Level-m scaled circulant factor matrices over the complex number field and the quaternion division algebra

  • Zhao-Lin Jiang
  • San-Yang Liu


The level-m scaled circulant factor matrix over the complex number field is introduced. Its diagonalization and spectral decomposition and representation are discussed. An explicit formula for the entries of the inverse of a level-m scaled circulant factor matrix is presented. Finally, an algorithm for finding the inverse of such matrices over the quaternion division algebra is given.

AMS Mathematics Subject Classification

15A21 65F15 

Key words and phrases

Level-m scaled circulant factor matrix diagonalization spectral decomposition inverse quaternion division algebra 


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  1. 1.
    Jeffrey L. Stuart,Diagonally scaled permutations and circulant matrices, Linear Algebra and its Appl.212/213 (1994), 397–411.CrossRefMathSciNetGoogle Scholar
  2. 2.
    R. E. Cline,Generalized inverses of certain Toeplitz matrices, Linear Algebra and its Appl.8 (1974), 25–33.MATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Jiang Zhaolin and Guo Yunrui,The explicit expressions of level-k circulant matrices of type and their some properties, Chinese Quarterly Journal of Mathematics11 (1996), No. 3, 103–110.Google Scholar
  4. 4.
    Jiang Zhaolin and Zhou Zhangxin,Circulant Matrices, Chengdu Technology University Publishing Company, Chengdu, 1999, (in Chinese)Google Scholar
  5. 5.
    P. Davis,Circulant Matrices, Wiley, New York, 1979.MATHGoogle Scholar
  6. 6.
    P. Dean,Atomic vibrations in solids, J. Inst. Math. Appl.3 (1967), 98–165.CrossRefGoogle Scholar
  7. 7.
    A. Wilde,Differential equations involving circulant matrices, Rocky Mount. J. Math.13 (1983), No. 1.Google Scholar
  8. 8.
    J. Ruiz-Claeyssen, M. Davila, and T. Tsukazan,Factor block circulant and periodic solutions of undamped matrix differential equations, Mat. Apl. Comput.3 (1983), No. 1.Google Scholar
  9. 9.
    F. J. Mac Williams, N. J. A. Sloane,The Theory Error-Correcting Codes, North-Holland, Amsterdam, 1981.Google Scholar
  10. 10.
    J. C. R. Claeyssen and L. A. S. Leal,Diagonalization and spectral decomposition of factor block circulant matrices, Linear Algebra and its Appl.99 (1988), 41–61.MATHCrossRefGoogle Scholar
  11. 11.
    G. Berman,Families of generalized weighing matrices, Canad. J. Math.30 (1978), 1016–1028.MATHMathSciNetGoogle Scholar
  12. 12.
    A. J. Hoffman and R. R. Singleton,On Moore graphs with diameters two and three, IBM J. Res. Develop.4 (1960), 497–504.MATHMathSciNetCrossRefGoogle Scholar
  13. 13.
    G. Johnson,A generalization of N-matrices, Linear Algebra Appl.48 (1982), 201–217.MATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    P. A. Leonard,Cyclic relative difference sets, Amer. Math. Monthly93 (1986), 106–111.MATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Predrag S. Stanimirović and Milan B. Tasić,Computing determinantal representation of generalized inverses, J. Appl. Math. and Computing(old KJCAM)9 (2002), No. 2, 349–360.MATHGoogle Scholar
  16. 16.
    Jae Heon Yun and Sang Wook Kim,A variant of block incomplete factorization preconditioners for a symmetric H-matrix, J. Appl. Math. and Computing(old KJCAM)8 (2001), No. 3, 481–496.MATHMathSciNetGoogle Scholar

Copyright information

© Korean Society for Computational and Applied Mathematics 2004

Authors and Affiliations

  1. 1.Department of Applied MathematicsXidian UniversityXi'anP. R. China
  2. 2.Department of MathematicsLinyi Teachers CollegeLinyi, ShandongP. R. China

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