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.