Abstract
We introduce new partial orders on the set S^+_n of positive definite matrices of dimension n derived from the affine-invariant geometry of S^+_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 geometry of S^+_n 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 S^+_n 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.
Highlights
IntroductionWell-defined notions of ordering of elements of a space are of fundamental importance to many areas of applied mathematics, including the theory of monotone functions and
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 G L(n)
Summary
Well-defined notions of ordering of elements of a space are of fundamental importance to many areas of applied mathematics, including the theory of monotone functions and. The aim is to generate cone fields that are invariant with respect to the homogeneous geometry, thereby defining partial orders built upon the underlying symmetries of the space. The geometry of invariant cone fields and causal structures on homogeneous spaces has been the subject of extensive studies from a Lie theoretic perspective; see [12,13,18], for instance. The space Sn+ forms a smooth manifold that can be viewed as a homogeneous space admitting a transitive action by the general linear group G L(n), which endows the space with an affine-invariant geometry as reviewed in Sect. The set Sn+ of symmetric positive definite matrices of dimension n has the structure of a homogeneous space with a transitive G L(n)-action.
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
