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A simple approach to the Perron-Frobenius theory for positive operators on general partially-ordered finite-dimensional linear spaces


Authors: Werner C. Rheinboldt and James S. Vandergraft
Journal: Math. Comp. 27 (1973), 139-145
MSC: Primary 15A48
DOI: https://doi.org/10.1090/S0025-5718-1973-0325650-4
MathSciNet review: 0325650
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Abstract: This paper presents simple proofs of the principal results of the Perron-Frobenius theory for linear mappings on finite-dimensional spaces which are nonnegative relative to a general partial ordering on the space. The principal tool for these proofs is an application of the theory of norms in finite dimensions to the study of order inequalities of the form $ Ax \leqq \alpha x,x \geqq 0$ where $ A \geqq 0$. This approach also permits the derivation of various inclusion and comparison theorems.


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DOI: https://doi.org/10.1090/S0025-5718-1973-0325650-4
Article copyright: © Copyright 1973 American Mathematical Society

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