On the Perron-Frobenius eigenvector for nonnegative integral matrices whose largest eigenvalue is integral
Let A be a k by k nonnegative matrix with spectral radius r. For \(\lambda\geq r\) it is shown that \(| \det (\lambda I-A)| \leq \lambda^ k-r^ k\). The proof proceeds by induction on k using an unusual formula for the derivative of det(\(\lambda\) I-A). The inequality is applied to the case where A is irreducible with r and all entries integral: if the Perron-Frobenius eigenvector has integral relatively prime coordinates, then each of them is at most \(r^{k-1}\).
scientific article; zbMATH DE number 4008581