We consider in this question a matrix $A \in M _ { n } ( \mathbb { C } )$ that is upper triangular, $$A = \left( \begin{array} { c c c c c }
a _ { 1,1 } & a _ { 1,2 } & \ldots & \ldots & a _ { 1 , n } \\
0 & a _ { 2,2 } & \ldots & \ldots & a _ { 2 , n } \\
\vdots & \ddots & \ddots & & \vdots \\
\vdots & & \ddots & \ddots & \vdots \\
0 & \ldots & \ldots & 0 & a _ { n , n }
\end{array} \right)$$ We assume that $$\forall i \in \llbracket 1 , n \rrbracket , \left| a _ { i , i } \right| < 1$$ For any real $b > 0$, we set $P _ { b } = \operatorname { diag } \left( 1 , b , b ^ { 2 } , \ldots , b ^ { n - 1 } \right) \in M _ { n } ( \mathbb { R } )$. a) Compute $P _ { b } ^ { - 1 } A P _ { b }$. What happens when $b$ tends to 0? b) Show that there exists $b > 0$ such that $$\left\| P _ { b } ^ { - 1 } A P _ { b } \right\| < 1$$ c) Deduce that the sequence $\left( A ^ { k } \right) _ { k \in \mathbb { N } ^ { * } }$ converges to 0.
We consider in this question a matrix $A \in M _ { n } ( \mathbb { C } )$ that is upper triangular,
$$A = \left( \begin{array} { c c c c c }
a _ { 1,1 } & a _ { 1,2 } & \ldots & \ldots & a _ { 1 , n } \\
0 & a _ { 2,2 } & \ldots & \ldots & a _ { 2 , n } \\
\vdots & \ddots & \ddots & & \vdots \\
\vdots & & \ddots & \ddots & \vdots \\
0 & \ldots & \ldots & 0 & a _ { n , n }
\end{array} \right)$$
We assume that
$$\forall i \in \llbracket 1 , n \rrbracket , \left| a _ { i , i } \right| < 1$$
For any real $b > 0$, we set $P _ { b } = \operatorname { diag } \left( 1 , b , b ^ { 2 } , \ldots , b ^ { n - 1 } \right) \in M _ { n } ( \mathbb { R } )$.\\
a) Compute $P _ { b } ^ { - 1 } A P _ { b }$. What happens when $b$ tends to 0?\\
b) Show that there exists $b > 0$ such that
$$\left\| P _ { b } ^ { - 1 } A P _ { b } \right\| < 1$$
c) Deduce that the sequence $\left( A ^ { k } \right) _ { k \in \mathbb { N } ^ { * } }$ converges to 0.