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• W2029258199 abstract "The Riemannian metric on the manifold of positive definite matrices is defined by a kernel function ϕ in the form KDϕ(H,K)=∑i,jϕ(λi,λj)-1TrPiHPjK when ∑iλiPi 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." @default.
• W2029258199 created "2016-06-24" @default.
• W2029258199 creator A5011761244 @default.
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• W2029258199 date "2009-06-01" @default.
• W2029258199 modified "2023-10-12" @default.
• W2029258199 title "Riemannian metrics on positive definite matrices related to means" @default.
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• W2029258199 doi "https://doi.org/10.1016/j.laa.2009.01.025" @default.
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