151. G-decompositions of matrices and related problems I
- Author
-
Farrukh Mukhamedov, Rasul Ganikhodzhaev, and Mansoor Saburov
- Subjects
15A51, 47H60, 46T05, 92B99 ,Numerical Analysis ,Class (set theory) ,Substochastic matrix ,Extreme points ,Algebra and Number Theory ,Stochastic matrix ,G-doubly stochastic operator ,Functional Analysis (math.FA) ,Mathematics - Functional Analysis ,G-decomposition ,Set (abstract data type) ,Combinatorics ,Matrix (mathematics) ,Quadratic equation ,FOS: Mathematics ,Mathematics - Combinatorics ,Discrete Mathematics and Combinatorics ,Symmetric matrix ,Combinatorics (math.CO) ,Geometry and Topology ,Nonnegative matrix ,Extreme point ,Mathematics - Abstract
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., 23 pages
- Published
- 2012