Riemannian geometry and matrix geometric means
作者： Rajendra Bhatia , John Holbrook
DOI: 10.1016/J.LAA.2005.08.025
关键词: Manifold 、 Geodesic 、 Matrix (mathematics) 、 Mathematics 、 Geometric probability 、 Geometric mean 、 Riemannian geometry 、 Geometric transformation 、 Riemannian manifold 、 Algebra 、 Pure mathematics 、 Algebra and Number Theory 、 Numerical analysis 、 Discrete Mathematics and Combinatorics 、 Geometry and topology
摘要: The geometric mean of two positive definite matrices has been defined in several ways and studied by authors, including Pusz Woronowicz, Ando. characterizations these authors do not readily extend to three it a long-standing problem define natural matrices. In some recent papers new understanding the achieved identifying A B as midpoint geodesic (with respect Riemannian metric) joining B. This suggests definitions for We explain necessary background explore properties candidates.
core.ac.uk
elsevier.com
sci-hub.se 参考文章(12)
George E. Trapp, Hermitian semidefinite matrix means and related matrix inequalities—an introduction Linear & Multilinear Algebra. ,vol. 16, pp. 113- 123 ,(1984) , 10.1080/03081088408817613
Maher Moakher, On the Averaging of Symmetric Positive-Definite Tensors Journal of Elasticity. ,vol. 82, pp. 273- 296 ,(2006) , 10.1007/S10659-005-9035-Z
Dénes Petz, Róbert Temesi, Means of Positive Numbers and Matrices SIAM Journal on Matrix Analysis and Applications. ,vol. 27, pp. 712- 720 ,(2005) , 10.1137/050621906
Jimmie D. Lawson, Yongdo Lim, The Geometric Mean, Matrices, Metrics, and More American Mathematical Monthly. ,vol. 108, pp. 797- 812 ,(2001) , 10.1080/00029890.2001.11919815
T. Ando, Concavity of certain maps on positive definite matrices and applications to Hadamard products Linear Algebra and its Applications. ,vol. 26, pp. 203- 241 ,(1979) , 10.1016/0024-3795(79)90179-4
Maher Moakher, A Differential Geometric Approach to the Geometric Mean of Symmetric Positive-Definite Matrices SIAM Journal on Matrix Analysis and Applications. ,vol. 26, pp. 735- 747 ,(2005) , 10.1137/S0895479803436937
W. Pusz, S.L. Woronowicz, Functional calculus for sesquilinear forms and the purification map Reports on Mathematical Physics. ,vol. 8, pp. 159- 170 ,(1975) , 10.1016/0034-4877(75)90061-0
Rajendra Bhatia, On the exponential metric increasing property Linear Algebra and its Applications. ,vol. 375, pp. 211- 220 ,(2003) , 10.1016/S0024-3795(03)00647-5
GUSTAVO CORACH, HORACIO PORTA, LÁZARO RECHT, GEODESICS AND OPERATOR MEANS IN THE SPACE OF POSITIVE OPERATORS International Journal of Mathematics. ,vol. 04, pp. 193- 202 ,(1993) , 10.1142/S0129167X9300011X
André Haefliger, Martin R. Bridson, Metric Spaces of Non-Positive Curvature ,(1999)