Sometimes I find definitions which say that something happens in a neighbourhood of a matrix.
For example a dynamical system generated by $x'=Ax, \ A \in \mathcal{M}(n)$ is structurally stable if there exists a neighbourhood $U$ of $A$ in $\mathcal{M}(n)$ such that $\forall B \in U$ dynamical systems given by $A$ and $B$ are topologically conjugate.
Could you tell me what is a neighbourhood of a matrix? I suppose it depends on the norm in $\mathcal{M}(n)$ - it is the norm of the linear mapping that this matrix gives.
Thank you!
In this case, a neighborhood of $A$ is any set $U$ of matrices for which there is a $\delta$ such that $U$ contains all matrices strictly within a distance $\delta$ of $A$.
There are a number of notions of distance which could be used here but when talking about neighborhoods for finite dimensional matrices the Frobenius norm is probably the right thing, so the distance from $A$ to $B$ is $\|A-B\|_F$ where $\|X\|_F=\sqrt{(\mathop{tr}(X^∗ X))}$
So in your example you know that if you have structural stability, there's some $\delta$ such that you can perturb $A$ by any $X$ with $\|X\|_F < \delta$ and you have a system conjugate to the original one. (Though I confess I don't know what topologically conjugate means.)