Let $m \in \mathbb{N}$, let $I_m \in \mathbb{R}^{m \times m}$ denote the identity matrix, and let $Q \in \mathbb{R}^{m \times m}$ be an [orthogonal matrix](/page/Orthogonal%20Matrix), meaning that $Q^\top Q=I_m$. Regard $Q$ as the [linear map](/page/Linear%20Map) $Q: \mathbb{R}^m \to \mathbb{R}^m$ represented by this matrix in the standard basis. Then

\begin{align*} \|Q\|_F=\sqrt{m}. \end{align*}