Let $A \in \mathbb{C}^{N \times N}$ be a D6159: Complex square matrix such that

(i)	\begin{equation} A^T = A \end{equation}
(ii)	$\lambda, \mu \in \mathbb{C}$ are each a D1207: Complex number
(iii)	\begin{equation} \lambda \neq \mu \end{equation}
(iv)	$z, w \in \mathbb{C}^{N \times 1} \setminus \{ \boldsymbol{0} \}$ are each a D6214: Complex matrix standard eigenvector sequence
(v)	\begin{equation} A z = \lambda z \end{equation}
(vi)	\begin{equation} A w = \mu w \end{equation}

We have \begin{equation} \lambda z^T w = (\lambda z)^T w = (A z)^T w = z^T A^T w = z^T (A w) = z^T (\mu w) = \mu z^T w \end{equation}