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Articles

On the Ando–Li–Mathias mean and the Karcher mean of positive definite matrices

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Pages 636-649 | Received 28 Aug 2013, Accepted 28 Jan 2014, Published online: 29 Apr 2014

References

  • Ando T, Li CK, Mathias R. Geometric means. Linear Algebra Appl. 2004;385:305–334.
  • Lawson J, Lim Y. A general framework for extending means to higher orders. Colloq. Math. 2008;113:191–221.
  • Bini DA, Meini B, Poloni F. An effective matrix geometric mean satisfying the Ando–Li–Mathias properties. Math. Comp. 2010;79:437–452.
  • Izumino S, Nakamura N. Geometric means of positive operators II. Sci. Math. Jpn. 2009;69:35–44.
  • Izumino S, Nakamura N. Geometric means of more than two operators. RIMS Kokyuroku. 2009;1632:14–27.
  • Moakher M. A differential geometric approch to the geometric mean of symmetric positive-definite matrices. SIAM J. Matrix Anal. Appl. 2005;26:735–747.
  • Bhatia R, Holbrook J. Riemannian geometry and matrix means. Linear Algebra Appl. 2006;413:594–618.
  • Bhatia R. Positive definite matrices. Princeton series in applied mathematics. Princeton (NJ): Princeton University Press; 2007.
  • Bhatia R, Karandikar RI. Monotonicity of the matrix geometric mean. Math. Ann. 2012;353:1453–1467.
  • Lim Y, Palfia M. Matrix power means and the Karcher mean. J. Funct. Anal. 2012;262:1498–1514.
  • Lawson J, Lim Y. Karcher means and Karcher equations of positive definite operators. Trans. Am. Math. Soc., Ser. B. 2014;1:1–22.
  • Ando T, Hiai F. Log-majorization and complementary Golden–Thompson type inequalities. Linear Algebra Appl. 1994;197–198:113–131.
  • Yamazaki T. Riemannian mean and matrix inequalities related to the Ando–Hiai inequality and chaotic order. Oper. Matrices. 2012;6:577–588.
  • Bhagwat KV, Subramanian R. Inequalities between means of positive operators. Math. Proc. Camb. Phil. Soc. 1978;83:393–401.
  • Fujii JI, Fujii M, Seo Y. The Golden–Thompson–Segal type inequalities related to the weighted geometric mean due to Lawson–Lim. J. Math. Inequal. 2009;3:511–518.
  • Fujii M, Nakamoto R. A geometric mean in the Furuta inequality. Sci. Math. Jpn. 2002;55:615–621.
  • Seo Y. Norm inequalities for the chaotically geometric mean and its reverse. J. Math. Inequal. 2007;1:39–43.
  • Seo Y. Generalized Pólya–Szegö type inequalities for some non-commutative geometric means. Linear Algebra Appl. 2013;438:1711–1726.
  • Bhatia R, Grover P. Norm inequalities related to the matrix geometric mean. Linear Algebra Appl. 2012;437:726–733.
  • Arsigny V, Fillard P, Pennec X, Ayache N. Geometric means in a novel vector space structure on symmetric positive-definite matrices. SIAM J. Matrix Anal. Appl. 2007;29:328–347.
  • Bhatia R. Matrix analysis. New York (NY): Springer-Verlag; 1997.
  • Specht W. Zur Theorie der elementaren Mittel. Math. Z. 1960;74:91–98.
  • Furuta T, Mićić Hot J, Pečarić J, Seo Y. Mond–Pečarić method in operator inequalities. Monographs in Inequalities 1. Zagreb: Element; 2005.
  • Tominaga M. Specht’s ratio in the Young inequality. Sci. Math. Jpn. 2002;55:585–588.
  • Alić M, Bullen PS, Pečarić J, Volenec V. On the geometric.arithmeticmean inequality for matrices. Math. Commun. 1997;2:125–128.
  • Fujii JI, Fujii M, Nakamura M, Pečarić JE, Seo Y. A reverse of the weighted geometric mean due to Lawson–Lim. Linear Algebra Appl. 2007;427:272–284.
  • Yamazaki T, Yanagida M. Characterizations of chaotic order associated with Kantorovich inequality. Sci. Math. 1999;2:37–50.
  • Hiai F, Petz D. Riemannian metrics on positive definite matrices related to means. II, Linear Algebra Appl. 2012;436:2117–2136.
  • Fujii JI, Nakamura M. J. Pec̈arić and Y. Seo, Bounds for the ratio and difference between parallel sum and series via Mond-Pec̈arić method. Math. Inequal. Appl. 2006;9:749–759.
  • Lim Y. Riemannian distances between Geometric means. SIAM J. Matrix Anal. Appl. 2013;34:932–945.

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