# Some Results for Hermitian Matrix

Lemma 1: All eigenvalues of a Hermitian matrix are real.

Proof: Let $A$ be Hermitian and $\lambda$ and $x$ be an eigenvalue and the corresponding eigenvector of $A$. We have

$\lambda^* x^H x = (x^H A^H) x = x^H (A x)=\lambda x^Hx$. Thus we have $\lambda^*=\lambda$ as $x^H x$ is real in general. $\Box$

Lemma 2: Two eigenvectors of a Hermitian matrix are orthogonal
to each other if their eigenvalues are different.

Proof: Let $x$ and $y$ be two eigenvectors of a Hermitian matrix $A$. Let $\lambda$ and $\mu$ be the respective eigenvalues and $\lambda \neq \mu$. We have

$\lambda^*x^Hy=(x^HA^H)y=x^H(Ay)=\mu x^H y$. Thus we have $(\lambda^* - \mu)x^Hy = (\lambda -\mu)x^Hy=0$, where the first equality is from Lemma 1. Since $\lambda\neq \mu$, we have $x^Hy=0$. That is, $x$ is orthogonal to $y$. $\Box$