On Perron–Frobenius property of matrices having some negative entries

We extend the theory of nonnegative matrices to the matrices that have some negative entries. We present and prove some properties which give us information, when a matrix possesses a Perron–Frobenius eigenpair. We apply also this theory by proposing the Perron–Frobenius splitting for the solution o...

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Bibliographic Details
Published in:Linear algebra and its applications 2006-01, Vol.412 (2), p.132-153
Main Author: Noutsos, Dimitrios
Format: Article
Language:English
Subjects:
Algebra
Exact sciences and technology
Linear and multilinear algebra, matrix theory
Mathematics
Nonnegative matrices
Perron–Frobenius splitting
Perron–Frobenius theorem
Sciences and techniques of general use
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Summary:We extend the theory of nonnegative matrices to the matrices that have some negative entries. We present and prove some properties which give us information, when a matrix possesses a Perron–Frobenius eigenpair. We apply also this theory by proposing the Perron–Frobenius splitting for the solution of the linear system Ax = b by classical iterative methods. Perron–Frobenius splittings constitute an extension of the well known regular splittings, weak regular splittings and nonnegative splittings. Convergence and comparison properties are given and proved.
ISSN:0024-3795
1873-1856
DOI:10.1016/j.laa.2005.06.037