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Czechoslovak Mathematical Journal

, Volume 66, Issue 3, pp 777–792 | Cite as

Geometry and inequalities of geometric mean

  • Trung Hoa Dinh
  • Sima Ahsani
  • Tin-Yau Tam
Article
  • 105 Downloads

Abstract

We study some geometric properties associated with the t-geometric means A t B:= A 1/2(A −1/2 BA −1/2) t A 1/2 of two n × n positive definite matrices A and B. Some geodesical convexity results with respect to the Riemannian structure of the n × n positive definite matrices are obtained. Several norm inequalities with geometric mean are obtained. In particular, we generalize a recent result of Audenaert (2015). Numerical counterexamples are given for some inequality questions. A conjecture on the geometric mean inequality regarding m pairs of positive definite matrices is posted.

Keywords

geometric mean positive definite matrix log majorization geodesics geodesically convex geodesic convex hull unitarily invariant norm 

MSC 2010

15A45 15B48 

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Copyright information

© Institute of Mathematics of the Academy of Sciences of the Czech Republic, Praha, Czech Republic 2016

Authors and Affiliations

  1. 1.Division of Computational Mathematics and Engineering, Institute for Computational ScienceTon Duc Thang UniversityHo Chi Minh CityVietnam
  2. 2.Faculty of Civil EngineeringTon Duc Thang UniversityHo Chi Minh CityVietnam
  3. 3.Department of Mathematics and StatisticsAuburn UniversityDuncan Dr, AuburnUSA

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