Bhatia, Rajendra ; Holbrook, John
(2006)
*Riemannian geometry and matrix geometric means*
Linear Algebra and its Applications, 413
(2-3).
pp. 594-618.
ISSN 0024-3795

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Official URL: http://linkinghub.elsevier.com/retrieve/pii/S00243...

Related URL: http://dx.doi.org/10.1016/j.laa.2005.08.025

## Abstract

The geometric mean of two positive definite matrices has been defined in several ways and studied by several authors, including Pusz and Woronowicz, and Ando. The characterizations by these authors do not readily extend to three matrices and it has been a long-standing problem to define a natural geometric mean of three positive definite matrices. In some recent papers new understanding of the geometric mean of two positive definite matrices has been achieved by identifying the geometric mean of A and B as the midpoint of the geodesic (with respect to a natural Riemannian metric) joining A and B. This suggests some natural definitions for a geometric mean of three positive definite matrices. We explain the necessary geometric background and explore the properties of some of these candidates.

Item Type: | Article |
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Source: | Copyright of this article belongs to Elsevier Science. |

Keywords: | Positive Definite Matrix; Geometric Mean; Riemannian Manifold; Semi-parallelogram Law; Gradient |

ID Code: | 2601 |

Deposited On: | 08 Oct 2010 07:19 |

Last Modified: | 08 Oct 2010 07:19 |

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