Equation Solving Generalized Inverses

Part of the Developments in Mathematics book series (DEVM, volume 53)


There are various ways to introduce the generalized inverses. We introduce them by considering the problem of solving systems of linear equations.


  1. 1.
    G.W. Stewart, Introduction to Matrix Computation (Academic Press, New York, 1973)zbMATHGoogle Scholar
  2. 2.
    C.C. MacDuffe, The Theory of Matrices (Chelsea, New York, 1956)Google Scholar
  3. 3.
    N.S. Urquhart, Computation of generalized inverse matrtices which satisfy specified conditions. SIAM Rev. 10, 216–218 (1968)MathSciNetCrossRefGoogle Scholar
  4. 4.
    S. Zlobec, An explicit form of the Moore-Penrose inverse of an arbitrary complex matrix. SIAM Rev. 12, 132–134 (1970)MathSciNetCrossRefGoogle Scholar
  5. 5.
    A. Ben-Israel, T.N.E. Greville, Generalized Inverses: Theory and Applications, 2nd edn. (Springer Verlag, New York, 2003)zbMATHGoogle Scholar
  6. 6.
    Y. Chen, Generalized Bott-Duffin inverse and its applications. Appl. Math. J. Chinese Univ. 4, 247–257 (1989). in ChineseGoogle Scholar
  7. 7.
    S.L. Campbell, C.D. Meyer Jr., Generalized Inverses of Linear Transformations (Pitman, London, 1979)zbMATHGoogle Scholar
  8. 8.
    X. He, W. Sun. Introduction to Generalized Inverses of Matrices. (Jiangsu Science and Technology Press, 1990). in ChineseGoogle Scholar
  9. 9.
    Y. Chen, The generalized Bott-Duffin inverse and its applications. Linear Algebra Appl. 134, 71–91 (1990)MathSciNetCrossRefGoogle Scholar
  10. 10.
    C.L. Lawson, R.J. Hanson, Solving Least Squares Problems (Prentice-Hall Inc, Englewood Cliffs, N.J., 1974)zbMATHGoogle Scholar
  11. 11.
    Å. Björck, Numerical Methods for Least Squares Problems (SIAM, Philadelphia, 1996)CrossRefGoogle Scholar
  12. 12.
    M. Wei, Supremum and Stability of Weighted Pseudoinverses and Weighted Least Squares Problems Analysis and Computations (Nova Science Publisher Inc, Huntington, NY, 2001)zbMATHGoogle Scholar
  13. 13.
    L. Eldén, A weighted pseudoinverse, generalized singular values and constrained least squares problems. BIT 22, 487–502 (1982)MathSciNetCrossRefGoogle Scholar
  14. 14.
    C. Bu, W. Gu, J. Zhou, Y. Wei, On matrices whose Moore-Penrose inverses are ray unique. Linear Multilinear Algebra 64(6), 1236–1243 (2016)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Y. Chen, On the weighted projector and weighted generalized inverse matrices. Acta Math. Appl. Sinica 6, 282–291 (1983). in ChineseMathSciNetzbMATHGoogle Scholar
  16. 16.
    L. Sun, B. Zheng, C. Bu, Y. Wei, Moore-Penrose inverse of tensors via Einstein product. Linear Multilinear Algebra 64(4), 686–698 (2016)MathSciNetCrossRefGoogle Scholar
  17. 17.
    J. Ji, Y. Wei, Weighted Moore-Penrose inverses and fundamental theorem of even-order tensors with Einstein product. Front. Math. China 12(6), 1319–1337 (2017)MathSciNetCrossRefGoogle Scholar
  18. 18.
    C. Deng, Y. Wei, Further results on the Moore-Penrose invertibility of projectors and its applications. Linear Multilinear Algebra 60(1), 109–129 (2012)MathSciNetCrossRefGoogle Scholar
  19. 19.
    D.S. Djordjević, P.S. Stanimirović, Y. Wei, The representation and approximations of outer generalized inverses. Acta Math. Hungar. 104, 1–26 (2004)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Y. Wei, H. Wu, On the perturbation and subproper splittings for the generalized inverse \(A_{T, S}^{(2)}\) of rectangular matrix \(A\). J. Comput. Appl. Math. 137, 317–329 (2001)Google Scholar
  21. 21.
    Y. Wei, H. Wu, (\(T\)-\(S\)) splitting methods for computing the generalized inverse \(A_{T, S}^{(2)}\) of rectangular systems. Int. J. Comput. Math. 77, 401–424 (2001)Google Scholar
  22. 22.
    Y. Wei, N. Zhang, Condition number related with generalized inverse \(A_{T, S}^{(2)}\) and constrained linear systems. J. Comput. Appl. Math. 157, 57–72 (2003)MathSciNetCrossRefGoogle Scholar
  23. 23.
    M. Wei, B. Zhang, Structures and uniqueness conditions of MK-weighted pseudoinverses. BIT 34, 437–450 (1994)MathSciNetCrossRefGoogle Scholar
  24. 24.
    J. Shao, H. Shan, The solution of a problem on matrices having signed generalized inverses. Linear Algebra Appl. 345, 43–70 (2002)MathSciNetCrossRefGoogle Scholar
  25. 25.
    J. Zhou, C. Bu, Y. Wei, Group inverse for block matrices and some related sign analysis. Linear Multilinear Algebra 60, 669–681 (2012)MathSciNetCrossRefGoogle Scholar
  26. 26.
    J. Zhou, C. Bu, Y. Wei, Some block matrices with signed Drazin inverses. Linear Algebra Appl. 437, 1779–1792 (2012)MathSciNetCrossRefGoogle Scholar
  27. 27.
    G.W. Stewart, On scaled projections and pseudoinverses. Linear Algebra Appl. 112, 189–193 (1989)MathSciNetCrossRefGoogle Scholar
  28. 28.
    M. Wei, Upper bound and stability of scaled pseudoinverses. Numer. Math. 72(2), 285–293 (1995)MathSciNetCrossRefGoogle Scholar
  29. 29.
    O.M. Baksalary, G. Trenkler, Core inverse of matrices. Linear Multilinear Algebra 58(5–6), 681–697 (2010)MathSciNetCrossRefGoogle Scholar
  30. 30.
    D. Rakić, N. Dinčić, D. Djordjević, Group, Moore-Penrose, core and dual core inverse in rings with involution. Linear Algebra Appl. 463, 115–133 (2014)MathSciNetCrossRefGoogle Scholar
  31. 31.
    Y. Wei, P. Xie, L. Zhang, Tikhonov regularization and randomized GSVD. SIAM J. Matrix Anal. Appl. 37(2), 649–675 (2016)MathSciNetCrossRefGoogle Scholar
  32. 32.
    F.T. Luk, S. Qiao, Analysis of a recursive least squares signal processing algorithm. SIAM J. Sci. Stat. Comput. 10, 407–418 (1989)MathSciNetCrossRefGoogle Scholar
  33. 33.
    F. Hsuan, P. Langenberg, A. Getson, The \(\{2\}\)-inverse with applications in statistics. Linear Algebra Appl. 70, 241–248 (1985)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Nature Singapore Pte Ltd. and Science Press 2018

Authors and Affiliations

  1. 1.Department of MathematicsShanghai Normal UniversityShanghaiChina
  2. 2.Department of MathematicsFudan UniversityShanghaiChina
  3. 3.Department of Computing and SoftwareMcMaster UniversityHamiltonCanada

Personalised recommendations