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.
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Jeffrey L. Stuart,Diagonally scaled permutations and circulant matrices, Linear Algebra and its Appl.212/213 (1994), 397–411.
R. E. Cline,Generalized inverses of certain Toeplitz matrices, Linear Algebra and its Appl.8 (1974), 25–33.
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.
Jiang Zhaolin and Zhou Zhangxin,Circulant Matrices, Chengdu Technology University Publishing Company, Chengdu, 1999, (in Chinese)
P. Davis,Circulant Matrices, Wiley, New York, 1979.
P. Dean,Atomic vibrations in solids, J. Inst. Math. Appl.3 (1967), 98–165.
A. Wilde,Differential equations involving circulant matrices, Rocky Mount. J. Math.13 (1983), No. 1.
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.
F. J. Mac Williams, N. J. A. Sloane,The Theory Error-Correcting Codes, North-Holland, Amsterdam, 1981.
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.
G. Berman,Families of generalized weighing matrices, Canad. J. Math.30 (1978), 1016–1028.
A. J. Hoffman and R. R. Singleton,On Moore graphs with diameters two and three, IBM J. Res. Develop.4 (1960), 497–504.
G. Johnson,A generalization of N-matrices, Linear Algebra Appl.48 (1982), 201–217.
P. A. Leonard,Cyclic relative difference sets, Amer. Math. Monthly93 (1986), 106–111.
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.
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.
Foundation item: the National Natural Science Foundation of China (69972036)
Jiang Zhaolin1,2 received his BS and MS from Qufu Normal University in 1986 and 1993, respectively. Since 1986 he has been at the Linyi Teachers College. Since 2000 he has been at the Xidian University for Ph.D. His research interests focus on the numerical linear algebra and matrix theory.
Liu Sanyang1 received his BS and MS and Ph.D from Shaaxi Normal University and Xidian University and Xi'an Jiaotong University in 1982 and 1984 and 1989, respectively. Since 1987 he has been at the Xidian University. His research interests focus on the multiobjective programming, combinatorial optimization, convex analysis and matrix theory.
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Jiang, Z., Liu, S. Level-m scaled circulant factor matrices over the complex number field and the quaternion division algebra. JAMC 14, 81–96 (2004). https://doi.org/10.1007/BF02936100
AMS Mathematics Subject Classification
Key words and phrases
- Level-m scaled circulant factor matrix
- spectral decomposition
- quaternion division algebra