6th International Young Scientist Congress (IYSC-2020) will be Postponed to 8th and 9th May 2021 Due to COVID-19. 10th International Science Congress (ISC-2020).  International E-publication: Publish Projects, Dissertation, Theses, Books, Souvenir, Conference Proceeding with ISBN.  International E-Bulletin: Information/News regarding: Academics and Research

Reverse Order Laws for the con-s-k-EP Weighted Generalized Inverses

Author Affiliations

  • 1Ramanujan Research Centre, Department of Mathematics, Govt. Arts College (Autonomous), Kumbakonam, Tamilnadu, INDIA
  • 2Ramanujan Research Centre, Department of Mathematics, Govt. Arts College (Autonomous), Kumbakonam, Tamilnadu, INDIA

Res. J. Mathematical & Statistical Sci., Volume 1, Issue (8), Pages 1-7, September,12 (2013)

Abstract

If A is a con s-k-EP matrix, then the reverse order laws for the con-s-k-EP weighted generalized inverse of A ( with respect to the given matrices M,N) is a matrix which satisfies AAA = A, AAA = A and that MAA and that AAN are symmetric under certain conditions on M,N. It is shown that the weighted generalized inverse exists if and only if A N A MA = A, in which case the inverse is NAM. When M.N are identity matrices, this reduces to the well known result that the weighted generalized inverse of a con-s-k-EP matrix when it exists, must be A.

References

  1. Rao C.R. and Mitra S.K., Generalized Inverse of Matrices and Its Applications , Wiley and Sons, New York, (1971)
  2. Krishnamoorthy S., Gunasekaran K. and Muthugobal B.K.N., Con-s-k- EP matrices , Journal of Mathematical Sciences andEngineering Applications, 5(1), 353-364 (2011)
  3. Ben-Israel A. and Greville T.N.E., Generalized Inverses: Theory and Applications, 2nd Edition, Springer, New York, 2003.
  4. Harte R.E., Invertibility and singularity, Dekker (1988)
  5. Wang G., Wei, Y. and Qiao S., Generalized inverses: theory and computat ions, Science Press, (2003)
  6. Greville T.N.E., Note on the generalized inverse of a matrix product, SIAM Rev. 8, 518-521 (1966)
  7. Xiong Z. and Qin Y., Mixed type reverse order laws for the generalized inverses of an operator product, Arab. J. Sci. Eng.,36, 475-486 (2011)
  8. Zheng B. and Xiong Z., The reverse order law for {1,2,3}- and {1,2,4}- inverses of multiple matrix product, LinearMultilinear Algebra, 58 , 765-782 (2010)
  9. Zheng B. and Xiong Z., On reverse order laws for the weighted generalized inverse, Arab. J. Sci. Eng., 34,(2A), 195-203(2009)
  10. Xiong Z. and Zheng B., The reverse order law for {1,2,3} and {1,2,4}- inverses of a two- matrix product, Appl. Math. Lett.,21 , 649-655 (2008)
  11. Cvetkovic-Ilic D. and Harte R., Reverse order laws in C*-algebras, Linear Algebra Appl. 434, 1388-1394 (2011)
  12. Cvetkovic-Ilic D.S. and Pavlovic V., A comment on some recent results concerning the reverse order law for {1,3,4}-inverses, App. Math. Comput., 217, 105-109 (2010)
  13. Sun W. and Wei Y., Inverse order rule for weighted generalized inverse, SIAM. J. Matrix Anal. Appl., 19(3), 772-775 (1998)
  14. Liu D. and Yang H., The reverse order law for {1,3,4}- inverse of the product of two matrices, Appl. Math. Comput., 215(12), 4293- 4303 (2010)
  15. Sun W. and Wei Y., Triple reverse- order law for weighted generalized inverses, Appl. Math. Comput., 125(2-3), 221-229(2002)