Let $T > 0$ be a real number and $n \in \mathbb{N}^*$ be a natural integer. Let $A : \mathbb{R} \rightarrow \mathscr{M}_n(\mathbb{C})$ be a continuous application on $\mathbb{R}$ and $T$-periodic. We consider the differential system $$X'(t) = A(t) X(t) \tag{2}$$ where $X$ is a function from $\mathbb{R}$ to $\mathbb{C}^n$, of class $\mathscr{C}^1$ on $\mathbb{R}$. Show that there exists $\mu \in \mathbb{C}^*$ and a non-zero solution $Y \in \mathscr{C}^1(\mathbb{R}, \mathbb{C}^n)$ of (2) such that $$\forall t \in \mathbb{R}, \quad Y(t+T) = \mu Y(t)$$
Let $T > 0$ be a real number and $n \in \mathbb{N}^*$ be a natural integer. Let $A : \mathbb{R} \rightarrow \mathscr{M}_n(\mathbb{C})$ be a continuous application on $\mathbb{R}$ and $T$-periodic. We consider the differential system
$$X'(t) = A(t) X(t) \tag{2}$$
where $X$ is a function from $\mathbb{R}$ to $\mathbb{C}^n$, of class $\mathscr{C}^1$ on $\mathbb{R}$.
Show that there exists $\mu \in \mathbb{C}^*$ and a non-zero solution $Y \in \mathscr{C}^1(\mathbb{R}, \mathbb{C}^n)$ of (2) such that
$$\forall t \in \mathbb{R}, \quad Y(t+T) = \mu Y(t)$$