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Ordering positive definite matrices

Mostajeran, CS and Sepulchre, R Ordering positive definite matrices. Springer Information Geometry. (Unpublished)

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

Item Type: Article
Divisions: Div F > Control
Depositing User: Cron Job
Date Deposited: 09 May 2018 20:10
Last Modified: 02 Mar 2021 07:30