Let $n$ and $p$ be two integers greater than or equal to 2. We fix a sequence $\left( A _ { k } \right) _ { k \in \mathbb { N } }$ of matrices of $\mathrm { GL } _ { n } ( \mathbb { C } )$ which is $p$-periodic. The sequence $\left( \Phi _ { k } \right) _ { k \in \mathbb { N } }$ is defined by $\Phi _ { 0 } = I _ { n }$ and $\Phi _ { k + 1 } = A _ { k } \Phi _ { k }$. We denote by $B$ a matrix belonging to $\mathrm { GL } _ { n } ( \mathbb { C } )$ satisfying $B ^ { p } = \Phi _ { p }$. Prove that there exists a unique sequence $\left( P _ { k } \right) _ { k \in \mathbb { N } } \in \left( \mathrm { GL } _ { n } ( \mathbb { C } ) \right) ^ { \mathbb { N } }$, periodic of period $p$, such that $$\forall k \in \mathbb { N } , \quad \Phi _ { k } = P _ { k } B ^ { k }$$
Let $n$ and $p$ be two integers greater than or equal to 2. We fix a sequence $\left( A _ { k } \right) _ { k \in \mathbb { N } }$ of matrices of $\mathrm { GL } _ { n } ( \mathbb { C } )$ which is $p$-periodic. The sequence $\left( \Phi _ { k } \right) _ { k \in \mathbb { N } }$ is defined by $\Phi _ { 0 } = I _ { n }$ and $\Phi _ { k + 1 } = A _ { k } \Phi _ { k }$. We denote by $B$ a matrix belonging to $\mathrm { GL } _ { n } ( \mathbb { C } )$ satisfying $B ^ { p } = \Phi _ { p }$. Prove that there exists a unique sequence $\left( P _ { k } \right) _ { k \in \mathbb { N } } \in \left( \mathrm { GL } _ { n } ( \mathbb { C } ) \right) ^ { \mathbb { N } }$, periodic of period $p$, such that
$$\forall k \in \mathbb { N } , \quad \Phi _ { k } = P _ { k } B ^ { k }$$