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The influence of the marked reduced graph of a nonnegative matrix on the Jordan form and on related properties: a survey. (English) Zbl 0613.15017
The author gives a survey, beginning with Frobenius, of certain properties of a nonnegative matrix which are related to, or can be expressed by, properties of the marked reduced graph of the matrix. For a nonnegative square matrix P with Perron-Frobenius root \(\rho\) the areas surveyed are: (1) positivity properties of the eigenvectors and generalized eigenvectors of P; (2) the nonnegative solutions x of \((\lambda I-P)x=b\), for a given nonnegative vector b and given real number \(\lambda\) ; (3) the Jordan blocks associated with \(\rho\) in the Jordan canonical form of P; (4) the growth of the elements of \(P^ m\) as m grows.
Reviewer: F.J.Gaines

MSC:
15B48 Positive matrices and their generalizations; cones of matrices
15A21 Canonical forms, reductions, classification
15A18 Eigenvalues, singular values, and eigenvectors
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