$G-$Decompositions of Matrices and Related Problems I

TL;DR

This paper introduces the concept of G-decompositions of matrices, characterizes symmetric matrices with such decompositions in terms of stochastic matrices, and explores the geometric structure of related matrix sets.

Contribution

It defines G-decompositions for matrices and provides a characterization of symmetric matrices with these decompositions using extremal points and geometric analysis.

Findings

Symmetric matrix A_m has a G-decomposition in stochastic matrices iff A_m belongs to set U^m.

Characterization of matrices in set U_m and U^m via extremal points.

Facilitates study of Birkhoff's problem for quadratic G-doubly stochastic operators.

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.