The Generalized Inverse Inequalities for Symmetric Nonnegative Definite Matrices

Let A be symmetric positive definite matrix, B symmetric nonnegative definite matrix. If the difference of A and B is positive definite, then the difference of A^-1 and B^-1 is also positive definite. If A, B are all symmetric nonnegative definite matrices, Milliken and Akdeniz (1977) proved that they also have this relationship if only the ranks of the two matrices are same. That is the difference of A⁺ and B⁺ is symmetric nonnegative definite, where A⁺ is Penrose-Moore inverse matrix of A. Wang (2010) improved this inequality and extended by his result Belmega´s (2009) a theorem. In this paper, we give some inequalities of the sum and the product for symmetric nonnegative definite matrix. They all extend Milliken and Akdeniz´s (1977) and Wang´s (2010) results.