# What is the relationship between $l_1$-norm in matrices?

It is well-known that the $l_1$-norm defined for $A \in M_n$ by \begin{equation} ||A||_1=\displaystyle \sum _{i,j=1}^n|a_{ij}| \quad (1) \end{equation} is a matrix norm. On the other hand the maximum column sum matrix norm $||.||_1$ is defined on $M_n$ by \begin{equation} ||A||_1=\max _{1\leq j\leq n}\sum _{i=1}^n |a_{ij}| \quad (2) \end{equation} which is induced by the $l_1$-norm on $\mathbb{C}^n$ and hence is a matrix norm. What is the relationship or difference between $(1)$ and $(2)$?

• From what I understand it's just a matter of who you ask. The second is a matrix $p$-norm, the first is kind of just 1 norm if you vectorize it. They're useful in different contexts. As for relationship, you can play around with bounds, basically they're very similar to vector bounds between $\|\cdot\|_\infty$ and $\|\cdot\|_1$. – Y. S. Dec 26 '17 at 6:39
• I think it depends what kind of relationship you are looking for... FYI, every norm is equivalent in finite dimensional vector space. – induction601 Dec 26 '17 at 7:19