Acta Mathematica Sinica

, Volume 21, Issue 2, pp 323–334 | Cite as

A System of Four Matrix Equations over von Neumann Regular Rings and Its Applications

  • Qing Wen WangEmail author


We consider the system of four linear matrix equations A 1 X = C 1, XB 2 = C 2, A 3 XB 3 = C 3 and A 4 XB 4 = C 4 over ℛ, an arbitrary von Neumann regular ring with identity. A necessary and sufficient condition for the existence and the expression of the general solution to the system are derived. As applications, necessary and sufficient conditions are given for the system of matrix equations A 1 X = C 1 and A 3 X = C 3 to have a bisymmetric solution, the system of matrix equations A 1 X = C 1 and A 3 XB 3 = C 3 to have a perselfconjugate solution over ℛ with an involution and char ℛ ≠2, respectively. The representations of such solutions are also presented. Moreover, some auxiliary results on other systems over ℛ are obtained. The previous known results on some systems of matrix equations are special cases of the new results.


von Neumann regular ring System of matrix equations Perselfconjugate matrix Centrosymmetric matrix Bisymmetric matrix 

MR (2000) Subject Classification

15A06 15A24 15A33 15A57 15A09 16E50 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Wang, Q. W.: A system of matrix equations and a linear matrix equation over arbitrary regular rings with identity. Linear Algebra Appl., 384, 43–54 (2004)CrossRefMathSciNetGoogle Scholar
  2. 2.
    Bhimasankaram, P.: Common solutions to the linear matrix equations AX = B,CX = D, and EXF = G. Sankhya Ser. A, 38, 404–409 (1976)MathSciNetGoogle Scholar
  3. 3.
    Aitken, A. C.: Determinants and Matrices, Oliver and Boyd, Edinburgh, 1939Google Scholar
  4. 4.
    Datta, L., Morgera, S. D.: On the reducibility of centrosymmetric matrices–applications in engineering problems. Circuits Systems Sig. Proc., 8(1), 71–96 (1989)CrossRefGoogle Scholar
  5. 5.
    Cantoni, A., Butler, P.: Eigenvalues and eigenvectors of symmetric centrosymmetric matrices. Linear Algebra Appl., 13, 275–288 (1976)CrossRefMathSciNetGoogle Scholar
  6. 6.
    Weaver, J. R.: Centrosymmetric (cross–symmetric) matrices, their basic properties, eigenvalues, eigenvectors. Amer. Math. Monthly, 92, 711–717 (1985)CrossRefMathSciNetGoogle Scholar
  7. 7.
    Lee, A.: Centrohermitian and skew–centrohermitian matrices. Linear Algebra Appl., 29, 205–210 (1980)CrossRefMathSciNetGoogle Scholar
  8. 8.
    Hell, R. D., Bates, R. G., Waters, S. R.: On centrohermitian matrices. SIAM J. Matrix Anal. Appl., 11(1), 128–133 (1990)CrossRefMathSciNetGoogle Scholar
  9. 9.
    Hell, R. D., Bates, R. G., Waters, S. R.: On perhermitian matrices. SIAM J. Matrix Anal. Appl., 11(2), 173–179 (1990)CrossRefMathSciNetGoogle Scholar
  10. 10.
    Russell, M. Reid: Some eigenvalues properties of persymmetric matrices. SIAM Rev., 39, 313–316 (1997)CrossRefMathSciNetGoogle Scholar
  11. 11.
    Andrew, A. L.: Centrosymmetric matrices. SIAM Rev., 40, 697–698 (1998)CrossRefMathSciNetGoogle Scholar
  12. 12.
    Pressman, I. S.: Matrices with multiple symmetry properties: applications of centrohermitian and perhermitian matrices. Linear Algebra Appl., 284, 239–258 (1998)CrossRefMathSciNetGoogle Scholar
  13. 13.
    Melman, A.: Symmetric centrosymmetric matrix–vector multiplication. Linear Algebra Appl., 320, 193–198 (2000)CrossRefMathSciNetGoogle Scholar
  14. 14.
    Tao, D., Yasuda, M.: A spectral characterization of generalized real symmetric centrosymmetric and generalized real symmetric skew–centrosymmetric matrices. SIAM J. Matrix Anal. Appl., 23(3), 885–895 (2002)CrossRefGoogle Scholar
  15. 15.
    Khatri, C. G., Mitra, S. K.: Hermitian and nonnegative definite solutions of linear matrix equations. SIAM J. Appl. Math., 31, 578–585 (1976)CrossRefMathSciNetGoogle Scholar
  16. 16.
    Vetter, W. J.: Vector structures and solutions of linear matrix equations. Linear Algebra Appl., 9, 181–188 (1975)CrossRefMathSciNetGoogle Scholar
  17. 17.
    Magnus, J. R., Neudecker, H.: The elimination matrix: Some lemmas and applications. SIAM J. Algebraic Discrete Methods, 1, 422–428 (1980)MathSciNetGoogle Scholar
  18. 18.
    Chu, K. E.: Symmetric solutions of linear matrix equations by matrix decomposition. Linear Algebra Appl., 119, 35–50 (1989)CrossRefMathSciNetGoogle Scholar
  19. 19.
    Henk Don, F. J.: On the symmetric solutions of a linear matrix equation. Linear Algebra Appl., 93, 1–7 (1987)CrossRefMathSciNetGoogle Scholar
  20. 20.
    Dai, H.: On the symmetric solution of linear matrix equations. Linear Algebra Appl., 131, 1–7 (1990)CrossRefMathSciNetGoogle Scholar
  21. 21.
    Wang, Q. W., Sun, J. H., Li, S. Z.: Consistency for bi(skew)symmetric solutions to systems of generalized Sylvester equations over a finite central algebra. Linear Algebra Appl., 353, 169–182 (2002)CrossRefMathSciNetGoogle Scholar
  22. 22.
    Wang, Q. W.,Wang, A. Y., Li, S. Z.: Bi(skew)symmetric and bipositive semidefinite solutions to a system of linear matrix equations over division rings. Math. Sci. Res. J., 6(7), 333–339 (2002)MathSciNetGoogle Scholar
  23. 23.
    Wang, Q. W., Tian, Y. G., Li, S. Z.: Roth’s theorems for centroselfconjugate solutions to systems of matrix equations over a finite dimensional central algebra. Southeast Asian Bulletin of Mathematics, 27, 929–938 (2004)MathSciNetGoogle Scholar
  24. 24.
    Wang, Q. W., Li, S. Z.: The persymmetric and perskewsymmetric solutions to sets of matrix equations over a finite central algebra. Acta Math Sinica, Chinese Series, 47(1), 27–34 (2004)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2005

Authors and Affiliations

  1. 1.Department of MathematicsShanghai UniversityShanghai 200444P. R. China

Personalised recommendations