群逆的仿射组合表示

The Representation of Group Inverse with Affine Combination

中文关键词:  群逆  Cramer法则  仿射组合.

英文关键词:group inverses  cramer rule  affine combination.

中文摘要:

      本文首先给出行列式运算的两个等式,接着利用群逆的满秩分解表达式$A_g=F(GF)^{-2}G$和非奇异线性方程组$Ax=b$求解的Cramer法则，给出了方阵$A$的群逆$A_g$的仿射组合表达式\[A_g=\sum_{(I,J)\in {\cal N}(A)}\frac{1}{\nu^2}\det(A)_{IJ}\det(A)_{JI}\widehat{A^{-1}_{IJ}}\]的又一直接证明方法，最后用数值例子检验了结论的正确性.

英文摘要:

      In this paper, we first give two equalities in the operation of determinant. Using the expression of group inverse with full-rank factorization $A_g=F(GF)^{-2}G$ and the Cramer rule of the nonsingular linear system $Ax=b$, we present a new method to prove the representation of group inverse with affine combination $$A_g=\sum_{(I,J)\in {\cal N}(A)}\frac{1}{\nu^2}\det(A)_{IJ}\widehat{{\rm adj}A_{JI}}.$$ A numerical example is given to demonstrate that the formula is efficient.