G-decompositions of matrices and related problems I

In the present paper we introduce a notion of G-decompositions of matrices. Main result of the paper is that a symmetric matrix Am has a G-decomposition in the class of stochastic (resp. substochastic) matrices if and only if Am belongs to the set Um (resp. Um). To prove the main result, we study extremal points and geometrical structures of the sets Um, Um. Note that such kind of investigations enables to study Birkhoff's problem for quadratic G-doubly stochastic operators.