gyrovector spaces on the open convex cone of positive definite matrices

‎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 möbius 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.