Throughout this part, $A$ is a strictly positive matrix in $M _ { n } ( \mathbb { R } )$. We use the notation from question 17: $w_0$, $v_0$, $F = \left\{ x \in \mathbb { C } ^ { n } \mid { } ^ { t } x w _ { 0 } = 0 \right\}$, and $\mathbb { C } ^ { n } = F \oplus \mathbb { C } v _ { 0 }$. a) We denote by $\psi$ the endomorphism of $F$ defined as the restriction of $\varphi _ { A }$ to $F$. Show that all eigenvalues of $\psi$ have modulus strictly less than $\rho ( A )$. Deduce that $\rho ( A )$ is a simple root of the characteristic polynomial of $A$ and that $$\operatorname { ker } \left( A - \rho ( A ) I _ { n } \right) = \mathbb { C } v _ { 0 }$$ b) Show that if $x \in F , \lim _ { k \rightarrow + \infty } \frac { A ^ { k } x } { \rho ( A ) ^ { k } } = 0$. c) Let $x$ be a positive non-zero vector. Determine the limit of $\frac { A ^ { k } x } { \rho ( A ) ^ { k } }$ when $k$ tends to $+ \infty$.
Throughout this part, $A$ is a strictly positive matrix in $M _ { n } ( \mathbb { R } )$. We use the notation from question 17: $w_0$, $v_0$, $F = \left\{ x \in \mathbb { C } ^ { n } \mid { } ^ { t } x w _ { 0 } = 0 \right\}$, and $\mathbb { C } ^ { n } = F \oplus \mathbb { C } v _ { 0 }$.\\
a) We denote by $\psi$ the endomorphism of $F$ defined as the restriction of $\varphi _ { A }$ to $F$. Show that all eigenvalues of $\psi$ have modulus strictly less than $\rho ( A )$. Deduce that $\rho ( A )$ is a simple root of the characteristic polynomial of $A$ and that
$$\operatorname { ker } \left( A - \rho ( A ) I _ { n } \right) = \mathbb { C } v _ { 0 }$$
b) Show that if $x \in F , \lim _ { k \rightarrow + \infty } \frac { A ^ { k } x } { \rho ( A ) ^ { k } } = 0$.\\
c) Let $x$ be a positive non-zero vector. Determine the limit of $\frac { A ^ { k } x } { \rho ( A ) ^ { k } }$ when $k$ tends to $+ \infty$.