On an equality and four inequalities for generalized inverses of Hermitian matrices
Yonge Tian · Electronic Journal of Linear Algebra · 2012
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 --P N -P * , 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 -= P N -P * to hold, and the four matrix inequalities A -> (≥, (≥, (≥, <, ≤) A -+ B -in the Löwner partial ordering to hold, respectively.