Riemannian metrics on positive definite matrices related to means

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 , j ϕ ( λ i , λ j ) - 1 Tr P i HP j K when ∑ i λ i P 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 statistical 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.