In this video in 5:15 there's a proof that every hermitian matrix has real eigenvalues. I don't understand the step: $<\vec x, L\vec y>=<L^*\vec x,\vec y>$.
I know that I can pull out scalars from an inner product and those scalars come out as complex adjugates if they are the 2nd member of the inner product, but don't understand how that applies to matrices.
With inner product do you mean $\langle x, y \rangle = x^T y $ ?
Because in that case
$ \langle Ax, y \rangle = (Ax)^T y = x^T A^T y = x^T (A^* y) = \langle x, A^* y \rangle$