G-decompositions of matrices and quadratic doubly stochastic operators
G.D Birkhoff characterized the set of extreme doubly stochastic matrices. Namely his result states as follows: the set of extreme points of the set of doubly stochastic matrices coincides with the set of all permutations matrices. One can consider a generalization of Birkhoff's result for nonl...
| Main Authors: | , , |
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| Format: | Conference or Workshop Item |
| Language: | English |
| Published: |
2011
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| Subjects: | |
| Online Access: | http://irep.iium.edu.my/12726/ http://irep.iium.edu.my/12726/ http://irep.iium.edu.my/12726/1/isasm2011_2.pdf |
| Summary: | G.D Birkhoff characterized the set of extreme doubly stochastic matrices. Namely his result states as follows: the set of extreme points of the set of doubly stochastic
matrices coincides with the set of all permutations matrices. One can consider a generalization of Birkhoff's result for nonlinear doubly stochastic operators. Among all
nonlinear operators, the simplest one is a quadratic operator. In this work we introduce a notion of G-decomposition of matrices which enables to study Birkhoff's problem for quadratic G-doubly stochastic operators. We find necessary and sufficient conditions for the matrices having G-decomposition in the class of stochastic and substochastic matrices. We study geometrical structures of the set of those matrices. Moreover, we investigate extreme points of the sets of matrices having G-decompositions. |
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