Column-sum and row-sum norms of a 2×2 matrix

Idea. These are the operator norms induced by the $\ell^1$ and $\ell^\infty$ norms on $\mathbb{R}^n$; both have simple closed formulas.

Step 1: absolute values. $|A|=\begin{pmatrix}1&2\\3&4\end{pmatrix}$.

Step 2: columns. Column sums: $1+3=4$ and $2+4=6$. Hence $\|A\|_1=\max(4,6)=6$.

Step 3: rows. Row sums: $1+2=3$ and $3+4=7$. Hence $\|A\|_\infty=\max(3,7)=7$.

Check with a test vector. For $x=(0,1)$, $|x|_1=1$ and $Ax=(-2,4)$ with $|Ax|_1=6$: the bound $\|A\|_1=6$ is attained. For $x=(1,1)$, $|x|_\infty=1$ and $Ax=(-1,7)$ with $|Ax|_\infty=7$: the bound $\|A\|_\infty=7$ is attained.

Remark. Note that $\|A^T\|_1=\|A\|_\infty$ in general. For comparison, the Frobenius norm is $\sqrt{1+4+9+16}=\sqrt{30}\approx5.48$ and the operator (spectral) norm is about $5.12$. The inequality $\|A\|^2\le\|A\|_1\|A\|_\infty$ ($26.2\le42$) is proved in the problem "Schur test" below.