The block independence in the generalized inverse AT,S(2) for some ordered matrices and applications

In this paper, the definition of block independence in the generalized inverse AT,S(2) is firstly given, and then a necessary and sufficient condition for some ordered matrices to be block independent in the generalized inverse AT,S(2) is derived. As an application, a necessary and sufficient condition forA1+A2+⋯+AkT,S(2)=A1T1,S1(2)+A2T2,S2(2)+⋯+AkTk,Sk(2)is proved. Moreover, some results are shown with respect to the Moore–Penrose inverse, the Weighted Moore–Penrose inverse and the Drazin inverse, respectively.