Matrix norm is defined as
\begin{aligned}\|A\|&=\sup \left\{{\frac {\|Ax\|}{\|x\|}}:x\in K^{n}{\text{ with }}x\neq 0\right\}.\end{aligned}
How can we derive the following calculating formulas
\begin{aligned}\|A\|_{1}=\max _{1\leq j\leq n}\sum _{i=1}^{m}|a_{ij}|\end{aligned}
\begin{aligned}\|A\|_{\infty }=\max _{1\leq i\leq m}\sum _{j=1}^{n}|a_{ij}|\end{aligned}
\begin{aligned} \|A\|_{2}=\sigma _{\max }(A)\end{aligned}