If $x$ and $y$ are eigenvectors of a Hermitian matrix corresponding to distinct eigenvalues, then $x$ and $y$ are orthogonal with respect to the standard inner product on $\mathbb{C}^{n\times1}$.
What I know that $x$ and $y$ are hermitian if $x = x^*$ and $y = y^*$ and that the standard inner product of $\mathbb{C}^{n\times1}$ is $y^*x$. But I don't know how to relate the orthogonality with respect to the standard inner product $\mathbb{C}^{n\times1}$.