Convexity of the inverse and Moore-Penrose inverse

Convexity properties of the inverse of positive definite matrices and the Moore-Penrose inverse of nonnegative definite matrices with respect to the partial ordering induced by nonnegative definiteness are studied. For the positive definite case null-space characterizations are derived, and lead naturally to a concept of strong convexity of a matrix function, extending the conventional concept of strict convexity. The positive definite results are shown to allow for a unified analysis of problems in reproducing kernel Hilbert space theory and inequalities involving matrix means. The main results comprise a detailed study of the convexity properties of the Moore-Penrose inverse, providing extensions and generalizations of all the earlier work in this area.