Proof of Theorem ERMCP

\begin{align*} A\conjugate{\vect{x}}&=\conjugate{A}\conjugate{\vect{x}}&&\text{$A$ has real entries}\\ &=\conjugate{A\vect{x}}&&\knowl{./knowls/theorem.MMCC.knowl}{\text{Theorem MMCC}}\\ &=\conjugate{\lambda\vect{x}}&&\knowl{./knowls/definition.EEM.knowl}{\text{Definition EEM}}\\ &=\conjugate{\lambda}\conjugate{\vect{x}}&&\knowl{./knowls/theorem.CRSM.knowl}{\text{Theorem CRSM}} \end{align*}

So $\conjugate{\vect{x}}$ is an eigenvector of $A$ for the eigenvalue $\conjugate{\lambda}$.