Gyrovector Spaces on the Open Convex Cone of Positive Definite Matrices

Abstract

‎In this article we review an algebraic definition of the gyrogroup and a simplified version of the gyrovector space with two fundamental examples on the open ball of finite-dimensional Euclidean spaces‎, ‎which are the Einstein and M"{o}bius gyrovector spaces‎. ‎We introduce the structure of gyrovector space and the gyroline on the open convex cone of positive definite matrices and explore its interesting applications on the set of invertible density matrices‎. ‎Finally we give an example of the gyrovector space on the unit ball of Hermitian matrices‎.