Applied Mathematics and Mechanics

, Volume 16, Issue 12, pp 1133–1141 | Cite as

The explicit solution of the matrix equation AX−XB=C

  • Chen Yuming
  • Xiao Heng


Almost all of the existing results on the explicit solutions of the matrix equation AX−XB=C are obtained under the condition that A and B have no eigenvalues in common. For both symmetric or skewsymmetric matrices A and B, we shall give out the explicit general solutions of this equation by using the notions of eigenprojections. The results we obtained are applicable not only to any cases of eigenvalues regardless of their multiplicities, but also to the discussion of the general case of this equation.

Key words

matrix equation explicit solution eigenprojection matrix square-product 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    W. E. Roth, The equationAX−YB=C andAX−XB=C in matrices,Proc. Amer. Math. Soc.,97, (1952), 392–396.CrossRefGoogle Scholar
  2. [2]
    S. Barnett and C. Storey, Some applications of Liapunov matrix equations,J. Inst. Math. Appl.,4, 1 (1968), 33–42.MATHGoogle Scholar
  3. [3]
    A. Jameson, Solution of the equationAX+XB=C by inversion of anM×M orN×N matrix,SIAM J. Appl. Math.,16, (1968), 1020–1023.MATHMathSciNetCrossRefGoogle Scholar
  4. [4]
    P. Lancaster, Explicit solution of linear matrix equations,SIAM Rev.,12 (1970), 544–566.MATHMathSciNetCrossRefGoogle Scholar
  5. [5]
    D. H. Carlson and B. N. Datta, The Liapunov matrix equationSA+A * S=S * B * BS,Linear Algebra Appl.,28 (1979), 43–53.MATHMathSciNetCrossRefGoogle Scholar
  6. [6]
    Eurice de Souza and S. P. Bhattacharyya, Controllability, observability and the solution ofAX−XB=C, Linear Algebra Appl.,39 (1981), 167–188.MATHMathSciNetCrossRefGoogle Scholar
  7. [7]
    T. E. Djaferis and S. K. Mitter, Algebraic methods for the study of some linear matrix equations,Linear Algebra Appl.,44, (1982), 125–142.MATHMathSciNetCrossRefGoogle Scholar
  8. [8]
    J. K. John Jones and C. Lew, Solutions of Liapunov matrix equationBX−XA=C, IEEE Trans, Automatic Control. AC-27 (1982), 464–466.CrossRefGoogle Scholar
  9. [9]
    Gao Weixing, Continued-fraction solution of matrix equationAX−XB=C, Scientia Sinica Ser. A.,32, (1989), 1025–1035.Google Scholar
  10. [10]
    H. K. Wimmer, Linear matrix equation: the module theoretic approach.Linear Algebra Appl.,120 (1989), 149–164.MATHMathSciNetCrossRefGoogle Scholar
  11. [11]
    Ma Er-chieh, A finite series solution of the matrix equationAX−XB=C, SIAM J. Appl., Math.,14, (1966), 490–495.MathSciNetCrossRefGoogle Scholar
  12. [12]
    B. N. Datta and K. Datta, The matrix equationXA=A T X and an associated algorithm for solving the inertia and stability problems,Linear Algebra Appl.,97 (1987), 103–119.MATHMathSciNetCrossRefGoogle Scholar
  13. [13]
    Guo Zhongheng, T. H. Lehman, Liang Haoyun and C.-S. Man, Twirl tensors and the tensor equationAX−XA=C,J. Elasticity,27, 2 (1992), 227–245.MathSciNetCrossRefMATHGoogle Scholar
  14. [14]
    C. D. Luehr and M. B. Rubin, The significance of projectors in the spectral representation of symmetric second order tensors,Comput. Methods Appl. Mech. Engrg.,84 (1990), 243–246.MATHMathSciNetCrossRefGoogle Scholar
  15. [15]
    Guo Zhongheng, Li Jianbo, Xiao Heng and Chen Yuming, Intrinsic solution to then-dimensional tensor equation ∑r=1m Um−r×Ur−1=C.Comput. Methods Appl. Mech. Engrg.,115 (1994), 359–364.MathSciNetCrossRefGoogle Scholar

Copyright information

© SU Shanghai, China 1995

Authors and Affiliations

  • Chen Yuming
    • 1
  • Xiao Heng
    • 2
  1. 1.Department of Applied MathematicsHuman UniversityChangshaP. R. China
  2. 2.Department of MathematicsPeking UniversityBeijingP. R. China

Personalised recommendations