Let $A \in \mathbb{R}^{N \times N}$ be a D4571: Real matrix such that

(i)	\begin{equation} A^T = A \end{equation}
(ii)	$\lambda \in \mathbb{C}$ is a D1207: Complex number
(iii)	$x \in \mathbb{R}^{N \times 1} \setminus \{ \boldsymbol{0} \}$ is a D5200: Real column matrix
(iv)	\begin{equation} A x = \lambda x \end{equation}

Then \begin{equation} \lambda \in \mathbb{R} \end{equation}