Riemannian metrics on positive definite matrices related to means
| dc.creator | Hiai, F. | |
| dc.creator | Petz, D. | |
| dc.date | 2008-09-29 | |
| dc.date | 2008-11-08 | |
| dc.date.accessioned | 2026-07-07T10:16:38Z | |
| dc.date.available | 2026-07-07T10:16:38Z | |
| dc.description | The Riemannian metric on the manifold of positive definite matrices is defined by a kernel function $ϕ$ in the form $K_D^ϕ(H,K)=\sum_{i,j}ϕ(λ_i,λ_j)^{-1} Tr P_iHP_jK$ when $\sum_iλ_iP_i$ 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 $D\mapsto G(D)$ is a kernel metric as well. Several Riemannian geometries of the literature are particular cases, for example, the Fisher-Rao metric for multivariate Gaussian distributions and the quantum Fisher information. In the paper the case $ϕ(x,y)=M(x,y)^θ$ is mostly studied when $M(x,y)$ is a mean of the positive numbers $x$ and $y$. There are results about the geodesic curves and geodesic distances. The geometric mean, the logarithmic mean and the root mean are important cases. | |
| dc.description | 28 pages | |
| dc.identifier | https://arxiv.org/abs/0809.4974 | |
| dc.identifier | http://arxiv.org/abs/0809.4974 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/173575 | |
| dc.subject | Mathematical Physics | |
| dc.subject | Functional Analysis | |
| dc.subject | 15A45; 15A48; 53B21; 53C22 | |
| dc.title | Riemannian metrics on positive definite matrices related to means | |
| dc.type | text |