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arXiv · 1712.02573

Ordering Positive Definite Matrices

Cyrus Mostajeran·Rodolphe Sepulchre
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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.

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Cyrus Mostajeran, Rodolphe Sepulchre. 2018-05-03. Ordering Positive Definite Matrices. https://doi.org/10.1007/s41884-018-0003-7

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