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Geometry and inequalities of geometric mean

Trung Hoa Dinh; Sima Ahsani; Tin-Yau Tam

Czechoslovak Mathematical Journal (2016)

  • Volume: 66, Issue: 3, page 777-792
  • ISSN: 0011-4642

Abstract

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We study some geometric properties associated with the t -geometric means A ♯ t B : = A 1 / 2 ( A - 1 / 2 B A - 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.

How to cite

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Dinh, Trung Hoa, Ahsani, Sima, and Tam, Tin-Yau. "Geometry and inequalities of geometric mean." Czechoslovak Mathematical Journal 66.3 (2016): 777-792. <http://eudml.org/doc/286783>.

@article{Dinh2016,
abstract = {We study some geometric properties associated with the $t$-geometric means $A\sharp _\{t\}B := A^\{1/2\}(A^\{-1/2\}BA^\{-1/2\})^\{t\}A^\{1/2\}$ of two $n\times n$ positive definite matrices $A$ and $B$. Some geodesical convexity results with respect to the Riemannian structure of the $n\times n$ 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.},
author = {Dinh, Trung Hoa, Ahsani, Sima, Tam, Tin-Yau},
journal = {Czechoslovak Mathematical Journal},
keywords = {geometric mean; positive definite matrix; log majorization; geodesics; geodesically convex; geodesic convex hull; unitarily invariant norm},
language = {eng},
number = {3},
pages = {777-792},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Geometry and inequalities of geometric mean},
url = {http://eudml.org/doc/286783},
volume = {66},
year = {2016},
}

TY - JOUR
AU - Dinh, Trung Hoa
AU - Ahsani, Sima
AU - Tam, Tin-Yau
TI - Geometry and inequalities of geometric mean
JO - Czechoslovak Mathematical Journal
PY - 2016
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 66
IS - 3
SP - 777
EP - 792
AB - We study some geometric properties associated with the $t$-geometric means $A\sharp _{t}B := A^{1/2}(A^{-1/2}BA^{-1/2})^{t}A^{1/2}$ of two $n\times n$ positive definite matrices $A$ and $B$. Some geodesical convexity results with respect to the Riemannian structure of the $n\times n$ 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.
LA - eng
KW - geometric mean; positive definite matrix; log majorization; geodesics; geodesically convex; geodesic convex hull; unitarily invariant norm
UR - http://eudml.org/doc/286783
ER -

References

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  1. Ando, T., 10.1016/0024-3795(79)90179-4, Linear Algebra Appl. 26 (1979), 203-241. (1979) Zbl0495.15018MR0535686DOI10.1016/0024-3795(79)90179-4
  2. Ando, T., Hiai, F., Log majorization and complementary Golden-Thompson type inequalities, Linear Algebra Appl. 197/198 (1994), 113-131. (1994) Zbl0793.15011MR1275611
  3. Araki, H., 10.1007/BF01045887, Lett. Math. Phys. 19 (1990), 167-170. (1990) Zbl0705.47020MR1039525DOI10.1007/BF01045887
  4. Audenaert, K. M. R., A norm inequality for pairs of commuting positive semidefinite matrices, Electron. J. Linear Algebra (electronic only) 30 (2015), 80-84. (2015) Zbl1326.15030MR3318430
  5. Bhatia, R., 10.1007/978-3-642-30232-9_2, Matrix Information Geometry Springer, Berlin F. Nielsen et al. (2013), 35-51. (2013) Zbl1271.15019MR2964446DOI10.1007/978-3-642-30232-9_2
  6. Bhatia, R., Postitive Definite Matrices, Princeton Series in Applied Mathematics Princeton University Press, Princeton (2007). (2007) MR3443454
  7. Bhatia, R., Matrix Analysis, Graduate Texts in Mathematics 169 Springer, New York (1997). (1997) MR1477662
  8. Bhatia, R., Grover, P., Norm inequalities related to the matrix geometric mean, Linear Algebra Appl. 437 (2012), 726-733. (2012) Zbl1252.15023MR2921731
  9. Bourin, J.-C., 10.1142/S0129167X09005509, Int. J. Math. 20 (2009), 679-691. (2009) Zbl1181.15030MR2541930DOI10.1142/S0129167X09005509
  10. Bourin, J.-C., Uchiyama, M., A matrix subadditivity inequality for f ( A + B ) and f ( A ) + f ( B ) , Linear Algebra Appl. 423 (2007), 512-518. (2007) Zbl1123.15013MR2312422
  11. Fiedler, M., Pták, V., A new positive definite geometric mean of two positive definite matrices, Linear Algebra Appl. 251 (1997), 1-20. (1997) Zbl0872.15014MR1421263
  12. Hayajneh, S., Kittaneh, F., 10.1017/S0004972712001104, Bull. Aust. Math. Soc. 88 (2013), 384-389. (2013) Zbl1287.47011MR3189289DOI10.1017/S0004972712001104
  13. Lin, M., Remarks on two recent results of Audenaert, Linear Algebra Appl. 489 (2016), 24-29. (2016) Zbl1326.15033MR3421835
  14. Lin, M., 10.7153/oam-09-54, Oper. Matrices 9 (2015), 917-924. (2015) MR3447594DOI10.7153/oam-09-54
  15. Marshall, A. W., Olkin, I., Arnold, B. C., Inequalities: Theory of Majorization and Its Applications, Springer Series in Statistics Springer, New York (2011). (2011) Zbl1219.26003MR2759813
  16. Papadopoulos, A., Metric Spaces, Convexity and Nonpositive Curvature, IRMA Lectures in Mathematics and Theoretical Physics 6 European Mathematical Society, Zürich (2005). (2005) Zbl1115.53002MR2132506
  17. Pusz, W., Woronowicz, S. L., 10.1016/0034-4877(75)90061-0, Rep. Math. Phys. 8 (1975), 159-170. (1975) Zbl0327.46032MR0420302DOI10.1016/0034-4877(75)90061-0
  18. Thompson, R. C., 10.1137/0132003, SIAM J. Appl. Math. 32 (1977), 39-63. (1977) Zbl0361.15009MR0424847DOI10.1137/0132003
  19. Zhan, X., 10.1007/b83956, Lecture Notes in Mathematics 1790 Springer, Berlin (2002). (2002) Zbl1018.15016MR1927396DOI10.1007/b83956

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