Aequationes mathematicae

, Volume 85, Issue 3, pp 505–517 | Cite as

Basic reverse order law and its equivalencies



In this paper we present new results related to various equivalencies of the reverse order law \({(AB)^{\dag} = B^{\dag} A^{\dag}}\) for the Moore–Penrose inverse for operators on Hilbert spaces. Some finite dimensional results given by Tian (Int J Math Educ Sci Technol 37(3):331–339, 2007) are extended to infinite dimensional settings; also some new more general relations are proved.

Mathematics Subject Classification (2000)

47A05 15A09 


Moore–Penrose inverse reverse order law 


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  1. 1.
    Ben-Israel A., Greville T.N.E.: Generalized Inverses: Theory and Applications, 2nd edn. Springer, New York (2003)MATHGoogle Scholar
  2. 2.
    Bouldin R.H.: The pseudo-inverse of a product. SIAM J. Appl. Math. 25, 489–495 (1973)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Bouldin R.H.: Generalized inverses and factorizations, Recent applications of generalized inverses. Pitman Ser. Res. Notes Math. 66, 233–248 (1982)MathSciNetGoogle Scholar
  4. 4.
    Caradus S.R.: Generalized inverses and operator theory, Queen’s paper in pure and applied mathematics. Queen’s University, Kingston (1978)Google Scholar
  5. 5.
    Dinčić N.Č., Djordjević D.S., Mosić D.: Mixed-type reverse order law and its equivalencies. Studia Math. 204, 123–136 (2011)MathSciNetMATHCrossRefGoogle Scholar
  6. 6.
    Djordjević D.S.: Unified approach to the reverse order rule for generalized inverses. Acta Sci. Math. (Szeged) 167, 761–776 (2001)Google Scholar
  7. 7.
    Djordjević D.S., Dinčić N.Č.: Reverse order law for the Moore–Penrose inverse. J. Math. Anal. Appl. 361(1), 252–261 (2010)MathSciNetMATHCrossRefGoogle Scholar
  8. 8.
    Djordjević D.S., Rakočević V.: Lectures on Generalized Inverses. Faculty of Sciences and Mathematics, University of Niš, Niš (2008)Google Scholar
  9. 9.
    Greville T.N.E.: Note on the generalized inverse of a matrix product. SIAM Rev. 8, 518–521 (1966)MathSciNetMATHCrossRefGoogle Scholar
  10. 10.
    Hartwig R.E.: The reverse order law revisited. Linear Algebra Appl. 76, 241–246 (1986)MathSciNetMATHCrossRefGoogle Scholar
  11. 11.
    Izumino S.: The product of operators with closed range and an extension of the reverse order law. J. Tohoku Math. 34, 43–52 (1982)MathSciNetMATHCrossRefGoogle Scholar
  12. 12.
    Koliha J.J., Djordjević D.S., Cvetković Ilić D.: Moore–Penrose inverse in rings with involution. Linear Algebra Appl. 426, 371–381 (2007)MathSciNetMATHCrossRefGoogle Scholar
  13. 13.
    Tian Y.: The equivalence between \({(AB)^{\dag} = B^{\dag} A^{\dag}}\) and other mixed-type reverse-order laws. Int. J. Math. Educ. Sci. Technol. 37(3), 331–339 (2007)CrossRefGoogle Scholar
  14. 14.
    Tian Y.: Using rank formulas to characterize equalities for Moore–Penrose inverses of matrix products. Appl. Math. Comput. 147, 581–600 (2004)MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Springer Basel AG 2012

Authors and Affiliations

  • Nebojša Č. Dinčić
    • 1
  • Dragan S. Djordjević
    • 1
  1. 1.Faculty of Sciences and MathematicsUniversity of NišNišSerbia

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