Let $n$ and $p$ be two integers greater than or equal to 2. We fix throughout this part a sequence $\left( A _ { k } \right) _ { k \in \mathbb { N } }$ of matrices of $\mathrm { GL } _ { n } ( \mathbb { C } )$ which we assume to be $p$-periodic, that is such that $\forall k \in \mathbb { N } , A _ { k + p } = A _ { k }$. We denote by $\operatorname { Sol }$ (III.1) the set of sequences $\left( Y _ { k } \right) _ { k \in \mathbb { N } }$ of vectors of $\mathbb { C } ^ { n }$ satisfying the recurrence relation $$\forall k \in \mathbb { N } , \quad Y _ { k + 1 } = A _ { k } Y _ { k }$$ Justify that we define a sequence $\left( \Phi _ { k } \right) _ { k \in \mathbb { N } }$ of matrices of $\mathrm { GL } _ { n } ( \mathbb { C } )$ by setting $\left\{ \begin{array} { l } \Phi _ { 0 } = I _ { n } \\ \Phi _ { k + 1 } = A _ { k } \Phi _ { k } \quad \forall k \in \mathbb { N } \end{array} \right.$ and that $\left( Y _ { k } \right) _ { k \in \mathbb { N } } \in \operatorname { Sol }$ (III.1) if and only if $\forall k \in \mathbb { N } , Y _ { k } = \Phi _ { k } Y _ { 0 }$.
Let $n$ and $p$ be two integers greater than or equal to 2. We fix throughout this part a sequence $\left( A _ { k } \right) _ { k \in \mathbb { N } }$ of matrices of $\mathrm { GL } _ { n } ( \mathbb { C } )$ which we assume to be $p$-periodic, that is such that $\forall k \in \mathbb { N } , A _ { k + p } = A _ { k }$. We denote by $\operatorname { Sol }$ (III.1) the set of sequences $\left( Y _ { k } \right) _ { k \in \mathbb { N } }$ of vectors of $\mathbb { C } ^ { n }$ satisfying the recurrence relation
$$\forall k \in \mathbb { N } , \quad Y _ { k + 1 } = A _ { k } Y _ { k }$$
Justify that we define a sequence $\left( \Phi _ { k } \right) _ { k \in \mathbb { N } }$ of matrices of $\mathrm { GL } _ { n } ( \mathbb { C } )$ by setting $\left\{ \begin{array} { l } \Phi _ { 0 } = I _ { n } \\ \Phi _ { k + 1 } = A _ { k } \Phi _ { k } \quad \forall k \in \mathbb { N } \end{array} \right.$ and that $\left( Y _ { k } \right) _ { k \in \mathbb { N } } \in \operatorname { Sol }$ (III.1) if and only if $\forall k \in \mathbb { N } , Y _ { k } = \Phi _ { k } Y _ { 0 }$.