Let $A \geqslant 0$ in $\mathcal{M}_n(\mathbb{R})$, an irreducible matrix. We denote $r$ its spectral radius. Let $p \geqslant 1$ be the coefficient of imprimitivity of $A$. Let $\chi_A(X) = X^n + c_{k_1}X^{n-k_1} + c_{k_1}X^{n-k_2} + \cdots + c_{k_s}X^{n-k_s}$ be its characteristic polynomial, written according to decreasing powers and showing only the nonzero coefficients $c_k$. We will show that $p$ is the gcd of the integers $k_1, k_2, \ldots, k_s$. Conversely, we assume by contradiction that the $k_j$ are all divisible by $qp$, with $q \geqslant 2$. We set $\beta = \mathrm{e}^{2\mathrm{i}\pi/(qp)}$ (so $\beta^q = \omega$). Show that $\beta r$ is an eigenvalue of $A$ and conclude.
Let $A \geqslant 0$ in $\mathcal{M}_n(\mathbb{R})$, an irreducible matrix. We denote $r$ its spectral radius. Let $p \geqslant 1$ be the coefficient of imprimitivity of $A$. Let $\chi_A(X) = X^n + c_{k_1}X^{n-k_1} + c_{k_1}X^{n-k_2} + \cdots + c_{k_s}X^{n-k_s}$ be its characteristic polynomial, written according to decreasing powers and showing only the nonzero coefficients $c_k$. We will show that $p$ is the gcd of the integers $k_1, k_2, \ldots, k_s$.
Conversely, we assume by contradiction that the $k_j$ are all divisible by $qp$, with $q \geqslant 2$. We set $\beta = \mathrm{e}^{2\mathrm{i}\pi/(qp)}$ (so $\beta^q = \omega$). Show that $\beta r$ is an eigenvalue of $A$ and conclude.