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 $$P_{i,j} = \frac{M_{i,j} h_j}{\lambda h_i}.$$ Let $n \geqslant 1$. Give an expression for the coefficients of $P^n$ in terms of the coefficients of $M^n$, $h$ and $\lambda$.
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
$$P_{i,j} = \frac{M_{i,j} h_j}{\lambda h_i}.$$
Let $n \geqslant 1$. Give an expression for the coefficients of $P^n$ in terms of the coefficients of $M^n$, $h$ and $\lambda$.