Ordering Positive Definite Matrices

TL;DR

This paper introduces new partial orders on positive-definite matrices based on affine-invariant cone fields and explores their implications for monotone functions, extending classical theorems like L"owner-Heinz.

Contribution

It develops a geometric framework for partial orders on $S^+_n$ and extends the L"owner-Heinz theorem using differential positivity.

Findings

New partial orders on $S^+_n$ derived from homogeneous geometry.

Extension of L"owner-Heinz theorem via affine-invariant cone fields.

Analysis of monotone functions within the geometric order framework.

Abstract

We introduce new partial 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 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 L\"owner-Heinz theorem derived using differential positivity with respect to affine-invariant cone fields.