A short and beautifully simple matrix theoretical proof of the negative curvature of the set of positive definite real or complex matrices
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
of positive definite matrices, as well as a generalized exponential metric increasing property for symmetric gauge functions
is shown to also be a metric space of non-positive curvature in any Finsler metric
. The last section derives the Golden-Thompson inequality from these results and investigates related majorization results.