= Solution
Write a complex <eigenvector> as $v=u+iw$, where $u,w\in\mathbb R^n$. Since $M$ and its <eigenvalue> $\lambda$ are real,
$$
Mu=\lambda u,\qquad Mw=\lambda w.
$$
At least one of $u,w$ is nonzero, and it is a real <eigenvector> for the same <eigenvalue>. More generally every complex <eigenvector> is a complex linear combination of real ones: its real and imaginary parts lie in the real <eigenspace>. Thus \b[each <eigenspace> has a <basis> chosen over the reals]. The conclusion does not assert that every arbitrary complex <eigenvector> is itself real.
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