Let $m,n \in \mathbb{N}$. For each [linear map](/page/Linear%20Map) $T \in \mathcal{L}(\mathbb{R}^m,\mathbb{R}^n)$, let $A(T)=(A(T)_{ij}) \in \mathbb{R}^{n\times m}$ denote the standard matrix of $T$ with respect to the standard bases of $\mathbb{R}^m$ and $\mathbb{R}^n$, so that $T x=A(T)x$ for every $x\in\mathbb{R}^m$. Define

\begin{align*} \|T\|_F := \left(\sum_{i=1}^{n}\sum_{j=1}^{m} A(T)_{ij}^2\right)^{1/2}. \end{align*}

Then the map

\begin{align*} \|\cdot\|_F:\mathcal{L}(\mathbb{R}^m,\mathbb{R}^n)\to[0,\infty) \end{align*}

is a norm on the real [vector space](/page/Vector%20Space) $\mathcal{L}(\mathbb{R}^m,\mathbb{R}^n)$.