特殊分块矩阵的群逆的表示

The representation of the group inverses for the special partitioned matrices

1983年,Campbell提出寻找形如M=[ABC0]的2×2分块矩阵广义逆的表达形式的问题,至今没有得到完全解决,设Cm×n是所有m×n复矩阵的集合,设A∈Cn×n,令A*为A的共轭转置。文中主要研究形为[AAA*0](其中A为幂等阵)的分块矩阵的群逆问题,一方面利用群逆的定义及其存在的充分必要条件证明形如A AA*[]0的分块矩阵的群逆的存在性;另一方面,应用群逆的求解公式M#=M(M3)(1)M及分块矩阵的一系列初等变换给出上述分块矩阵群逆的一般表示公式。

In 1983, Campbell discussed the problem of expressing the generalized inverse of a 2 × 2 partitioned block matrix M=[A B C 0] , but the problem has not yet been satisfactorily solved by scientists C 0 and researchers. Let On be the set of all m × n matrices over the field of complex numbers. Suppose A C and let A be the conjugate transpose of A. The group inverse of block matrices in the form of [A B C 0], where A is idempotent matrix. On one hand, making use of the definition and the existential [A B C 0]condition of the group inverses, the existence is proved of the form A 0 ; on the other hand, using the operating formula of group inverses M＊ M（M3）（〉M and a series of the elementary transformation of block matrices, the general eupressions of the group inverses are obtained for block matrices above.