Let $H$ be a complex [Hilbert space](/page/Hilbert%20Space), let $I_H:H\to H$ be the identity operator on $H$, let $N \in \mathcal{L}(H)$ be normal, and let $N^*\in\mathcal{L}(H)$ denote the Hilbert-space adjoint of $N$. Let $\lambda,\mu \in \mathbb{C}$, and let $x,y \in H$. If $Nx=\lambda x$, $Ny=\mu y$, and $\lambda \ne \mu$, then

\begin{align*} (x,y)_H=0. \end{align*}