Throughout this part, $A$ is a strictly positive matrix in $M _ { n } ( \mathbb { R } )$. Let $\lambda$ be an eigenvalue of $A$ with modulus $\rho ( A )$ and let $x \in \mathbb { C } ^ { n } \backslash \{ 0 \}$ be an eigenvector of $A$ associated with $\lambda$. We define the positive non-zero vector $v _ { 0 }$ by $\left( v _ { 0 } \right) _ { i } = \left| x _ { i } \right|$ for $1 \leqslant i \leqslant n$. a) Show that $A v _ { 0 } \geqslant \rho ( A ) v _ { 0 }$, then that $$A v _ { 0 } = \rho ( A ) v _ { 0 }$$ b) Deduce that $\rho ( A ) > 0$ and $$\forall i \in \llbracket 1 , n \rrbracket , \left( v _ { 0 } \right) _ { i } > 0 .$$ c) Show that $x$ is collinear with $v _ { 0 }$. Deduce that $\lambda = \rho ( A )$.
Throughout this part, $A$ is a strictly positive matrix in $M _ { n } ( \mathbb { R } )$.\\
Let $\lambda$ be an eigenvalue of $A$ with modulus $\rho ( A )$ and let $x \in \mathbb { C } ^ { n } \backslash \{ 0 \}$ be an eigenvector of $A$ associated with $\lambda$. We define the positive non-zero vector $v _ { 0 }$ by $\left( v _ { 0 } \right) _ { i } = \left| x _ { i } \right|$ for $1 \leqslant i \leqslant n$.\\
a) Show that $A v _ { 0 } \geqslant \rho ( A ) v _ { 0 }$, then that
$$A v _ { 0 } = \rho ( A ) v _ { 0 }$$
b) Deduce that $\rho ( A ) > 0$ and
$$\forall i \in \llbracket 1 , n \rrbracket , \left( v _ { 0 } \right) _ { i } > 0 .$$
c) Show that $x$ is collinear with $v _ { 0 }$. Deduce that $\lambda = \rho ( A )$.