Riemannian metrics on positive definite matrices related to means

2008/09/29 by Hiai, F., Petz, D.
#15A45 #15A48 #53B21 #53C22 #FOS: Mathematics #FOS: Physical sciences #Functional Analysis (math.FA) #Mathematical Physics (math-ph)

paper · doi:10.48550/arxiv.0809.4974

Abstract

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)=∑_i,jϕ(λ_i,λ_j)^-1 Tr P_iHP_jK when ∑_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↦ 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.

Related

• The Riemannian median of positive-definite matrices
• The geometry of mixed-Euclidean metrics on symmetric positive definite matrices
• Wasserstein Riemannian Geometry of Positive Definite Matrices
• Geometries on the cone of positive-definite matrices derived from the power potential and their relation to the power means
• On the Bures-Wasserstein distance between positive definite matrices