| Title:
|
Geometry and inequalities of geometric mean (English) |
| Author:
|
Dinh, Trung Hoa |
| Author:
|
Ahsani, Sima |
| Author:
|
Tam, Tin-Yau |
| Language:
|
English |
| Journal:
|
Czechoslovak Mathematical Journal |
| ISSN:
|
0011-4642 (print) |
| ISSN:
|
1572-9141 (online) |
| Volume:
|
66 |
| Issue:
|
3 |
| Year:
|
2016 |
| Pages:
|
777-792 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
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. (English) |
| Keyword:
|
geometric mean |
| Keyword:
|
positive definite matrix |
| Keyword:
|
log majorization |
| Keyword:
|
geodesics |
| Keyword:
|
geodesically convex |
| Keyword:
|
geodesic convex hull |
| Keyword:
|
unitarily invariant norm |
| MSC:
|
15A45 |
| MSC:
|
15B48 |
| idZBL:
|
Zbl 06644033 |
| idMR:
|
MR3556867 |
| DOI:
|
10.1007/s10587-016-0292-8 |
| . |
| Date available:
|
2016-10-01T15:24:02Z |
| Last updated:
|
2023-10-28 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/145871 |
| . |
| Reference:
|
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