For any primitive matrix $A$ in $\mathcal{M}_n(\mathbb{R})$, we denote $r$ the spectral radius of $A$, $L = xy^\top$ where $x > 0$ is a direction vector of $\operatorname{Ker}(A - rI_n)$ and $y > 0$ is a direction vector of $\operatorname{Ker}(A^\top - rI_n)$ with $y^\top x = 1$. We set $B = A - rL$, and we have shown $\rho(B) < r$ and $\forall m \in \mathbb{N}^*, (A - rL)^m = A^m - r^m L$. Deduce from the above (and from subsection IV.A) that $\lim_{m \rightarrow +\infty} \left(\frac{1}{r}A\right)^m = L$.
For any primitive matrix $A$ in $\mathcal{M}_n(\mathbb{R})$, we denote $r$ the spectral radius of $A$, $L = xy^\top$ where $x > 0$ is a direction vector of $\operatorname{Ker}(A - rI_n)$ and $y > 0$ is a direction vector of $\operatorname{Ker}(A^\top - rI_n)$ with $y^\top x = 1$. We set $B = A - rL$, and we have shown $\rho(B) < r$ and $\forall m \in \mathbb{N}^*, (A - rL)^m = A^m - r^m L$.
Deduce from the above (and from subsection IV.A) that $\lim_{m \rightarrow +\infty} \left(\frac{1}{r}A\right)^m = L$.