Lemma 1: All eigenvalues of a Hermitian matrix are real.
Proof: Let $latex A$ be Hermitian and $latex \lambda$ and $latex x$ be an eigenvalue and the corresponding eigenvector of $latex A$. We have
$latex \lambda^* x^H x = (x^H A^H) x = x^H (A x)=\lambda x^Hx$. Thus we have $latex \lambda^*=\lambda$ as $latex x^H x$ is real in general. $latex \Box$
Lemma 2: Two eigenvectors of a Hermitian matrix are orthogonal to each other if their eigenvalues are different.
Proof: Let $latex x$ and $latex y$ be two eigenvectors of a Hermitian matrix $latex A$. Let $latex \lambda$ and $latex \mu$ be the respective eigenvalues and $latex \lambda \neq \mu$. We have
$latex \lambda^*x^Hy=(x^HA^H)y=x^H(Ay)=\mu x^H y$. Thus we have $latex (\lambda^* – \mu)x^Hy = (\lambda -\mu)x^Hy=0$, where the first equality is from Lemma 1. Since $latex \lambda\neq \mu$, we have $latex x^Hy=0$. That is, $latex x$ is orthogonal to $latex y$. $latex \Box$