I'm currently working on a specific problem in a periodic structure, for which I obtain a set of eigenvectors that I would like to plot. It is probably easiest if I give you an explicit example, then I can try and phrase what I want to do. An example of a matrix I would have is
$\left( \begin{array}{ccc} 0.819436 & -0.405293 & -0.405293 \\ -\text{7.072120666862247$\times 10^{-14}$} & -0.707107 & 0.707107 \\ -0.573171 & -0.579429 & -0.579429 \\ \end{array} \right)$
The data goes from left to right, in the sense that the $(1,3)$ entry is the third component of the first eigenvector.
So, how do I currently visualize this:
This is nice in the sense that I can see the direction and the magnitude of each component. However, what is not clear here is the periodic structure of my problem: here component 1 is placed next to 2, which is placed next to three. However in my setup the components are in a circle: 3 is next to 1, which is next to 2, which is next to 3. So my question is if someone can help me figure out an intelligent way to represent the data in a figure in which the magnitude, the direction (positive or negative) and the adjacency is clear. The example I'm giving here is for 3, but I should be able to generalize it to 8. To extend on this: with component I mean the components of each eigenvector. So in this example the 0.8 is the first component of the first eigenvector, -0.4 the second component of the first, and the next -0.4 the third. So in this case I want to somehow visualize that the first component is twice as large as the other two, and in the other direction. But right now it is plotted in a two dimensional line; I somehow want the visual part of the third component to wrap back, so that it is again next to the first one.
Something I thought of myself is perhaps making disks, where each component would take up 1/nth of the disk and the height is the amplitude, and up or down from the equator for the sign. Perhaps this is not the most clear way of doing this, so I'm very open to suggestions.
One note is that I will most likely have to make these figures in Mathematica, but I think most designs should be possible there too, right? And I'd first like to think of designs before actual code, although that might turn out impractical.
Some words on why/for what this is being done: the eigenvectors give insight into the mode structure of a system of coupled resonators, which are of interest in circuit QED. The system is periodic, so the visualization should make it clear that the third component (which refers to the field at the third resonator) is equidistant from the first component and the second component in our setup. Right now, the periodicity is not apparent. What/who for.. currently bachelor thesis, but my supervisor would like to use the visual for actual academic publications, if he is satisfied with it. The way it looks now was too unintuitive in his opinion.
A bit of an ugly drawing for sure, but maybe something like this would be an idea? This would be for a vector of length 4, with the first component zero, the second component, say, $-\sqrt{0.25}$, the third $\sqrt{0.5}$ and the fourth again $-\sqrt{0.25}$. Now the colors are a bit too much here, clearly, but how would you feel about something like this? This would then replace the cubes in the rectangular box, so a cylinder (or disk maybe) for each eigenvector.
