Solution
= Solution
Let $Mv=\lambda v$ with $v\in\mathbb C^n$ nonzero. For a real <symmetric matrix>, $M^*=M$, where the star denotes the <conjugate transpose>. Consequently $v^*Mv$ equals its <complex conjugate> and is real. The <eigenvalue equation> gives
$$
\boxed{\lambda=\frac{v^*Mv}{v^*v}\in\mathbb R,}
$$
since $v^*v>0$. Allowing a complex <eigenvector> in this argument is essential: reality of an <eigenvalue> must be proved, rather than assumed in advance.