Riemannian metrics on positive de nite matrices related to means Fumio Hiai 1 and D
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چکیده
The Riemannian metric on the manifold of positive de nite matrices is de ned by a kernel function in the formK D(H;K) = P i;j ( i; j) TrPiHPjK when P i iPi 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! 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 (x; y) =M(x; y) 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. AMS classi cation: 15A45; 15A48; 53B21; 53C22
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Riemannian metrics on positive de nite matrices related to means
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