Information Geometry of Positive Matrices and Isospectral Flow

A di erential-geometric structure of the manifold PD(n) of all the n n positive-de nite matrices is studied by means of informatin geometry. It is given by a pair of dual at a ne connections with an invariant Riemannian metric and a divergence function. Special attention is paid to the geometry induced in the isospectral submanifolds and the natural gradient ow induced in it. Its relation to the Brockett ow, which is related to a completely integrable dynamical system, is elucidated from the point of view of information geometry. It is interesting to remark that intrinsic relations exist among inner methods of optimization and analysis, completely integrable dynamical systems and dually at information geometry.