We are given two matrices $A$ and $B$ in $\mathcal{M}_n(\mathbf{K})$. We assume that $A$ and $B$ commute. We define an application $$\begin{aligned} g : \mathbf{R} & \rightarrow \mathcal{M}_n(\mathbf{K}) \\ t & \longmapsto g(t) = e^{t(A+B)} e^{-tB} \end{aligned}$$ $\mathbf{2}$ ▷ Show that the application $g$, and the application $f_A$ defined in the preamble, are solutions to the same Cauchy problem. Deduce from this a proof of the relation $$\forall t \in \mathbf{R} \quad e^{t(A+B)} = e^{tA} e^{tB}.$$
We are given two matrices $A$ and $B$ in $\mathcal{M}_n(\mathbf{K})$. We assume that $A$ and $B$ commute.
We define an application
$$\begin{aligned} g : \mathbf{R} & \rightarrow \mathcal{M}_n(\mathbf{K}) \\ t & \longmapsto g(t) = e^{t(A+B)} e^{-tB} \end{aligned}$$
$\mathbf{2}$ ▷ Show that the application $g$, and the application $f_A$ defined in the preamble, are solutions to the same Cauchy problem. Deduce from this a proof of the relation
$$\forall t \in \mathbf{R} \quad e^{t(A+B)} = e^{tA} e^{tB}.$$