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.