Theorem 5 (Perron–Frobenius)

a) If $a_{i,j}\geqslant 0$ for all $i$ and $j$ in $\{1, . . . , n\}$, then the real number $\varrho$ is an eigenvalue of A;

b) If $a_{i,j}>0$ for all $i$ and $j$ in $\{1, . . . , n\}$, then $\varrho >0$ and the primary space of $u_A$ relative to $\varrho$, that is to say the nilespace of $A-\varrho 1_{\mathbf{R}^n}$, is of dimension 1 and is generated by a vector $x_0\in (\mathbf{R}_+^*)^n$. There exists no other eigenvalue of A having an eigenvector in $\mathbf{R}^n_+$, and all complex eigenvalues $\lambda \not =\varrho$ of A satisfy $|\lambda |< \varrho$.

If $\varrho = 0$, assertion a) is elementary, for 0 is then an eigenvalue of A. If $\varrho >0$, assertion a) follows from theorem 4 in view of lemma 7. Assertion b) is, for the same reason, a consequence of proposition 8.