体上带有和与差形式子块的分块矩阵的群逆

Group inverses of block matrices with a sub-block in the form of sum or difference over skew fields

对于体上n阶方阵A,称满足方程AXA=A,XAX=X,AX=XA的n阶方阵X为矩阵A的群逆。分块矩阵的群逆的存在性和表达式的研究不仅有重要的理论意义,而且有广泛的应用价值。分块矩阵(CAB0)的群逆存在性和表达式是一个未解决的问题。主要给出体上分块矩阵(CAB0)(其中A,B群逆存在且C=±(A+B),或者A,B群逆存在且C=±(A-B))的群逆存在的充分必要条件和表达式。

For an n×n matrix A over a skew field,the n×n matrix X is called the group inverse of A,if it satisfying the matrix equations AXA=A,XAX=X,AX=XA.Investigating the existence and representation of group inverses for block matrices not only have important theoretical significance but also extensive application value.At present,the existence and representation of group inverse for block matrix（CAB0）is still an open problem.Necessary and sufficient conditions for existence and representation of group inverse of the block matrix（CAB0）over skew fields is given for the cases：（i）A#,B# exist and C=±（A＋B）;and（ii）A#,B# exist and C=±（A-B））.