Let $A \in \mathbb{R}^{n \times n}$, and consider the autonomous linear system

\begin{align*} \dot{x}(t)=Ax(t) \end{align*}

on $\mathbb{R}^n$, with equilibrium point $0 \in \mathbb{R}^n$. If every complex eigenvalue $\lambda \in \mathbb{C}$ of $A$ satisfies $\operatorname{Re}(\lambda)<0$, then $0$ is an asymptotically stable equilibrium. If $A$ has a complex eigenvalue $\lambda \in \mathbb{C}$ satisfying $\operatorname{Re}(\lambda)>0$, then $0$ is not Lyapunov stable.