
$$a=|E_x|\cos(\varphi_x)$$
$$b=|E_x|\sin(\varphi_x)$$
$$c=|E_y|\cos(\varphi_y)$$
$$d=|E_y|\sin(\varphi_y)$$
Then by comparison with the linked math.SE question the accepted answer states that the major and minor axes point on the ellipse (which is centred on the origin) fulfill
$$\omega t={1\over2}\arctan{2(ab+cd)\over(a^2+c^2)-(b^2+d^2)}+{k\pi\over2}\qquad(0\leq k\leq3)\ .$$
Expansion required by the OP
Indeed this quantity is the angle $\theta$ in the expansion appearing in the OPs question. However depending on which value for $k$ is chosen in may be $\pi/2 - \theta$, a case distinction depending on the sector of the arctan is necessary.
This of course also yields $\hat{u}$ and $\hat{v}$, so the expansion can now be easily obtained by projecting the Jones vector this basis.
The formulae may be along, but constitute a close form solution to the problem up to the case distinction of choosing $k$. I do not see how a simpler solution can exist, since the formula give for the angle given does not seem algebraically reducible.