Generalization of the Hilbert metric to the space of positive definite matrices
type: Item
instance of: scholarly article
title: Generalization of the Hilbert metric to the space of positive definite matrices
zbMATH Open document ID: 0824.53019
author: Carlangelo Liverani
author: Maciej P. Wojtkowski
published in: Pacific Journal of Mathematics
publication date: 1995-05-30T00:00:00Z
review text: Graphs of symmetric linear maps from \(\mathbb{R}^ n\) to \(\mathbb{R}^ n\) are Lagrangian subspaces in the standard linear symplectic space \(\mathbb{R}^ n \times \mathbb{R}^ n\). The authors call a Lagrangian subspace positive, if it is a graph of a positive definite linear map. Further, they call a linear symplectic map monotone, if it maps positive Lagrangian subspaces onto positive Lagrangian subspaces. The goal of this paper is to introduce a natural Finsler metric in the manifold of positive definite matrices which, in addition to being contracted by the action of any monotone map, has striking geometric properties. In particular, they obtain that the coefficient of least contraction is equal to the image. This is the same relation obtained by Birkhoff for the Hilbert projective metric. The authors introduce a generalization of the Hilbert projective metric to the space of positive definite matrices which they view as part of the Lagrangian Grassmannian.
reviewed by: Zdeněk Jankovský
Mathematics Subject Classification ID: 53B40
Mathematics Subject Classification ID: 53C15
Mathematics Subject Classification ID: 15A99
zbMATH DE Number: 759245
zbMATH Keywords: standard linear symplectic space
zbMATH Keywords: positive Lagrangian subspaces
zbMATH Keywords: manifold of positive definite matrices
zbMATH Keywords: Hilbert projective metric
MaRDI profile type: Publication
full work available at URL: pjm.1994.166.339
OpenAlex ID: W2086412760
DOI: 10.2140/PJM.1994.166.339
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description: scientific article; zbMATH DE number 759245