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

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## Abstract

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## Preview

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## References

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

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