I have a discrete dynamical system $(Mat_n(\mathbb R), F)$ where $Mat_n(\mathbb R)$ denotes the space of $n\times n$ matrices with real entries, and $F:Mat_n(\mathbb R)\to Mat_n(\mathbb R)$ is a smooth function. The system evolves as $X_{n+1}=F(X_n)$ starting from an initialization $X_0$.
It turns out that $0$ is a fixed point. However, I am interested to know if $0$ is an attracting fixed point.
The theorems I see so far concern real valued dynamical system and derivatives of $F$ (when $F$ is real valued). My $F$ is firstly matrix valued and also a bit complicated.
What are some strategies I can try? Perhaps the very first thing is to try something like a fixed point theorem, but what is the right notion of derivative whose norm I want to show is upper bounded by 1? Is there a converse, like if the derivative norm (whatever that is) is larger than 1, then 0 is no longer an attracting fixed point?
If there is any comprehensive text that provides methods to prove such things, I would be more than grateful. Thanks a lot.