Generalization of the Hilbert metric to the space of positive definite matrices
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.
scientific article; zbMATH DE number 759245