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A NEW PROOF TO CONSTRUCT MULTIVARIABLE GEOMETRIC MEANS BY SYMMETRIZATION

  • KIM, SEJONG ;
  • PETZ, DENES
  • Received : 2015.01.13
  • Accepted : 2015.03.20
  • Published : 2015.05.30

Abstract

The original geometric mean of two positive definite operators A and B is given by A#B = A1/2(A-1/2BA-1/2)1/2A1/2. In this article we provide a new proof to construct from the two-variable geometric mean to the multivariable mean via symmetrization introduced by Lawson and Lim [5]. Finally we provide an algorithm to find three-variable geometric mean via symmetrization, which plays an important role to construct higher-order geometric means.

Keywords

positive definite operator;operator mean

References

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  2. J. Lawson and Y. Lim, Karhcer means and Karcher equations of positive definite operators, Trans. Amer. Math. Soc. Series B 1 (2014), 1-22. https://doi.org/10.1090/S2330-0000-2014-00003-4
  3. A. Ungar, Analytic Hyperbolic Geometry and Albert Einstein's Special Theory of Relativity, World Scientific, 2008.
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  5. S. K. Chakraborty and G. Panda, A higher order iterative algorithm for multivariate opti-mization problem, J. Appl. Math. & Informatics 32 (2014), 747-760. https://doi.org/10.14317/jami.2014.747
  6. F. Hiai and D. Petz, Introduction to Matrix Analysis and Applications, Hindustan Book Agancy and Springer, 2014.
  7. F. Kubo and T. Ando, Means of positive linear operators, Math. Ann. 246 (1980), 205-224. https://doi.org/10.1007/BF01371042

Cited by

  1. REMARKS ON CONVERGENCE OF INDUCTIVE MEANS vol.34, pp.3_4, 2016, https://doi.org/10.14317/jami.2016.28