Exercise 3.5.2

From the equation we have

To find the matrix with smallest “sum norm” $\parallel A{\text{∥}}_S=\left|a\right|+\left|b\right|+\left|c\right|+\left|d\right|$, since $a,b$ and $c,d$ are kind of independent of each other in terms of constraints. So we can find the minimum of $\left|a\right|+\left|b\right|$ and $\left|c\right|+\left|d\right|$ separately.

First notice, that since $3c+4d=0$, according to the figure I.16, this line will always cross the diamond generated by $\left|c\right|+\left|d\right|\le t$ regardless of the value of $t$. so the minimum value of $\left|c\right|+\left|d\right|$ is 0 when $c=d=0$.

For $3a+4b=1$, this line has intercepts with $a$-axis at $\frac{1}{3}$, and intercept with $b$-axis at $\frac{1}{4}$. Imagine the diamond generated by $\left|a\right|+\left|b\right|\le t$ gradually expands as $t$ increases from 0. It will first touch the intercept with $b$-axis at $\frac{1}{4}$, so the minimum of $\left|a\right|+\left|b\right|=\frac{1}{4}$ when $a=0,b=\frac{1}{4}$.