On an equality and four inequalities for generalized inverses of Hermitian matrices

A Hermitian matrix X is called a g-inverse of a Hermitian matrix A, denoted by A, if it satisfies AXA = A. In this paper, a group of explicit formulas are established for calculating the global maximum and minimum ranks and inertias of the difference A − PNP , where both A and N are Hermitian g-inverses of two Hermitian matrices A and N , respectively. As a consequence, necessary and sufficient conditions are derived for the matrix equality A = PNP ∗ to hold, and the four matrix inequalities A > (≥, <, ≤)PNP ∗ in the Löwner partial ordering to hold, respectively. In addition, necessary and sufficient conditions are established for the Hermitian matrix equality A = PNP ∗ to hold, and the four Hermitian matrix inequalities A > (≥, < , ≤)PNP ∗ to hold, respectively, where (·) denotes the Moore-Penrose inverse of a matrix. As applications, identifying conditions are given for the additive decomposition of a Hermitian g-inverse C = A + B (parallel sum of two Hermitian matrices) to hold, as well as the four matrix inequalities C > (≥, <, ≤)A + B in the Löwner partial ordering to hold, respectively.