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 an application continuous 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 $M(t) = Q(t)\exp(tB)$ for some $B \in \mathscr{M}_n(\mathbb{C})$ and $T$-periodic $Q$. We admit that $B = D + N$ where $D$ is diagonalizable, $N$ is nilpotent, $DN = ND$, and $D = P\Delta P^{-1}$ with $\Delta = \operatorname{Diag}(\lambda_1, \lambda_2, \ldots, \lambda_n)$. Let $Z_1(t), \ldots, Z_n(t)$ be the columns of $M(t)P$. For all $0 \leqslant i \leqslant n-1$, $1 \leqslant k \leqslant n$ and $t \in \mathbb{R}$, we denote by $R_{i,k}(t)$ the $k$-th column of the matrix $\frac{1}{i!} Q(t) N^i P$. Show that for all $k \in \{1, 2, \ldots, n\}$, we have $$Z_k(t) = e^{\lambda_k t} \left(\sum_{i=0}^{n-1} t^i R_{i,k}(t)\right)$$ and verify that the applications $R_{i,k}$ are continuous on $\mathbb{R}$ and $T$-periodic.
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 an application continuous 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 $M(t) = Q(t)\exp(tB)$ for some $B \in \mathscr{M}_n(\mathbb{C})$ and $T$-periodic $Q$. We admit that $B = D + N$ where $D$ is diagonalizable, $N$ is nilpotent, $DN = ND$, and $D = P\Delta P^{-1}$ with $\Delta = \operatorname{Diag}(\lambda_1, \lambda_2, \ldots, \lambda_n)$. Let $Z_1(t), \ldots, Z_n(t)$ be the columns of $M(t)P$.
For all $0 \leqslant i \leqslant n-1$, $1 \leqslant k \leqslant n$ and $t \in \mathbb{R}$, we denote by $R_{i,k}(t)$ the $k$-th column of the matrix $\frac{1}{i!} Q(t) N^i P$. Show that for all $k \in \{1, 2, \ldots, n\}$, we have
$$Z_k(t) = e^{\lambda_k t} \left(\sum_{i=0}^{n-1} t^i R_{i,k}(t)\right)$$
and verify that the applications $R_{i,k}$ are continuous on $\mathbb{R}$ and $T$-periodic.