Q: Metric space contractions Consider $(\mathbb{C}^n ,||\cdot||_{1})$ where $||x||_{1} = \sum\limits_{i=1}^{n} |x_{i}|$ and $d_{1}(x,y) = ||x-y||_{1} = \sum\limits_{i=1}^n |x_{i} - y_{i}|$.
I want to find a condition such that when $f: \mathbb{C} \rightarrow \mathbb{C}, f(x) = Ax + b$ is a contracting map.
Note $||\cdot||_{1}$ is norm base $1$ and $A = (a_{ij})$.
I can see that it contracts if $||A|| < 1$. Is their any other conditions I missed that make this a contracting map? Does $A$ have to be anything specific? 

A: If by $\|A\|$ you mean the operator norm of $A$ with respect to this norm, then yes, you're set.  If you mean $\|A\|_1 = \sum_{i,j = 1}^n |a_{ij}|$, then you are incorrect.
