Throughout this part, $A$ is a strictly positive matrix in $M _ { n } ( \mathbb { R } )$. Suppose that there exist a non-negative real $\mu$ and a positive non-zero vector $w$ such that $A w \geqslant \mu w$. a) Show that for all natural integer $k , A ^ { k } w \geqslant \mu ^ { k } w$. Deduce that $\rho ( A ) \geqslant \mu$. b) Show that if $A w > \mu w$, then $\rho ( A ) > \mu$. c) We now suppose that in the system of inequalities $A w \geqslant \mu w$, the $k$-th inequality is strict, that is $$\sum _ { j = 1 } ^ { n } a _ { k j } w _ { j } > \mu w _ { k } .$$ Show that there exists $\epsilon > 0$ such that, by setting $w _ { j } ^ { \prime } = w _ { j }$ if $j \neq k$ and $w _ { k } ^ { \prime } = w _ { k } + \epsilon$, we have $A w ^ { \prime } > \mu w ^ { \prime }$. Deduce that $\rho ( A ) > \mu$.
Throughout this part, $A$ is a strictly positive matrix in $M _ { n } ( \mathbb { R } )$.\\
Suppose that there exist a non-negative real $\mu$ and a positive non-zero vector $w$ such that $A w \geqslant \mu w$.\\
a) Show that for all natural integer $k , A ^ { k } w \geqslant \mu ^ { k } w$. Deduce that $\rho ( A ) \geqslant \mu$.\\
b) Show that if $A w > \mu w$, then $\rho ( A ) > \mu$.\\
c) We now suppose that in the system of inequalities $A w \geqslant \mu w$, the $k$-th inequality is strict, that is
$$\sum _ { j = 1 } ^ { n } a _ { k j } w _ { j } > \mu w _ { k } .$$
Show that there exists $\epsilon > 0$ such that, by setting $w _ { j } ^ { \prime } = w _ { j }$ if $j \neq k$ and $w _ { k } ^ { \prime } = w _ { k } + \epsilon$, we have $A w ^ { \prime } > \mu w ^ { \prime }$. Deduce that $\rho ( A ) > \mu$.