# Intuition and Symmetries in Electromagnetism: Eigenstates of Four Antennas
## 2 Problem Formulation  
### Setting Up of the Problem  
Let \(S_{1}\), \(S_{2}\), \(S_{3}\) and \(S_{4}\) be four sources of self-amplitudes \(E_{1}\), \(E_{2}\), \(E_{3}\) and \(E_{4}\) of four antennas as given in Figure 1[13, 14]. Each eigenstate of symmetry (under the condition of the mirror effect) has an amplitude \(\vec{E}_{1}\), \(\vec{E}_{2}\), \(\vec{E}_{3}\) and \(\vec{E}_{4}\), as we are going to explain [13, 14, 19]. **Proper states:**  
Depending on the direction (ox): we can put up a magnetic wall or an electric wall. Similarly for the direction (oy), we can set up a magnetic wall or an electric wall, as described in Figure 1[16, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34]. The combination between the two axes (ox) and (oy) allows us to establish four states of amplitude mirroring, which can be summarized as follows (see Table 1 of [13, 14])[12, 13, 14, 15, 20, 21, 22, 35]: \begin{tabular}{c c c} States & Walls & Amplitudes \\  & & of sources \\
1 & electric (ox)\(\backslash\)magnetic (oy) & 1 1 1 1 \\
2 & magnetic (ox)\(\backslash\)magnetic (oy) & 1 1 1 \\
3 & electric (ox)\(\backslash\)electric (oy) & 1 1 1 - 1 \\
4 & magnetic (ox)\(\backslash\)electric (oy) & 1 1 - 1 1 \\ \end{tabular}   
Similar to the theory of point group symmetry [8], the amplitudes of the sources are defined as the tables (Tables 9.3. and 9.4. of [8]), extensible to other types of symmetries (given in the same reference [8]). Note that the type of these symmetries is explained in the same tables. For example in our case, the orthogonal excitations are of group \(C_{2v}\) (see Figure 9.4. of [8]). The character table for the \(C_{2v}\) symmetry point group is given below (identical to the arrangement of the source amplitudes):  
Figure 1: Four antenna configuration with a combination of electrical and magnetic symmetries [13, 14]. \begin{tabular}{c c c c c c} States & \(C_{2v}\) & \(E\) & \(C_{2}^{z}\) & \(\sigma_{v}(x,z)\) & \(\sigma_{v}^{{}^{\prime}}(y,z)\) \\
1 & \(u_{1}\) & \(1\) & \(1\) & \(1\) & \(1\) \\
2 & \(u_{2}\) & \(1\) & \(1\) & \(-1\) & \(-1\) \\
3 & \(u_{3}\) & \(1\) & \(-1\) & \(1\) & \(-1\) \\
4 & \(u_{4}\) & \(1\) & \(-1\) & \(-1\) & \(1\) \\  & \(h=4\) & \(l_{1}=1\) & \(l_{2}=1\) & \(l_{3}=1\) & \(l_{4}=1\) \\ \end{tabular}   
The covering operations of the group \(C_{2v}\) are: the identity \(E\), \(C_{2}\) rotation around the z-axis, \(\sigma_{v}\) plane of symmetry about the x-z-plane and \(\sigma_{v}^{{}^{\prime}}\) plane of symmetry about the y-z-plane. The symmetry based on the combinations of electric-magnetic walls [13,14] is identical to the point group symmetry theory [8]. Both verify the case of four antennas symmetry [8,13,14]. Let us now return to the amplitudes of the sources in the first table, By normalizing the states, we have:  
\[u_{1}= \begin{pmatrix}\frac{1}{2}\\ \frac{1}{2}\\ \frac{1}{2}\\ \frac{1}{2}\\ \end{pmatrix} ,u_{2}= \begin{pmatrix}\frac{1}{2}\\ \frac{1}{2}\\ -\frac{1}{2}\\ -\frac{1}{2}\\ \end{pmatrix} ,u_{3}= \begin{pmatrix}\frac{1}{2}\\ -\frac{1}{2}\\ \frac{1}{2}\\ -\frac{1}{2}\\ \end{pmatrix} \text{ and }u_{4}= \begin{pmatrix}\frac{1}{2}\\ -\frac{1}{2}\\ -\frac{1}{2}\\ \frac{1}{2}\\ \end{pmatrix} \]  
\(u_{1}\), \(u_{2}\), \(u_{3}\) and \(u_{4}\) are orthonormal vectors. Then, using the theorem of superposition, any state can be written [13,14]:  
\[E=\sum_{i=1}E_{i}V_{i}=\sum_{i=1}\bar{E}_{i}U_{i} \tag{1}\]  
where  
\[V_{1}= \begin{pmatrix}1\\ 0\\ 0\\ 0\\ \end{pmatrix} ,V_{2}= \begin{pmatrix}0\\ 1\\ 0\\ 0\\ \end{pmatrix} ,V_{3}= \begin{pmatrix}0\\ 0\\ 1\\ 0\\ \end{pmatrix} \text{ and }V_{4}= \begin{pmatrix}0\\ 0\\ 0\\ 1\\ \end{pmatrix} \] \[\Rightarrow E=\underbrace{ \begin{pmatrix}1&0&0&0\\ 0&1&0&0\\ 0&0&1&0\\ 0&0&0&1\\ \end{pmatrix} }_{\text{Proper vectors}} \begin{pmatrix}E_{1}\\ E_{2}\\ E_{3}\\ E_{4}\end{pmatrix} \] \[=\underbrace{ \begin{pmatrix}E_{1}\\ E_{2}\\ E_{3}\\ E_{4}\end{pmatrix} }_{\text{Eigen amplitudes of the antennas}},\text{ so},\]  
\[E= \begin{pmatrix}E_{1}\\ E_{2}\\ E_{3}\\ E_{4}\end{pmatrix} =\tfrac{1}{2}\