Riemannian metrics on positive definite matrices related to means. II

Fumio Hiai, D. Petz

Research output: Contribution to journalArticle

16 Citations (Scopus)

Abstract

On the manifold of positive definite matrices, a Riemannian metric is associated with a positive kernel function φ on (0,∞)×(0,∞) by defining KDφ(H,K)=∑ i,jφ( λi, λj) -1Tr PiHPjK, where D is a foot point with the spectral decomposition D=∑ i λiPi and H,K are Hermitian matrices (tangent vectors). We are concerned with the case φ(x,y)=M(x,y) θ where M(x,y) is a mean of scalars x,y>0. We clarify the isometric structure among such kernel metrics and discuss the convergence properties of geodesic distances and geodesic shortest curves along each isometric line of metrics. The metric corresponding to the square of the logarithmic mean shows up as the attractor of the whole metrics concerned.

Original languageEnglish
Pages (from-to)2117-2136
Number of pages20
JournalLinear Algebra and Its Applications
Volume436
Issue number7
DOIs
Publication statusPublished - Apr 1 2012

Fingerprint

Positive definite matrix
Riemannian Metric
Metric
Isometric
Logarithmic Mean
Decomposition
Tangent vector
Geodesic Distance
Spectral Decomposition
Hermitian matrix
Kernel Function
Convergence Properties
Geodesic
Attractor
Scalar
kernel
Curve
Line

Keywords

  • Geodesic distance
  • Geodesic shortest curve
  • Geometric mean
  • Logarithmic mean
  • Positive definite matrix
  • Riemannian metric
  • Symmetric homogeneous mean

ASJC Scopus subject areas

  • Algebra and Number Theory
  • Discrete Mathematics and Combinatorics
  • Geometry and Topology
  • Numerical Analysis

Cite this

Riemannian metrics on positive definite matrices related to means. II. / Hiai, Fumio; Petz, D.

In: Linear Algebra and Its Applications, Vol. 436, No. 7, 01.04.2012, p. 2117-2136.

Research output: Contribution to journalArticle

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