Skip to main content
SpringerLink
Log in

Least-Squares Solution with the Minimum-Norm for the Matrix Equation A T XB+B T X T A = D and Its Applications

  • Original Papers
  • Published:
Acta Mathematicae Applicatae Sinica, English Series Aims and scope Submit manuscript

Abstract

An efficient method based on the projection theorem, the generalized singular value decomposition and the canonical correlation decomposition is presented to find the least-squares solution with the minimum-norm for the matrix equation A T XB+B T X T A = D. Analytical solution to the matrix equation is also derived. Furthermore, we apply this result to determine the least-squares symmetric and sub-antisymmetric solution of the matrix equation C T XC = D with minimum-norm. Finally, some numerical results are reported to support the theories established in this paper.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Baksalary, J.K. Nonnegative definite and positive definite solutions to the matrix equation AXA* = B. Linear Multilinear Algebra, 16: 133–139 (1984)

    MATH  MathSciNet  Google Scholar 

  2. Braden, H.W. The equations A T X ± X T A = B. SIAM J. Matrix Anal. Appl., 20: 295–302 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  3. Chen, Y. Inverse scattering via Heisenberg's uncertainty principle. Inverse Problem, 13: 253–282 (1997)

    Article  MATH  Google Scholar 

  4. Chu, K. E. Symmetric solution of linear matrix equations by matrix decomposition. Linear Algebra Appl, 119: 35–50 (1989)

    Article  MATH  MathSciNet  Google Scholar 

  5. Dai, H., Lancaster, P. Linear matrix equations from an inverse problem of vibration theory. Linear Algebra Appl,, 246: 31–47 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  6. Deng, Y.B., Hu, X.Y., Zhang, L. The symmetric and symmetric positive semidefinite solutions of linear matrix equation B T XB = D on linear manifolds. Numer. Math.-A J. Chinese Univ. (English Series), 12: 186–192 (2003)

    MATH  Google Scholar 

  7. Fujioka, H., Hara, S. State convariance assignment problem with measurement noise: a unified approach based on a symmetric matrix equation. Linear Algebra Appl., 203–204: 579–605 (1994)

    Article  MathSciNet  Google Scholar 

  8. Golub, G.H., Van Loan, C.F. Matrix Computations. The Johns Hopkins Univ. Press, Baltimore, MD, 1997

  9. Golub, G.H., Zha, H. Perturbation analysis of the canonical correlations of matrix pairs. Linear Algebra Appl., 210: 3–28 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  10. Groβ, J. A note on the general Hermitian solution to AXA* = B. Bull. Malay. Math. Soc., 21: 57–62 (1998)

    Google Scholar 

  11. Groβ, J. Nonnegative-definite and positive-definite solutions to the matrix equation AXA* = B-revisited. Linear Algebra Appl., 321: 123–129 (2000)

    Article  MathSciNet  Google Scholar 

  12. Liao, A.P., Bai, Z.Z. Least-squares solutions of the matrix equation A T XA = D in bisymmetric matrix set. Math. Numerica Sinica, 24: 9-20 (2002) (in Chinese)

    Google Scholar 

  13. Liao, A.P., Bai, Z.Z. The constrained solutions of two matrix equations. Acta Mathematica Sinica, 18: 671–678 (2002)

    Article  MATH  Google Scholar 

  14. Liao, A. P., Bai, Z. Z., Lei, Y. Best approximate solution of matrix equation AXB + CY D = E. SIAM J. Matrix Anal. Appl., 27: 675–688 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  15. Paige, C.C., Saunders, M.A. Towards a generalized singular value decomposition. SIAM J. Numer. Anal., 18: 398–405 (1981)

    Article  MATH  MathSciNet  Google Scholar 

  16. Peng, Y.X., Hu, X.Y., Zhang, L. The symmetric and ortho-symmetric solutions of linear matrix equation A T XA = D and its optimal approximation. Numer. Math.-A J. Chinese Univ., 25: 372–377 (2003) (in Chinese)

    MATH  Google Scholar 

  17. Peng, Z.Y., Hu, X.Y., Zhang, L. One kind of inverse problems for symmetric and skew anti-symmetric matrices. Numer. Math.-A J. Chinese Univ., 25: 144–152 (2003) (in Chinese)

    MATH  Google Scholar 

  18. Sheng, Y.P., Xie, D.X. The solvability conditions for the inverse problem of symmetric and sub-antisymmetric matrices. math. Numerica Sinica, 26: 73–80 (2004) (in Chinese)

    Google Scholar 

  19. Stewart, G.W. Computing the CS-decomposition of a partional orthogonal matrix. Numer. Math., 40: 298–306 (1982)

    Article  Google Scholar 

  20. Stewart, G.W., Sun, J.G. Matrix Perturbation Theory. Academic Press, New York, 1990

  21. Shim, S.Y., Chen, Y. Least squares solutions of matrix equation AXB* + CYD* = E. SIAM J. Matrix Anal. Appl., 24: 802–808 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  22. Wang, R.S. Functional Analysis and Optimization Theory. Beijing Univ. of Aeronautics & Astronautics Press, Beijing, 2003 (in Chinese)

  23. Xie, D.X., Zhang, L. Least-squares solutions of XAX* = B over positive semidefinite matrices A. J. Comput Math., 21: 167–174 (2003)

    MATH  MathSciNet  Google Scholar 

  24. Xu, G.P., Wei, M.S., Zheng, D.S. On solutions of matrix equation AXB + CYD = F. Linear Algebra Appl., 279: 93–109 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  25. Xu, Z., Zhang, K.Y., Lu, Q. Fast Algorithms of matrices of Toeplitz Form. Northwest Industry Univ. Press, 1999 (in Chinese)

  26. Yasuda, K., Skelton, R.E. Assigning controllability and observability graminans in feedback control. J. Guidance Control, 14: 878–885 (1990)

    Article  MathSciNet  Google Scholar 

  27. Yuan, Y.X. The minimum norm solutions of two classes of matrix equations. Numer. Math.-A J. Chinese Univ., 24: 127–134 (2002) (in Chinese)

    MATH  Google Scholar 

  28. Zhou, F.Z., Zhang, L., Hu, X.Y. Least-squares solutions for inverse problems of centrosymmetric matrices. Comput. Math. Appl., 45: 1581–1589 (2003)

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. College of Mathematics and Econometrics, Hunan University, Changsha, 410082, China

    An-ping Liao, Yuan Lei & Xi-yan Hu

  2. Department of Mathematics and Information Sciences, Changsha University, Changsha, 410003, China

    An-ping Liao

Authors
  1. An-ping Liao
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Yuan Lei
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Xi-yan Hu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to An-ping Liao.

Additional information

Supported in part by Natural Science Fund of Hunan Province (No. 03JJY6028) and National Natural Science Foundation of China (No. 10171032).

Rights and permissions

Reprints and permissions

About this article

Cite this article

Liao, Ap., Lei, Y. & Hu, Xy. Least-Squares Solution with the Minimum-Norm for the Matrix Equation A T XB+B T X T A = D and Its Applications. Acta Mathematicae Applicatae Sinica, English Series 23, 269–280 (2007). https://doi.org/10.1007/s10255-007-0369-0

Download citation

  • Received:

  • Revised:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10255-007-0369-0

Keywords

2000 MR Subject Classification

Search

Navigation