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On the exponential metric increasing property. (English) Zbl 1052.15013
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.

MSC:
 15A45 Miscellaneous inequalities involving matrices 15B48 Positive matrices and their generalizations; cones of matrices 15A60 Norms of matrices, numerical range, applications of functional analysis to matrix theory 53B21 Methods of local Riemannian geometry 53B40 Local differential geometry of Finsler spaces and generalizations (areal metrics)
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