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 $$X'(t) = A(t) X(t) \tag{2}$$ Let $\mathscr{S}$ be the space of solutions in $\mathscr{C}^1(\mathbb{R}, \mathbb{C}^n)$ of (2). Let $(Y_1, Y_2, \ldots, Y_n)$ be a basis of $\mathscr{S}$. For $t \in \mathbb{R}$, let $M(t)$ be the matrix whose columns are $Y_1(t), \ldots, Y_n(t)$, and let $M(t) = Q(t)\exp(tB)$ be the normal form. We admit that there exist two matrices $D$ and $N$ of $\mathscr{M}_n(\mathbb{C})$ such that $D$ is diagonalizable, $N$ is nilpotent and $B = D + N$ and $DN = ND$. There exists a matrix $P \in \mathrm{GL}_n(\mathbb{C})$ and a diagonal matrix $\Delta$ such that $D = P\Delta P^{-1}$. For $t \in \mathbb{R}$, we denote by $Z_1(t), Z_2(t), \ldots, Z_n(t) \in \mathbb{C}^n$ the columns of the matrix $M(t)P$. Show that $(Z_1, Z_2, \ldots, Z_n)$ is a basis of the space $\mathscr{S}$.
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
$$X'(t) = A(t) X(t) \tag{2}$$
Let $\mathscr{S}$ be the space of solutions in $\mathscr{C}^1(\mathbb{R}, \mathbb{C}^n)$ of (2). Let $(Y_1, Y_2, \ldots, Y_n)$ be a basis of $\mathscr{S}$. For $t \in \mathbb{R}$, let $M(t)$ be the matrix whose columns are $Y_1(t), \ldots, Y_n(t)$, and let $M(t) = Q(t)\exp(tB)$ be the normal form. We admit that there exist two matrices $D$ and $N$ of $\mathscr{M}_n(\mathbb{C})$ such that $D$ is diagonalizable, $N$ is nilpotent and $B = D + N$ and $DN = ND$. There exists a matrix $P \in \mathrm{GL}_n(\mathbb{C})$ and a diagonal matrix $\Delta$ such that $D = P\Delta P^{-1}$.
For $t \in \mathbb{R}$, we denote by $Z_1(t), Z_2(t), \ldots, Z_n(t) \in \mathbb{C}^n$ the columns of the matrix $M(t)P$. Show that $(Z_1, Z_2, \ldots, Z_n)$ is a basis of the space $\mathscr{S}$.