In the area of information theory and systems science, positive definite matrices appear in connection with various concepts. Hence, mathematical problems in these area are often formulated as ones relating to the structural properties of positive definite matrices, or more formally as optimization problems on symmetric cones, which is a generalization of the set of positive definite matrices.
The research has aimed at exploiting the followings, with the assistance of information geometry and Jordan algebra theory:
(a) the information-geometric structure and properties of symmetric cones and their submanifolds,
(b) the relation between geometric structure and computational complexity of linear programming on symmetric cones,
(c) applicability of mathematical programming of this type to systems science, in particular, control theory.
As for the item (a), the results we obtained are the characterization of so-called dual connections by Jordan algebra (ref.), decomposition property of the divergence on the level surfaces defined by the characteristic function of symmetric cones (ref.), the relation of the level surfaces with affine geometry (ref.[l]) and definition of means on symmetric cones and the relation with dualistic geodesies (ref.) and so on.
As for the item (b), we have obtained a result that shows flat submanifolds in a symmetric cone with respect to the dual connections play a key role. However, we have not established a clear relation between complexity and curvatures yet. We shall keep on researching along this line.
As for the item (c), we develop applications of inequalities of symmetric matrices induced by the positive definite cone, i.e., matrix inequality, to control theory (ref. , ).