Lemma 1: All eigenvalues of a Hermitian matrix are real.
Proof: Let be Hermitian and and be an eigenvalue and the corresponding eigenvector of . We have
. Thus we have as is real in general.
Lemma 2: Two eigenvectors of a Hermitian matrix are orthogonal to each other if their eigenvalues are different.
Proof: Let and be two eigenvectors of a Hermitian matrix . Let and be the respective eigenvalues and . We have
. Thus we have , where the first equality is from Lemma 1. Since , we have . That is, is orthogonal to .