Gyrovector Spaces on the Open Convex Cone of Positive Definite Matrices

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 space, 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 see its interest...

Deconvolution Density Estimation on Spaces of Positive Definite Symmetric Matrices

Motivated by applications in microwave engineering and diffusion tensor imaging, we study the problem of deconvolution density estimation on the space of positive definite symmetric matrices. We develop a nonparametric estimator for the density function of a random sample of positive definite matrices. Our estimator is based on the Helgason-Fourier transform and its inversion, the natural tools...

ON f-CONNECTIONS OF POSITIVE DEFINITE MATRICES

In this paper, by using Mond-Pečarić method we provide some inequalities for connections of positive definite matrices. Next, we discuss specifications of the obtained results for some special cases. In doing so, we use α-arithmetic, α-geometric and α-harmonic operator means.

A Characterization of Matrix Groups That Act Transitively on the Cone of Positive Definite Matrices

It is well known that the group of all nonsingular lower block -triangular pxp matrices acts transitively on the cone p* of all positive definite pxp matrices. This result has been applied to obtain several major reSUlts in mUltivariate statistical distribution theory and decision theory. Here a converse is established: if a matrix group acts transitively on P*, then its group algebra must be (...

Riemannian geometry on positive definite matrices

The Riemannian metric on the manifold of positive definite matrices is defined by a kernel function φ in the form K D(H,K) = ∑ i,j φ(λi, λj) −1TrPiHPjK when ∑ i λiPi is the spectral decomposition of the foot point D and the Hermitian matrices H,K are tangent vectors. For such kernel metrics the tangent space has an orthogonal decomposition. The pull-back of a kernel metric under a mapping D 7→ ...

On The Frobenius Condition Number of Positive Definite Matrices