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 in $\mathrm { GL } _ { n } ( \mathbb { C } )$ satisfying $B ^ { p } = \Phi _ { p }$, and $\left( P _ { k } \right) _ { k \in \mathbb { N } }$ the unique $p$-periodic sequence in $\left( \mathrm { GL } _ { n } ( \mathbb { C } ) \right) ^ { \mathbb { N } }$ such that $\Phi _ { k } = P _ { k } B ^ { k }$ for all $k \in \mathbb{N}$. Let $\left( Y _ { k } \right) _ { k \in \mathbb { N } }$ be a solution of (III.1). Justify the existence of $M = \max _ { k \in \mathbb { N } } \left\| P _ { k } \right\| _ { 0 }$. Show that for all $k \in \mathbb { N } , \left\| \Phi _ { k } \right\| _ { 0 } \leqslant n M \left\| B ^ { k } \right\| _ { 0 }$.
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 in $\mathrm { GL } _ { n } ( \mathbb { C } )$ satisfying $B ^ { p } = \Phi _ { p }$, and $\left( P _ { k } \right) _ { k \in \mathbb { N } }$ the unique $p$-periodic sequence in $\left( \mathrm { GL } _ { n } ( \mathbb { C } ) \right) ^ { \mathbb { N } }$ such that $\Phi _ { k } = P _ { k } B ^ { k }$ for all $k \in \mathbb{N}$. Let $\left( Y _ { k } \right) _ { k \in \mathbb { N } }$ be a solution of (III.1). Justify the existence of $M = \max _ { k \in \mathbb { N } } \left\| P _ { k } \right\| _ { 0 }$. Show that for all $k \in \mathbb { N } , \left\| \Phi _ { k } \right\| _ { 0 } \leqslant n M \left\| B ^ { k } \right\| _ { 0 }$.