Theoretical Statistics and Mathematics Unit, ISI Delhi

On the exponential metric increasing property

by Rajendra Bhatia

The metric increasing property of the exponential map is known to be equivalent to the
fact that the set of positive definite matrices is a Riemannian manifold of nonpositive curvature. We show that this property is an easy consequence of the logarithmic-geometric mean
inequality for positive numbers. Operator versions of this inequality lead to a generalisation
of the exponential metric increasing property to all Schatten-von Neumann norms.

isid/ms/2002/14 [fulltext]

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