Consider the system $\bf{\dot{x}}$=$A_{3x3}$$\bf{x}$ You are given that $A$ is nonsingular and the unstable subspace of $A$ is trivial. Is this information sufficient to infer the stability of the equilibrium at the origin? Explain your reasoning.
Solution
This information is not sufficient because equilibrium point $\bf{x}$=$ 0$ of $\bf{\dot{x}}$=$A$$\bf{x}$ is stable if and only if all eigenvalues of $A$ satisty $Re[\lambda_{i}] \leq 0$ and for every eigenvalue with $Re[\lambda_{i}]=0$ and algebraic multiplicity $q_{i} \geq 2$, $rank(A-\lambda_i I)=n-q_i$ where $n$ is the dimension of $\bf{x}$. I'm not sure if my explanation is right