Eigenvalues and eigenvectors of the momentum current density dyadic

[m3mb3r]

Homework Statement

What are the eigenvalues and eigenvectors of the momentum
current density dyadic $$\overleftrightarrow{T}$$ (Maxwell tensor)? Then make use of these eigenvalues in finding the determinant of $$\overleftrightarrow{T}$$ and the trace of $$\overleftrightarrow{T}^2$$

Homework Equations

$$\overleftrightarrow{T}=\overleftrightarrow{1}U-\frac{1}{4\pi}(\overrightarrow{E}\overrightarrow{E}+\overrightarrow{B}\overrightarrow{B})$$
$$U=\frac{1}{8\pi}(\overrightarrow{|E}|^{2}+|\overrightarrow{B|^{2}})$$

The Attempt at a Solution

$$\overleftrightarrow{T}x=\lambda x$$

$$det(\overleftrightarrow{T}-\overleftrightarrow{1}\lambda)=0$$

$$det\left(\begin{array}{ccc} T_{11}-\lambda & T_{12} & T_{13}\\ T_{21} & T_{22}-\lambda & T_{23}\\ T31 & T_{32} & T_{33}-\lambda\end{array}\right)=0$$

Since the tensor is symmetric, we have $$T_{ij}=T_{ji}$$, (after simplifying) our equation become:

$$-\lambda^{3}+(T_{11}+T_{22}+T_{33})\lambda^{2}+(T_{12}^{2}+T_{23}^{2}+T_{31}^{2}-T_{11}T_{22}-T_{22}T_{33}-T_{33}T_{11})\lambda+T_{11}T_{22}T_{33}+2T_{12}T_{23}T_{31}-T_{11}T_{23}^{2}-T_{22}T_{31}^{2}-T_{33}T_{12}^{2}=0$$

with $$T_{ij}=\delta_{ij}U-\frac{1}{4\pi}(E_{i}E_{j}+B_{i}B_{j})$$ (from the definition of $$\overleftrightarrow{T})$$

The problem is that as I expand $$T_{ij}$$ the equation become more and more complicated. Am I doing the right thing here?
And later how to find the determinant of $$\overleftrightarrow{T}$$ and the trace of $$\overleftrightarrow{T}^2$$ after obtaining the eigenvalue?

Thanks

 

[m3mb3r]

Anybody got any idea?