Dualistic differential geometry of positive definite matrices and its applications to related problems Original Research Article

Analysis on the set of positive definite matrices is involved in many engineering problems. We develop a differential geometric theory of the set of positive definite matrices by means of a specific class of connections introduced into it. Consequently, various dualistic aspects of the set are elucidated. Next, using the theory, we derive new results and interesting geometrical interpretations for matrix approximation, positive definite matrix completion, and linear matrix inequality problems.