Suppose we have the $\textit{mode functions}$ given as $\textbf{u}_m(\textbf{x})$ which are defined by the following eigenvalue equation:
\begin{equation} \nabla^2\textbf{u}_m(\textbf{x}) = -k_m^2\textbf{u}_m(\textbf{x}) \end{equation}
Now, we suppose that this eigenvalue equation expresses a Hermitian eigenvalue problem, meaning that the mode functions that correspond to unequal eigenvalues must be orthogonal. Why is this the case? How can we say that the mode functions corresponding to unequal eigevalues are orthogonal and how can I understand this?
