Distinct eigenvalues of a Hermitian matrix have orthogonal eigenvectors.

Theorem: Eigenvectors corresponding to distinct eigenvalues of a Hermitian matrix are orthogonal. If $Av_1 = \lambda_1 v_1$ and $Av_2 = \lambda_2 v_2$ with $\lambda_1 \neq \lambda_2$, then $v_1^*v_2 = 0$.