### Abstract

On the manifold of positive definite matrices, a Riemannian metric ^{Kφ} is associated with a positive kernel function φ on (0,∞)×(0,∞) by defining KDφ(H,K)=∑ _{i,j}φ( ^{λi}, ^{λj}) ^{-1}Tr ^{PiHPj}K, 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 language | English |
---|---|

Pages (from-to) | 2117-2136 |

Number of pages | 20 |

Journal | Linear Algebra and Its Applications |

Volume | 436 |

Issue number | 7 |

DOIs | |

Publication status | Published - Apr 1 2012 |

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### 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

*Linear Algebra and Its Applications*,

*436*(7), 2117-2136. https://doi.org/10.1016/j.laa.2011.10.029

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

Research output: Contribution to journal › Article

*Linear Algebra and Its Applications*, vol. 436, no. 7, pp. 2117-2136. https://doi.org/10.1016/j.laa.2011.10.029

}

TY - JOUR

T1 - Riemannian metrics on positive definite matrices related to means. II

AU - Hiai, Fumio

AU - Petz, D.

PY - 2012/4/1

Y1 - 2012/4/1

N2 - On the manifold of positive definite matrices, a Riemannian metric Kφ 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.

AB - On the manifold of positive definite matrices, a Riemannian metric Kφ 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.

KW - Geodesic distance

KW - Geodesic shortest curve

KW - Geometric mean

KW - Logarithmic mean

KW - Positive definite matrix

KW - Riemannian metric

KW - Symmetric homogeneous mean

UR - http://www.scopus.com/inward/record.url?scp=84857108675&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84857108675&partnerID=8YFLogxK

U2 - 10.1016/j.laa.2011.10.029

DO - 10.1016/j.laa.2011.10.029

M3 - Article

VL - 436

SP - 2117

EP - 2136

JO - Linear Algebra and Its Applications

JF - Linear Algebra and Its Applications

SN - 0024-3795

IS - 7

ER -