广义Schur补可逆的一些分块矩阵的Drazin逆表示

Representations of the Drazin inverse of several block matrices with a nonsingular generalized Schur complement

分块矩阵的Drazin逆不仅在矩阵理论上被广泛研究而且在自动控制、广义系统、概率统计等方面有重要的应用.给出了当广义Schur补S=D-CADB可逆时,分块矩阵M=[AC BD] ∈Cn×n(A,D是方阵)在满足下列条件之一时的Drazin逆表示:1)BCAπ=O,BDCAπ=O,D2CAπ=O;2)CAπA2=O,CAπBC=O,CAπBD=O,CAπAB=O.这些结果推广了文献[9-10,12]的结论.

The Drazin inverse of block matrices has been widely studied in matrix theory. It has important applications in automatic control, generalized systems, probability statistics, etc. In this paper, assuming that the generalized Schur complement S = D -CA^D B is nonsingular, the representations for the Drazin inverse of M=[A B C D]∈Cn×n were given （where A and D are square matrices） with the blocks satisfying one of the following conditions：l ） BCAπ =O,BDCAπ =O,D2CAπ =O;2 CAπA2 =O,CAπBC =O,CAπBD = ,CAπAB =O. These results generalize some known ones in [ 9-10,12 ].