G-decompositions of matrices and quadratic doubly stochastic operators - Ganikhodzaev, Rasul; Mukhamedov, Farrukh; Saburov, Mansoor (2011)
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.