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On the Perron-Frobenius eigenvector for nonnegative integral matrices whose largest eigenvalue is integral. (English) Zbl 0622.15012

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

MSC:

15B48 Positive matrices and their generalizations; cones of matrices
15A42 Inequalities involving eigenvalues and eigenvectors
15A18 Eigenvalues, singular values, and eigenvectors
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References:

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