{
  "id": "0809.4974",
  "version": "v3",
  "published": "2008-09-29T14:13:16.000Z",
  "updated": "2008-11-08T00:37:03.000Z",
  "title": "Riemannian metrics on positive definite matrices related to means",
  "authors": [
    "F. Hiai",
    "D. Petz"
  ],
  "comment": "28 pages",
  "categories": [
    "math-ph",
    "math.FA",
    "math.MP"
  ],
  "abstract": "The Riemannian metric on the manifold of positive definite matrices is defined by a kernel function $\\phi$ in the form $K_D^\\phi(H,K)=\\sum_{i,j}\\phi(\\lambda_i,\\lambda_j)^{-1} Tr P_iHP_jK$ when $\\sum_i\\lambda_iP_i$ 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 $\\phi(x,y)=M(x,y)^\\theta$ 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.",
  "revisions": [
    {
      "version": "v3",
      "updated": "2008-11-08T00:37:03.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "15A45",
      "15A48",
      "53B21",
      "53C22"
    ],
    "keywords": [
      "positive definite matrices",
      "riemannian metric",
      "kernel metric",
      "quantum fisher information",
      "multivariate gaussian distributions"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 28,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2008arXiv0809.4974H"
    }
  }
}