Ordering Positive Definite Matrices

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 Löwner-Heinz theorem derived using differential positivity with respect to affine-invariant cone fields.

Cited by in corpus (6)

• On second-order cone positive systems
• Fast, asymptotically efficient, recursive estimation in a Riemannian manifold
• Monotonicity on homogeneous spaces
• Inductive Geometric Matrix Midranges
• Differential geometry with extreme eigenvalues in the positive semidefinite cone
• Generic behavior of differentially positive systems on a globally orderable Riemannian manifold