\(G\)-decompositions of matrices and related problems. I. (Q665955)

The authors introduce a notion of \(G\)-decompositions of matrices. The main result is that a real symmetric matrix \(A_m\) of degree \(m\) has a \(G\)-decomposition in the category of stochastic (respectively, substochastic) matrices if and only if \(A_m\in U^m \text{ (respectively, }U_m\)). To prove this result, they study the extreme points and the geometric structures of the sets \(U^m\) (respectively, \(U_m\)). The study enables one to study the Birkhoff problem for quadratic \(G\)-doubly stochastic operators.