I am studying optics and I met a strange statement in the section 2.3.4 (page 34) of Fundamentals of Nonlinear Optics by Peter E. Powers and Joseph W. Haus.
The relationship between $\vec{D},\ \vec{E}$ is (non-principal axis system):
$$ \vec{D}=\varepsilon_0 \begin{pmatrix} \varepsilon_{xx}&\varepsilon_{xy}&\varepsilon_{xz}\\ \varepsilon_{yx}&\varepsilon_{yy}&\varepsilon_{yz}\\ \varepsilon_{zx}&\varepsilon_{zy}&\varepsilon_{zz} \end{pmatrix} \vec{E} $$ $$ \overleftrightarrow{\varepsilon}= \begin{pmatrix} \varepsilon_{xx}&\varepsilon_{xy}&\varepsilon_{xz}\\ \varepsilon_{yx}&\varepsilon_{yy}&\varepsilon_{yz}\\ \varepsilon_{zx}&\varepsilon_{zy}&\varepsilon_{zz} \end{pmatrix} $$
They say for lossless media $\overleftrightarrow{\varepsilon}$, dielectric permitivity matrix is symmetric and all elements are real. So this matrix is a Hermitian matrix. I can not understand why the matrix is symmetric.
I know this matrix should be Hermitian because it means the existence of real eigenvalues. When we diagonalize the matrix the three principal axes have the real refractive indices.
How do we know the elements of $\varepsilon_{ij}$ should be real for lossless media?
