Inner Product

The linearized Boltzmann collision operator computed by electron_electron() is not Hermitian. However, we can define the auxiliary matrix

\[ L^\text{(s)}_{ij} = f^{(0)}_i (1 - f^{(0)}_i) \frac{dV_i} \mathbf{L}_{ij}\]

which is symmetric. By defining the inner product as

\[ \langle a | b \rangle \equiv \frac{1}{V}\sum_\mathbf{k} f^{(0)}(\epsilon_\mathbf{k}) (1 - f^{(0)}(\epsilon_\mathbf{k})) a^_\k b_\k\]

then the matrix computed by electron_electron() is Hermitian with respect to the above inner product.

@docs Ludwig.inner_product