BIT Numerical Mathematics

, Volume 34, Issue 3, pp 437–450 | Cite as

Structures and uniqueness conditions of MK-weighted pseudoinverses

  • Musheng Wei
  • Birong Zhang


For given matricesM, A andK of appropriate sizes, we study the following two kinds of MK-weighted pseudoinverses ofA. The first kind of weighted pseudoinverseX is characterized by the following four Moore-Penrose like conditions:
$$\begin{gathered} (1)_M MAXA = MA; \hfill \\ (2) XAX = X; \hfill \\ (3)_M \left( {M^H MAX} \right)^H = M^H MAX; \hfill \\ (4)_K \left( {K^H KXA} \right)^H = K^H KXA. \hfill \\ \end{gathered} $$
The second kind of weighted pseudoinverseX is characterized by
$$\mathop {\min }\limits_{X \in S} \parallel X\parallel _K , with S = \{ X: \parallel AX - I\parallel _M = \mathop {\min }\limits_{W \in C^{n \times s} } \parallel AW - I\parallel _M \} .$$
Structures of these MK-weighted pseudoinverses ofA are deduced, and the uniqueness conditions of such pseudoinverses are obtained.

AMS subject classification


Key words

Weighted pseudoinverse constrained least squares uniqueness 


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  1. 1.
    A. Albert,Regression and the Moore-Penrose Pseudoinverse Academic Press, New York, 1972.Google Scholar
  2. 2.
    A. Ben-Israel and T. N. E. Greville,Generalized Inverses: Theory and Applications John Wiley & Sons, New York, 1974.Google Scholar
  3. 3.
    S. L. Campbell and C. D. Meyer, Jr.,Generalized Inverses of Linear Transformations Pitman, London, San Francisco, 1979.Google Scholar
  4. 4.
    B. De Moor and H. Zha,A tree of generalizations of the ordinary singular value decomposition Linear Algebra Appl. 147 (1991), pp. 469–500.CrossRefGoogle Scholar
  5. 5.
    L. Eldén,Perturbation theory for the least squares problem with equality constraints SIAM J. Numer. Anal. 17 (1980), pp. 338–350.CrossRefGoogle Scholar
  6. 6.
    L. Eldén,A weighted pseudoinverse, generalized singular values, and constrained least squares problems BIT 22 (1982), pp. 487–502.Google Scholar
  7. 7.
    G. H. Golub and C. F. Van Loan,Matrix Computations 2nd Edit., The Johns Hopkins University Press, Baltimore, MD, 1989.Google Scholar
  8. 8.
    M. Gullikson and P.-Å. Wedin,Modifying the QR-decomposition to constrained and weighted linear least squares SIAM J. Matrix Anal. Appl. 13 (1992), pp. 1298–1313.CrossRefGoogle Scholar
  9. 9.
    Ö. Leringe and P.-Å. Wedin,Comparison between different methods to compute a vector x which minimizes ‖Ax−b‖ when Gx=h, Computer Science Rep., Lund Univ., Sweden (1970).Google Scholar
  10. 10.
    C. C. Paige and M. A. Saunders,Towards a generalized singular value decomposition SIAM J. Numer. Anal. 18 (1981), pp. 398–405.CrossRefGoogle Scholar
  11. 11.
    C. R. Rao and S. K. Mitra,Generalized Inverses of Matrices and Its Applications John Wiley & Sons, New York, 1971.Google Scholar
  12. 12.
    C. F. Van Loan,Generalizing the singular value decomposition SIAM J. Numer. Anal. 13 (1976), pp. 76–83.CrossRefGoogle Scholar
  13. 13.
    M. Wei,Algebraic properties of the rank-deficient equality constrained and weighted least squares problems Linear Algebra Appl. 161 (1992), pp. 27–43.CrossRefGoogle Scholar
  14. 14.
    M. Wei,Perturbation theory for the rank-deficient equality constrained least squares problem SIAM J. Numer. Anal. 29 (1992), pp. 1462–1481.CrossRefGoogle Scholar

Copyright information

© BIT Foundation 1994

Authors and Affiliations

  • Musheng Wei
    • 1
  • Birong Zhang
    • 2
  1. 1.Department of MathematicsEast China Normal UniversityShanghaiP.R. China
  2. 2.Danyang Teachers SchoolDanyangP.R. China

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