Proof. Suppose $\lambda$ is an eigenvalue of the self-adjoint matrix $A$ with non-zero eigenvector $v$ Then $Av = \lambda v$

$$ \lambda ^{\ast }v^{H}v=\left( \lambda v\right) ^{H}v=\left( Av\right) ^{H}v=v^{H}A^{H}v=v^{H}Av=v^{H}\lambda v=\lambda v^{H}v $$

Since $v$ is non-zero by assumption, $v^H v$ is non-zero as well and so $\lambda^{*}=\lambda$ meaning that $\lambda$ is real.