Geometry on Positive Definite Matrices Deformed by V-Potentials and Its Submanifold Structure

  • Atsumi OharaEmail author
  • Shinto Eguchi
Part of the Signals and Communication Technology book series (SCT)


In this paper we investigate dually flat structure of the space of positive definite matrices induced by a class of convex functions called V-potentials, from a viewpoint of information geometry. It is proved that the geometry is invariant under special linear group actions and naturally introduces a foliated structure. Each leaf is proved to be a homogeneous statistical manifold with a negative constant curvature and enjoy a special decomposition property of canonically defined divergence. As an application to statistics, we finally give the correspondence between the obtained geometry on the space and the one on elliptical distributions induced from a certain Bregman divergence.


Information geometry Divergence Elliptical distribution Negative constant curvature Affine differential geometry 



We thank the anonymous referees for their constructive comments and careful checks of the original manuscript.


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© Springer International Publishing Switzerland 2014

Authors and Affiliations

  1. 1.Department of Electrical and Electronics EngineeringUniversity of FukuiFukuiJapan
  2. 2.The Institute of Statistical MathematicsTachikawaJapan

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