Exercise 17, § 6

Let A be a real square matrix of type $(I,I)$, positive and irreducible. Let $y\in \mathbf{R}^I$ be a proper vector of A associated with an eigenvalue $\alpha$ such that $y\succcurlyeq 0$. We then have $\alpha =r$ (with the notations of exercise 12) and $y\succ 0$. (Consider an extremal vector $u$ for $^tA$, and calculate $\langle y|^tAu\rangle =\langle Ay|u\rangle .$)