Exercise 11, § 6

Let $A$ be a real matrix of type $(I,I)$, positive and irreducible. We denote $\psi \in \mathbf{R}[X]$ the minimal polynomial of A and put $m=$ deg($\psi$ ).

For every integer $q$, put $A^q= (a^{(q)}_{ij})_{i,j\in I}$.

For every $(i, j)\in I^2$, there exists an integer $q$ such that $1\leqslant q\leqslant m$ and $a^{(q)}_{ij}>0$. (Let $r$ be the remainder of the euclidean division of $(1 + X)^{n-1}$ by $\psi$; consider the matrix $r($A) and observe that it is strictly positive.)