Looking for a $2\times2$ real matrix $A$ with $Ax$ a contraction for the supremum norm, and not a contraction for the one norm

Question

I am looking for a $2\times2$ real matrix $A$, such that $ x\longmapsto Ax $ is a contraction considering $\|\cdot\|_\infty$ and a noncontraction considering $\|\cdot\|_1$.

I have now idea how to solve this. Thank you.

Answer 1

If I understand your question correctly, you can take for instance $$ A=\frac17\,\begin{bmatrix} 3&4\\ 3&4\end{bmatrix}. $$ Then $$ \|Ax\|_\infty=\left|\frac{3x_1+4x_2}7\right|\leq\frac{3\|x\|_\infty+4\|x\|_\infty}7\leq \|x\|_\infty. $$ So $A$ is a contraction for $\|\cdot\|_\infty$.

And $$ \|Ax\|_1=2\left|\frac{3x_1+4x_2}7\right|. $$ So if $x=\begin{bmatrix}0\\1\end{bmatrix}$, we have $\|x\|_1=1$ and $\|Ax\|_1=\frac87$.