$G-$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 $A_m$ has a $G-$decomposition in the class of stochastic (resp. substochastic) matrices if and only if $A_m$ belongs to the set ${\mathbf{U}}^m$ (resp. ${\mathbf{U}}_m$). To prove the main result, we study extremal points and geometrical structures of the sets ${\mathbf{U}}^m$, ${\mathbf{U}}_m$. Note that such kind of investigations enables to study Birkhoff's problem for quadratic $G-$doubly stochastic operators.