TITLE: Hyperbolic Systems ODE
QUESTION [6 upvotes]: Let $M_n$ the set of matrices of order $n \times n$ identified with $\mathbb{R^{n^2}}$ e $S=\{A \in M_n ; x'=Ax$ is hyperbolic$\}$. Show that $S$ is open and dense $M_n$.

REPLY [3 votes]: System
$$
\dot x=Ax
$$
is hyperbolic if and only if there are no eigenvalues of $A$ with zero real part. 
First, let me show that $S$ is open. Since eigenvalues are the roots of the characteristic polynomial, and the roots are continuous functions of the coefficients, then if for some $A$ there are no roots with zero real part, for some close $B$ this will be also true, and hence $S$ is open.
To show that $S$ is dense, take any non-hyperbolic matrix $A$ and assume that it is in Jordan Normal Form. Now consider $B=A+\epsilon I$. $B$ is hyperbolic and $\epsilon$-close to $A$, hence $S$ is dense.