What are some usual norms for matrices? I am familiar with norms on vectors and functions, but do there exist norms for spaces of matrices i.e. $A$ some $n \times m$ matrix?
If so, that would that imply matrices also form some sort of vector space?

Let $A=(a_{i,j})\in\mathcal M_{n,m}(\Bbb C)$. These some  norms on $\mathcal M_{n,m}(\Bbb C)$ which are equivalent


*

*$$\|A\|=\sum_{i,j}|a_{i,j}|$$

*$$\|A\|^2=\sum_{i,j}|a_{i,j}|^2$$

*$$\|A\|=\max_{i}\sum_{j}|a_{i,j}|$$

*$$\|A\|=\max_{j}\sum_{i}|a_{i,j}|$$
and of course we can see an analogy with the norms defined on $\Bbb C^{nm}$.

