Geometry and inequalities of geometric mean
- 105 Downloads
We study some geometric properties associated with the t-geometric means A ♯ t B:= A 1/2(A −1/2 BA −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.
Keywordsgeometric mean positive definite matrix log majorization geodesics geodesically convex geodesic convex hull unitarily invariant norm
MSC 201015A45 15B48
Unable to display preview. Download preview PDF.
- R. Bhatia: Postitive Definite Matrices. Princeton Series in Applied Mathematics, Princeton University Press, Princeton, 2007.Google Scholar
- R. Bhatia: Matrix Analysis. Graduate Texts in Mathematics 169, Springer, New York, 1997.Google Scholar
- A. W. Marshall, I. Olkin, B. C. Arnold: Inequalities: Theory of Majorization and Its Applications. Springer Series in Statistics, Springer, New York, 2011.Google Scholar
- A. Papadopoulos: Metric Spaces, Convexity and Nonpositive Curvature. IRMA Lectures in Mathematics and Theoretical Physics 6, European Mathematical Society, Zürich, 2005.Google Scholar
- X. Zhan: Matrix Inequalities. Lecture Notes inMathematics 1790, Springer, Berlin, 2002.Google Scholar