Let $M \in \mathscr{M}_d\left(\mathbb{R}_{+}\right)$. We assume that the matrix $M$ has an eigenvalue $\lambda > 0$ and that there exists $h \in \mathscr{M}_{d,1}\left(\mathbb{R}_{+}^{*}\right)$ a column vector such that: $$Mh = \lambda h.$$ We also assume that there exist $\nu \in \mathscr{P}$ and $c > 0$ such that for all $i,j \in \{1,\ldots,d\}$, $$M_{i,j} \geqslant c\nu_j.$$ We introduce the matrix $P \in \mathscr{M}_d\left(\mathbb{R}_{+}\right)$ defined for $1 \leqslant i,j \leqslant d$ by $$P_{i,j} = \frac{M_{i,j} h_j}{\lambda h_i}.$$ (a) Show that there exist $\mu \in \mathscr{P}$, $C > 0$ and $\gamma \in [0,1[$, such that $\mu P = \mu$ and for all $n \geqslant 0$, $$\sum_{i=1}^{d} \sum_{j=1}^{d} \left| \lambda^{-n} \left(M^n\right)_{i,j} - h_i \frac{\mu_j}{h_j} \right| \leqslant C\gamma^n.$$ (b) Prove that there exists a unique $\pi \in \mathscr{P}$ such that $\pi M = \lambda \pi$.
Let $M \in \mathscr{M}_d\left(\mathbb{R}_{+}\right)$. We assume that the matrix $M$ has an eigenvalue $\lambda > 0$ and that there exists $h \in \mathscr{M}_{d,1}\left(\mathbb{R}_{+}^{*}\right)$ a column vector such that:
$$Mh = \lambda h.$$
We also assume that there exist $\nu \in \mathscr{P}$ and $c > 0$ such that for all $i,j \in \{1,\ldots,d\}$,
$$M_{i,j} \geqslant c\nu_j.$$
We introduce the matrix $P \in \mathscr{M}_d\left(\mathbb{R}_{+}\right)$ defined for $1 \leqslant i,j \leqslant d$ by
$$P_{i,j} = \frac{M_{i,j} h_j}{\lambda h_i}.$$
(a) Show that there exist $\mu \in \mathscr{P}$, $C > 0$ and $\gamma \in [0,1[$, such that $\mu P = \mu$ and for all $n \geqslant 0$,
$$\sum_{i=1}^{d} \sum_{j=1}^{d} \left| \lambda^{-n} \left(M^n\right)_{i,j} - h_i \frac{\mu_j}{h_j} \right| \leqslant C\gamma^n.$$
(b) Prove that there exists a unique $\pi \in \mathscr{P}$ such that $\pi M = \lambda \pi$.