Affine-invariant orders on the set of positive-definite matrices

Abstract

We introduce a family of orders on the set S+n of positive-definite matrices of dimension n derived from the homogeneous geometry of S+n induced by the natural transitive action of the general linear group GL(n). 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 structure of S+n. We then revisit the well-known Löwner-Heinz theorem and provide an extension of this classical result derived using differential positivity with respect to affine-invariant cone fields.