Let $n\in \mathbb N$ with $n\geq 1$, let $A\in \mathbb R^{n\times n}$, and let $A_{\mathbb C}: \mathbb C^n\to \mathbb C^n$ denote the complex-[linear map](/page/Linear%20Map) represented by the same matrix entries as $A$. Let $|\cdot|_{\mathbb R^n}$ be the Euclidean norm on $\mathbb R^n$. Consider the linear autonomous system on $\mathbb R^n$ given by

\begin{align*} \frac{dx}{dt}=Ax. \end{align*}

Its solution semigroup is the family of linear maps $e^{tA}:\mathbb R^n\to\mathbb R^n$ for $t\geq 0$. The equilibrium $0\in \mathbb R^n$ is Lyapunov stable, meaning that for every $\varepsilon>0$ there exists $\delta>0$ such that $|x_0|_{\mathbb R^n}<\delta$ implies $|e^{tA}x_0|_{\mathbb R^n}<\varepsilon$ for every $t\geq 0$, if and only if every complex eigenvalue $\lambda\in \mathbb C$ of $A_{\mathbb C}$ satisfies $\operatorname{Re}(\lambda)\leq 0$, and every Jordan block in a [Jordan normal form](/theorems/864) of $A_{\mathbb C}$ over $\mathbb C$ corresponding to an eigenvalue $\lambda$ with $\operatorname{Re}(\lambda)=0$ has size $1$.