GENERALIZATION OF THE HILBERT METRIC TO THE SPACE OF POSITIVE DEFINITE MATRICES
GENERALIZATION OF THE HILBERT METRIC TO THE SPACE OF POSITIVE DEFINITE MATRICES
复制标题
希尔伯特度量在正定矩阵空间的推广
DOI：
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发表时间：
1994
期刊：
影响因子：
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通讯作者：
M. Wojtkowski
中科院分区：
文献类型：
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作者：
C. Liverani;M. Wojtkowski
我们介绍了一个推广的希尔伯特投影度量空间的正deenite矩阵，我们认为是拉格朗日格拉斯曼的一部分。x0。导论.在他对Kalman Bucylters Bougerol的处理中，B1]，B2]在被认为是黎曼对称空间的正Deenite矩阵集合上使用黎曼度量。从Rn到Rn的对称线性映射的图是标准线性辛空间Rn Rn中的拉格朗日子空间。如果一个拉格朗日子空间是正定线性映射的图，我们称它为正拉格朗日子空间。此外，我们称线性辛映射单调，如果它映射到正拉格朗日子空间的正拉格朗日子空间。Bougerol发现Kalman滤波理论中的辛矩阵是单调的。他表明，行动的任何单调映射流形上的积极拉格朗日子空间合同度量的黎曼对称空间。它是唯一的（规模）黎曼度量有这个属性。本文的目的是在正定矩阵流形中引入一个自然的Finsler度量，它除了被任何单调映射的作用所压缩外，还具有显著的几何性质.特别地，我们得到最小收缩系数等于象直径一半的双曲正切。这是由Birkhoo Bir]对希尔伯特投影度量得到的相同关系。在正正交的情况下，希尔伯特度量也只有芬斯勒（参见。W2]），它反映了锥的非光滑性。Hilbert度量推广到正定矩阵空间是不光滑的，这是很自然的，因为它在Lagrange格拉斯曼空间中的边界是不光滑的。在这篇论文写好之后，我们了解到，Vesentini Ves]早些时候从一个完全不同的观点讨论了这个度量，这个观点也是适用的。第二作者感谢ETH Z urich的Forschungsinstitut fur Mathe-matik的热情好客，这篇论文就是在那里写的。
We introduce a generalization of the Hilbert projective metric to the space of positive deenite matrices which we view as part of the Lagrangian Grass-mannian. x0. Introduction. In his treatment of Kalman Bucy lters Bougerol B1], B2] uses the Riemannian metric on the set of positive deenite matrices considered as a Riemannian symmetric space. Graphs of symmetric linear maps from R n to R n are Lagrangian subspaces in the standard linear symplectic space R n R n. We call a Lagrangian subspace positive if it is a graph of a positive deenite linear map. Further we call a linear symplectic map monotone if it maps positive Lagrangian subspaces onto positive Lagrangian subspaces. Bougerol discovered that the symplectic matrices in Kalman ltering theory are monotone. He shows that the action of any monotone map on the manifold of positive Lagrangian subspaces contracts the metric of the Riemannian symmetric space. It is the only (up to scale) Riemannian metric which has this property. The goal of this paper is to introduce a natural Finsler metric in the manifold of positive deenite matrices which in addition to being contracted by the action of any monotone map has striking geometric properties. In particular we obtain that the coeecient of least contraction is equal to the hyperbolic tangent of one half of the diameter of the image. This is the same relation which was obtained by Birkhoo Bir] for the Hilbert projective metric. In the case of the positive orthant the Hilbert metric is also only Finsler (cf. W2]) which reeects the nonsmoothness of the cone. It is natural that the generalization of the Hilbert metric to the space of positive deenite matrices is not smooth because its boundary in the Lagrangian Grassmannian is not smooth. After this paper was written we learned that this metric was discussed earlier by Vesentini Ves] from a completely diierent point of view which is applicable also The second author gratefully acknowledges the hospitality of Forschungsinstitut f ur Mathe-matik at ETH Z urich, where this paper was written.