两个复矩阵和的强-Drazin逆

The Strong Drazin Inverse of the Sum for Two Complex Matrices

一个复数矩阵A∈C n×n有强-Drazin逆,如果存在复数矩阵X∈C n×n满足X 2A=X,AX=XA,A-AX∈N(C n×n).这里X是唯一的,并且被称为矩阵A的强-Drazin逆.在本文中,若复矩阵A和B都具有强-Drazin逆,证明在条件A 2B=0,BAB=0或AB 2=0,ABA=0下,A+λB具有强-Drazin逆,并且给出了(A+λB)sD的表示形式.进一步应用该结论给出分块矩阵的一些对应结果.最后,给出例子来说明得到的结果.

A complex element A∈C n×n has strong Drazin inverse in case there is a complex element X∈C n×n satisfying X 2A=X,AX=XA,A-AX∈N(C n×n).Here X is unique,and is called the strong Drazin inverse of A∈C n×n.In this paper,if A and B both have strong Drazin inverses,(A+λB)is proved to have strong Drazin inverse under the conditions A 2B=0,BAB=0 or AB 2=0,ABA=0,and a formula for(A+λB)sD under these conditions is obtained.Some corresponding results of the block matrix are provided by the further application of this conclusion.Finally,numerical examples are given to illustrate the results.