Ashley, Jonathan On the Perron-Frobenius eigenvector for nonnegative integral matrices whose largest eigenvalue is integral. (English) Zbl 0622.15012 Linear Algebra Appl. 94, 103-108 (1987). 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 \(r^{k-1}\). Reviewer: J.Zemánek Cited in 8 Documents MSC: 15B48 Positive matrices and their generalizations; cones of matrices 15A42 Inequalities involving eigenvalues and eigenvectors 15A18 Eigenvalues, singular values, and eigenvectors Keywords:nonnegative matrix; spectral radius; Perron-Frobenius eigenvector PDFBibTeX XMLCite \textit{J. Ashley}, Linear Algebra Appl. 94, 103--108 (1987; Zbl 0622.15012) Full Text: DOI References: [1] Adler, R.; Coppersmith, D.; Hassner, M., Algorithms for sliding-block codes, IEEE Trans. Inform. Theory, IT-29, 5-22 (1983) · Zbl 0499.94009 [2] Gantmacher, F., The Theory of Matrices, Vol. II (1959), Chelsea: Chelsea New York · JFM 65.1131.03 [3] Marcus, B., Factors and extensions of full shifts, Monatsh. Math., 88, 239-247 (1979) · Zbl 0432.54036 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.