It can happen that the norm 1 of a matrix and the infinite norm are different?

Question

I have practiced some exercises with these two norms and in all of them I had the same result, until I tried with

$\begin{pmatrix} 5 & -3 & 2 \\ 4 & 8 & -4 \\ 2 & 6 & -1 \\ \end{pmatrix}$

$A_\infty =\frac{128}{9}$ $A_1 =\frac{119}{9}$

And the results were different, what makes me think I did something wrong (although I already reverb like 10 times). So It can happen that the norm 1 of a matrix and the infinite norm are different? Thanks :)

Answer 1

Yes, it can happen: $$A= \begin{pmatrix} 5 & -3 & 2 \\ 4 & 8 & -4 \\ 2 & 6 & -1 \\ \end{pmatrix}$$

In this case,

$\displaystyle\Vert A\Vert_1=\max\{5+4+2;|-3|+8+6;2+|-4|+|-1|\}=17$

and

$\Vert A\Vert_\infty=\max\{5+|-3|+2;4+8+|-4|;2+6+|-1|\}=16$

Answer 2

Yes, it can happen, though I would check the answer again, since $\|A\|_1$ is simply the maximum absolute column sum and $\|A\|_\infty$ is the absolute row sum and I expect them to be integer for your question.