In a certain anisotropic conductive material, the relationship between the current density $\vec j$ and the electric field $\vec E$ is given by: $$\vec j = \sigma_0\vec E + \sigma_1\vec n(\vec n\cdot\vec E)$$ where $\vec n$ is a constant unit vector.
i) Calculate the angle between the vectors $\vec j$ and $\vec E$ if the angle between $\vec E$ and $\vec n$ is α
ii) Now assume that $\vec n=\vec e_3$ and define a coordinate transformation ξ = x, η = y, ζ = γz where γ is a constant. For what value of γ does the conductivity tensor component take the form $\sigma_{ab} = \bar \sigmaδ_{ab}$ and what is the value of the constant $\bar\sigma$ in the new coordinate system?
My attempt:
I don't really know if I get it into the simplest possible form but i guess one way of solving i) would be:
$\vec E\cdot\vec j = |\vec E|\cdot|\vec j|\cdot cos(\phi)= \sigma_0\vec E^{2} + \sigma_1\vec n\cdot \vec E(\vec n\cdot\vec E) \implies \phi =arccos(\frac {\sigma_0|\vec E^{2}| + \sigma_1\cdot cos(α)\cdot|\vec E|\cdot cos(α)|\cdot|\vec E|} {|\vec E|\cdot|\vec j|})$
Is this the best way to solve this?
On ii) i am completely lost. What do the coordinate transformations mean? x, y and z are not even in the given expression $\vec j = \sigma_0\vec E + \sigma_1\vec n(\vec n\cdot\vec E)$. I have already found a matrix $\sigma$ that transforms $\vec E$ to $\vec j$. Do they want me to find eigenvectors and eigenvalues? Why?
EDIT:
I might have found the solutionnow but I am not sure so I would really appreciate verification by someone or comment on what is not correct.
Since n is just $\vec e_3$ i get $\vec j = \begin{bmatrix}\sigma_0&0&0\\0&\sigma_0&0\\0&0&\sigma_0+1\end{bmatrix}$.
$\vec E_3(\sigma_0+1)=\sigma_0\cdot\vec E_\zeta$. I don't really know why but I assume $\vec E_zeta = \gamma\cdot\vec E_3$ and therefore $\gamma=1+1/\sigma_0$ and $\bar\sigma = \sigma_0$. Is this correct?
Thanks in advance!