体上带有和与差形式子块的分块矩阵的群逆

摘要

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.分决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(C A B 0)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(C A B 0)(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 pres- ent, the existence and representation of group inverse for block matrix(C A B 0) is still an open problem.Necessary and sufficient conditions for existence and representation of group inverse of the block matrix(C A B 0) 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) ).