eigenvalues of a Hermitian matrix are real

The eigenvalues of a Hermitian (or self-adjoint) matrix are real.

Proof.

Suppose $\lambda$ is an eigenvalue of the self-adjoint matrix $A$ withnon-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.∎