Let $M \in M_{n}(\mathbf{R})$. We denote by (S) the differential system:
$$(\mathrm{S}) \quad X' = MX$$
where $X$ is a function from the variable $t$ in $\mathbf{R}$ to $\mathbf{R}^{n}$, differentiable on $\mathbf{R}$.

We consider the following assertions:
\begin{itemize}
\item[$\mathbf{A}_{1}$] $\chi_{M}$ is a Hurwitz polynomial;
\item[$\mathbf{A}_{2}$] The solutions of (S) tend to $0_{\mathbf{R}^{n}}$ as $t$ tends to $+\infty$;
\item[$\mathbf{A}_{3}$] There exist $\alpha > 0$ and $k > 0$ such that for every solution $\Phi$ of (S),
$$\forall t \geq 0 \quad : \quad \|\Phi(t)\| \leq k e^{-\alpha t} \|\Phi(0)\|.$$
\end{itemize}

Assume that $M \in M_{n}(\mathbf{R})$ is semi-simple. Prove that the assertions $\mathrm{A}_{1}$, $\mathrm{A}_{2}$ and $\mathrm{A}_{3}$ are equivalent.

Hint: one may start with $A_{3}$ implies $A_{2}$.