J. Aust. Math. Soc. 83 (2007), no. 3, pp. 423–437.
The generalized inverse A_{T,S}^{(2)} of a matrix over an associative ring
Yaoming Yu† Guorong Wang
College of Education Shanghai Normal University Shanghai 200234 People's Republic of China
yuyaoming@online.sh.cn		 College of Mathematics Science Shanghai Normal University Shanghai 200234 People's Republic of China
grwang@shnu.edu.cn
Received 6 July 2006; revised 3 January 2007
Communicated by J. Koliha
Supported by Science Foundation of Shanghai Municipal Education Commission (CW0519).

Abstract

In this paper we establish the definition of the generalized inverse A_{T,S}^{(2)} which is a \{2\} inverse of a matrix A with prescribed image T and kernel S over an associative ring, and give necessary and sufficient conditions for the existence of the generalized inverse A_{T,S}^{(1,2)} and some explicit expressions for A_{T,S}^{(1,2)} of a matrix A over an associative ring, which reduce to the group inverse or \{1\} inverses. In addition, we show that for an arbitrary matrix A over an associative ring, the Drazin inverse A_{d}, the group inverse A_{g} and the Moore–Penrose inverse A^{\dagger }, if they exist, are all the generalized inverse A_{T,S}^{(2)}.

Download the article in PDF format (size 144 Kb)

2000 Mathematics Subject Classification: primary 15A33, 15A09
(Metadata: XML, RSS, BibTeX)
†indicates author for correspondence

References

  1. F. W. Anderson and K. R. Full, Rings and Categories of Modules (Springer-Verlag, New York Heidelberg Berlin, 1973). MR1245487
  2. A. Ben-Israel and T. N. E. Greville, Generalized Inverses: Theory and Applications, 2nd edition (Springer Verlag, New York, 2003). MR1987382
  3. M. P. Drazin, ‘Pseudo-inverses in associative rings and semigroups’, Amer. Math. Monthly 65 (1958), 506–514. MR98762
  4. M. C. Gouveia and R. Puystjens, ‘About the group inverse and Moore-Penrose inverse of a product’, Linear Algebra Appl. 150 (1991), 361–369. MR1102077
  5. M. Z. Nashed (ed.), Generalized Inverses and Applications (Academic Press, New York, 1976). MR451661
  6. P. Patrício, ‘The Moore–Penrose inverse of von Neumann regular matrices over a ring’, Linear Algebra Appl. 332-334 (2001), 469–483. MR1839446
  7. R. Puystjens and M. C.Gouveia, ‘Drazin invertiblity for matrices over an arbitrary ring’, Linear Algebra Appl. 385 (2004), 105–116. MR2063351
  8. R. Puystjens and R. E. Hartwig, ‘The group inverse of a companion matrix’, Linear Multilinear Algebra 43 (1997), 137–150. MR1613187
  9. K. P. S. Bhaskara Rao, The Theory of Generalized Inverses over Commutative Rings, volume 17 of Algebra, logic and Applications Series (Taylor and Francis, London and New York, 2002). MR1919967
  10. G. Wang, Y. Wei and S. Qiao, Generalized Inverses: Theory and Computations (Science Press, Beijing/New York, 2004).
  11. Y. Wei, ‘A characterization and representation of the generalized inverse {A}_{T,S}^{(2)} and its applications’, Linear Algebra Appl. 280 (1998), 87–96. MR1645022
  12. Y. Yu and G. Wang, ‘The existence of Drazin inverses over integral domains’, J. Shanghai Normal University(NS) 32 (2003), 12–15.
  13. Y. Yu and G. Wang, ‘The generalized inverse A_{T,S}^{(2)} over commutative rings’, Linear Multilinear Algebra 53 (2005), 293–302. MR2160411