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Zbl 1052.15013
Bhatia, Rajendra
On the exponential metric increasing property.
(English)
[J] Linear Algebra Appl. 375, 211-220 (2003). ISSN 0024-3795

A short and beautifully simple matrix theoretical proof of the negative curvature of the set of positive definite real or complex matrices $P$ is given. It is based on the exponential operator of a matrix, the exponential metric increasing property, and the logarithmic mean and geometric-arithmetic mean inequalities for scalars and matrices. The Riemannn metric is studied on the manifold $P$ of positive definite matrices, as well as a generalized exponential metric increasing property for symmetric gauge functions $\Phi$. Consequently $P$ is shown to also be a metric space of non-positive curvature in any Finsler metric $\delta_\Phi$. The last section derives the Golden-Thompson inequality from these results and investigates related majorization results.
[Frank Uhlig (Auburn)]
MSC 2000:
*15A45 Miscellaneous inequalities involving matrices
15A48 Positive matrices and their generalizations
15A60 Appl. of functional analysis to matrix theory
53B21 Methods of Riemannian geometry
53B40 Finsler spaces and generalizations (local)

Keywords: exponential metric; positive definite matrix; matrix inequality; geometric mean; Riemannian manifold; negative curvature; matrix norm; symmetric gauge function; matrix exponential; majorization; exponential map; unitarily invariant norm

Cited in: Zbl 1153.15020

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