Solution
= Solution
An <eigenvalue> of $A$ is a scalar $\lambda$ for which $Az=\lambda z$ has a nonzero <eigenvector>. Allowing initially $z\in\mathbb C^m$, symmetry of the real matrix gives $A^*=A$, so $z^*Az$ is real. Since $z^*z>0$,
$$
\boxed{\lambda=\frac{z^*Az}{z^*z}\in\mathbb R.}
$$
This proves reality without presuming it when choosing the <eigenvector>. Once $\lambda$ is real, the real or imaginary part of $z$ supplies a nonzero real <eigenvector>.