| The Representation of Group Inverse with Affine Combination |
| Received:December 08, 2006 Revised:October 28, 2007 |
| Key Words:
group inverses cramer rule affine combination.
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| Fund Project:the Shanghai Science and Technology Committee (No.\,062112065); Shanghai Priority Academic Discipline Foundation; the University Young Teacher Sciences Foundation of Anhui Province (No.\,2006jq1220zd) and the PhD Program Scholarship Fund of ECNU 2007. |
| Author Name | Affiliation | | SHENG Xing Ping | School of Mathematics and Computational Science, Fuyang Normal College, Anhui 236032, China
Department of Mathematics, East China Normal University, Shanghai 200062, China | | CAI Jing | Department of Mathematics, East China Normal University, Shanghai 200062, China
School of Science, Huzhou Normal College, Zhejiang 313000, China | | CHEN Guo Liang | Department of Mathematics, East China Normal University, Shanghai 200062, China
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| Abstract: |
| 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. |
| Citation: |
| DOI:10.3770/j.issn:1000-341X.2009.02.014 |
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