Throughout this part, $A$ is a strictly positive matrix in $M _ { n } ( \mathbb { R } )$. By applying the previous results to the matrix ${ } ^ { t } A$, we obtain the existence of $w _ { 0 } \in \mathbb { R } ^ { n }$, whose all components are strictly positive, such that ${ } ^ { t } A w _ { 0 } = \rho ( A ) w _ { 0 }$. We set $$F = \left\{ x \in \mathbb { C } ^ { n } \mid { } ^ { t } x w _ { 0 } = 0 \right\}$$ a) Show that $F$ is a vector subspace of $\mathbb { C } ^ { n }$ stable by $\varphi _ { A }$, and that $$\mathbb { C } ^ { n } = F \oplus \mathbb { C } v _ { 0 }$$ b) Show that if $v$ is an eigenvector of $A$ associated with an eigenvalue $\mu \neq \rho ( A )$, then $v \in F$. Deduce property (iii): if $v$ is an eigenvector of $A$ whose all components are positive, then $v \in \operatorname { ker } ( A - \rho ( A ) I _ { n } )$.
Throughout this part, $A$ is a strictly positive matrix in $M _ { n } ( \mathbb { R } )$.\\
By applying the previous results to the matrix ${ } ^ { t } A$, we obtain the existence of $w _ { 0 } \in \mathbb { R } ^ { n }$, whose all components are strictly positive, such that ${ } ^ { t } A w _ { 0 } = \rho ( A ) w _ { 0 }$. We set
$$F = \left\{ x \in \mathbb { C } ^ { n } \mid { } ^ { t } x w _ { 0 } = 0 \right\}$$
a) Show that $F$ is a vector subspace of $\mathbb { C } ^ { n }$ stable by $\varphi _ { A }$, and that
$$\mathbb { C } ^ { n } = F \oplus \mathbb { C } v _ { 0 }$$
b) Show that if $v$ is an eigenvector of $A$ associated with an eigenvalue $\mu \neq \rho ( A )$, then $v \in F$. Deduce property (iii): if $v$ is an eigenvector of $A$ whose all components are positive, then $v \in \operatorname { ker } ( A - \rho ( A ) I _ { n } )$.