Remark

Assume that there exists an integer $k\geqslant 1$ such that all the coefficients of $A^k$ are $>0$ (such a matrix is sometimes called primitive). Then, by the cor. to Prop. 8, the spectral radius $\varrho$ is an eigenvalue of A, the primary space of $u_A$ relative to $\varrho$ is of dimension 1, and is generated by a vector $x_0\in (\mathbf{R}^*_+)^n$. Moreover, there exists no other complex eigenvalue of A admitting an eigenvector in $\mathbf{R}_+^n$ and all the complex eigenvalues $\lambda \not =\varrho$ of A satisfy $|\lambda |< \varrho$.