On the matrix convexity of the moore—penrose inverse and some applications
Demetrios G. Kaffes, Thomas Thazumpal Mathew, M. Bhaskara Rao, K. Subramanyam · Linear and Multilinear Algebra · 1989
It is well-known that if A and B are two positive definite matrices of the same order and 0 ≤ λ ≤ 1, then . It is easy to construct an example consisting of two positive semi-definite matrices for which the above inequality is not true when one replaces the inverse operation by Moore-Penrose inverse operation. In this paper we give necessary and sufficient conditions for the validity of the inequality for every 0 ≤ λ ≤ 1. As an application, we give a sufficient condition under which the inequality (EA)+ λ E (A +) is valid, where A is a square matrix of random variables which is almost surely positive semi-definite, generalizing the well-known result (EA)− ≤ EA−1 when A is almost surely positive definite.