  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.
