Affine-Invariant Orders on the Set of Positive-Definite Matrices

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 Lowner-Heinz theorem and provide an extension of this classical result derived using differential positivity with respect to affine-invariant cone fields.