stability of the equilibrium at the $\bf{x}=0$

Question

Consider the system $\bf{\dot{x}}$=$A$$\bf{x}$ in $\mathbb{R^6}$. You are given that the null-space of $A$ has dimension 2 and the stable subspace of $A$ has dimension 2 and that is an eigenvalue of A Is this information sufficient to infer the stability of the equilibrium at the origin?

Solution

We see that $i$ we have $Re[i]=0$ with algebraic multiplicity 2. Thus the $rank(A-iI)=6-2=4$. But I am not sure if this is sufficient to ensure stability of the equilibrium at the origin.

Answer 1

Generally, it is not because there could be an attractor that capture the trajectory outside the equilibrium. If somehow the state of the system gets into this attractor, it is impossible to get out.