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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