Abstract
We introduce a family of 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 revisit the well-known Löwner-Heinz theorem and provide an extension of this classical result derived using differential positivity with respect to affine-invariant cone fields.
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Mostajeran, C., Sepulchre, R. (2017). Affine-Invariant Orders on the Set of Positive-Definite Matrices. In: Nielsen, F., Barbaresco, F. (eds) Geometric Science of Information. GSI 2017. Lecture Notes in Computer Science(), vol 10589. Springer, Cham. https://doi.org/10.1007/978-3-319-68445-1_71
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DOI: https://doi.org/10.1007/978-3-319-68445-1_71
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