Ordering positive definite matrices

We introduce new partial orders on the set Sn+ of positive definite matrices of dimension n derived from the affine-invariant geometry of Sn+. The orders are induced by affine-invariant cone fields, which arise naturally from a local analysis of the orders that are compatible with the homogeneous geometry of Sn+ defined by the natural transitive action of the general linear group GL(n). We then take a geometric approach to the study of monotone functions on Sn+ and establish a number of relevant results, including an extension of the well-known Löwner-Heinz theorem derived using differential positivity with respect to affine-invariant cone fields. © 2018, The Author(s).