Let $m,n$ be positive integers and let $k\in\{\mathbb{R},\mathbb{C}\}$. Equip $k^n$ and $k^m$ with their Euclidean norms. For $A=(A_{ij})\in M_{m\times n}(k)$, define the Frobenius norm by $\|A\|_F:=\left(\sum_{i=1}^{m}\sum_{j=1}^{n}|A_{ij}|^2\right)^{1/2}$ and define the induced Euclidean operator norm by $\|A\|_{\mathrm{op}}:=\sup\{|Ax|:x\in k^n,\ |x|=1\}$, where $A$ is viewed as a [linear map](/page/Linear%20Map) $A:k^n\to k^m$. Then, for every $A\in M_{m\times n}(k)$,

\begin{align*} \|A\|_{\mathrm{op}}\leq \|A\|_F\leq \sqrt{n}\,\|A\|_{\mathrm{op}}. \end{align*}