Exercise 19, § 6

Let $A$ be a real square matrix of type $(I,I)$, positive and irreducible; let $\varrho$ be its spectral radius. Let $z\in \mathbf{R}^I$ such that $z\succcurlyeq 0$ and $z\not = 0$.

If $\varrho z\preccurlyeq Az$ (resp. $\varrho z\succcurlyeq Az$), then $\varrho z=Az$ and $z\succ 0$.