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 $\mathbf{R}$.

Let $T \in M_{n}(\mathbf{R})$. We assume that $M$ is similar to $T$ in $M_{n}(\mathbf{R})$ and we denote by $(\mathrm{S}^{*})$ the differential system
$$(\mathrm{S}^{*}) \quad Y' = TY$$

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}

Let $T \in M_{n}(\mathbf{R})$. We assume that $T$ satisfies the following condition:
$$(\mathrm{C}) \quad \exists \beta \in \mathbf{R}_{+}^{*}, \forall X \in \mathbf{R}^{n} : \langle TX, X \rangle \leq -\beta \|X\|^{2}.$$

Prove that $\mathrm{A}_{3}$ is true with $k = 1$ for every solution $\Phi$ of $(\mathrm{S}^{*})$.

Hint: one may introduce the function $t \mapsto e^{2\beta t} \|\Phi(t)\|^{2}$.