Intereting Posts

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Congruence modulo p
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If $f,g$ are analytic in the unit disk, and $|f|^2+|g|^2=1$, then $f,g$ constant.
Prove that an equation has no elementary solution
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Existence of acyclic coverings for a given sheaf
Unramification of a prime ideal in an order of a finite Galois extension of an algebraic number field

Proof: Let eigenvalue $\lambda \neq 0$ such as

$$\textbf{A}\vec{v} = \lambda\vec{v}$$

$$\Rightarrow (\textbf{A}\vec{v})^\ast = (\lambda\vec{v})^\ast$$

$$\Rightarrow (\vec{v}^\ast\textbf{A}^\ast)=(\lambda^\ast\vec{v}^\ast)$$

Right-multiply both sides by $\color{orangered}{\vec{v}}$$$\Rightarrow (\vec{v}^\ast\textbf{A}^\ast \color{orangered}{\vec{v}} )=(\lambda^\ast\vec{v}^\ast \color{orangered}{\vec{v}} )$$

$$\textbf{A}^\ast=\textbf{A}$$

$$\Rightarrow(\vec{v}^\ast\textbf{A}\vec{v})=(\lambda^\ast\vec{v}^\ast\vec{v})$$

$$\Rightarrow(\vec{v}^\ast\lambda\vec{v}) = (\lambda^\ast\vec{v}^\ast\vec{v})$$

$$\Rightarrow(\lambda\vec{v}^\ast\vec{v}) = (\lambda^\ast\vec{v}^\ast\vec{v})$$

$$\Rightarrow \lambda = \lambda^\ast$$

$$\Rightarrow \lambda\in\mathbb{R}$$

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The only thing missing is to note that, since $v$ is non-zero, you also have $v^*v\neq 0$, therefore you can conclude that $\lambda$ is real.

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