Geometry and inequalities of geometric mean

Czechoslovak Mathematical Journal

Trung ¹  Hoa ² 

[1] Division of Computational Mathematics and Engineering, Institute for Computational Science, Ton Duc Thang University, 19 Nguyen Huu Tho Street, Tan Phong Ward, District 7, Ho Chi Minh City, Vietnam, and Faculty of Civil Engineering, Ton Duc Thang University, 19 Nguyen Huu Tho Street, Tan Phong Ward, District 7, Ho Chi Minh City, Vietnam;Tin-Yau Tam, Department of Mathematics and Statistics, Auburn University, Duncan Dr, Auburn, Alabama 36849, USA

关键词: geometric mean;    positive definite matrix;    log majorization;    geodesics;    geodesically convex;    geodesic convex hull;    unitarily invariant norm;   

DOI  :  

学科分类：数学（综合）

来源: Akademie Ved Ceske Republiky

【 摘 要 】

We study some geometric properties associated with the $t$-geometric means $A\sharp_tB := A^{1/2}(A^{-1/2}BA^{-1/2})^tA^{1/2}$ 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.

【 授权许可】

Unknown