Given an autonomous ode $\dot{x}=F(x)$ with $0$ being its equilibrium point, and all eigenvalues of $DF(0)$ have non-zero real parts.
I have learned that if the real parts of eigenvalues are all negative , then the system is asymptotically stable near $0$. Now I guess that if one of the eigenvalues has positive real part then the system is unstable near $0$, but I can't give a proof, can anyone help me?
Many thanks!