# What am I understanding wrong about how matrix-norm works?

I'm learning how norm works today. I think I understood how a vector norm works and now I'm trying to understand how the matrix-norm works. I can't understand why the $p=1$ is the "maximum absolute column sum of the matrix". So, here's the definition of the matrix-norm:

I wanted to use a very simple example, the same the professor gave me today $$A = \begin{pmatrix} 1 & 10 & 3 \\ -5 & -1 & 0 \\ 3i & 2 & 0 \end{pmatrix}$$

So, from the definition I thought that I could pick any $x \in K^3$, so I pick $x=(1, 1, 1) \Rightarrow ||x||_{p=1}=3$. Doing $||Ax||$ gets me $(1, 10, 3)$, and the norm from that is $1+10+3=14$ divided per $||x||$, $14/3$. And this makes no sense at all! I should only be able to get 3 results $(9, 13, 3)$. How do I get to these results? From the way I understood I could get unlimited results.

The norm is defined to be the maximum possible value you can get as you range over all $x$. For this particular $x$, you get (EDIT: $25/3$) (taking the $p = 1$ norm), but there are $x$ out there such that you get larger, so there's still only one norm of $A$. If you take the standard basis, this maximum must be attained at one of the basis elements, which is why you only need to check three numbers. So the $p = 1$ norm of your matrix is $13$.
I thought the $p=1$ norm was 13, from the second column? – Clash Apr 23 '12 at 19:23
You're right, sorry, I was taking the $p = 2$ norm. Same argument, though. I'll fix my answer. – user29743 Apr 23 '12 at 19:28
Oh jesus, I just noticed my mistake! In my, case for $x=(1,1,1)$, $14/3$ is actually completely wrong! $||Ax||$ is actually $(14, -6, 3i+2)$ and the norm is $(14+6+5)/3=25/3$. I was doing to the matrix multiplication incorrectly! Thanks for the insight with the basis elements. Using $x=(0,1,0)$ will get me $(10, -1, 2)$ and it's norm is $13$. In fact, I now see why the basis elements will always get me the sum of the absolute values of a column. – Clash Apr 23 '12 at 19:36