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On General Factorizations for n-D Polynomial Matrices

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Abstract

Multivariate (n-D) polynomial matrix factorizations are basic research subjects in multidimensional (n-D) systems and signal processing. In this paper, several results on general matrix factorizations are provided for extracting a matrix factor from a given n-D polynomial matrix whose lower order minors satisfy certain conditions. These results are further generalizations of previous results in (Lin et al. in Circuits Syst. Signal Process. 20(6):601–618, 2001). As a consequence, the application range of the constructive algorithm in (Lin et al. in Circuits Syst. Signal Process. 20(6):601–618, 2001) has been greatly extended. Three examples are worked out in detail to show the practical value of the proposed method for obtaining general factorizations for a class of n-D polynomial matrices.

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References

  1. N.K. Bose, Applied Multidimensional Systems Theory (Van Nostrand Reinhold, New York, 1982)

    MATH  Google Scholar 

  2. N.K. Bose, C. Charoenlarpnopparut, Multivariate matrix factorization: new results. Presented at MNTS 98, Padova, Italy, pp. 97–100 (July, 1998)

  3. N.K. Bose, B. Buchberger, J.P. Guiver, Multidimensional Systems Theory and Applications (Kluwer Academic, Dordrecht, 2003)

    MATH  Google Scholar 

  4. C. Charoenlarpnopparut, N.K. Bose, Multidimensional FIR filter bank design using Gröbner bases. IEEE Trans. Circuits Syst. II: Analog. Digit. Signal Process. 46, 1475–1486 (1999)

    Article  MATH  Google Scholar 

  5. E. Fomasini, M.E. Valcher, nD polynomial matrix with applications to multidimensional signal analysis. Multidimens. Syst. Signal Process. 8, 387–408 (1997)

    Article  Google Scholar 

  6. A. Fabianska, A. Quadrat, Applications of the Qullen–Suslin theorem to multidimensional systems theory, in Gröbner Bases in Control Theory and Signal Processing, ed. by H. Park, G. Regensburger (Walter de Gruyter, Berlin, 2006), pp. 23–106

    Google Scholar 

  7. J.P. Guiver, N.K. Bose, Polynomial matrix primitive factorization over arbitrary coefficient field and related results. IEEE Trans. Circuits Syst. 29, 649–657 (1982)

    Article  MATH  MathSciNet  Google Scholar 

  8. G.M. Greuel, G. Pfister, A Singular Introduction to Commutative Algebra, 2nd edn. (Springer, Berlin, 2008)

    Google Scholar 

  9. S. Kleon, U. Oberst, Transfer operators and state spaces for discrete multidimensional linear systems. Acta Appl. Math. 57, 1–82 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  10. T.Y. Lam, Serre’s Problem on Projective Module (Springer, Berlin, 2006)

    Book  Google Scholar 

  11. Z. Lin, On primitive factorizations for n-D polynomial matrices, in Proc. IEEE Symp. Circuits Syst., Chicago, IL (1993), pp. 595–598

    Google Scholar 

  12. Z. Lin, On primitive factorizations for 3-D polynomial matrices. IEEE Trans. Circuits Syst. 39(12), 1024–1027 (1992)

    Article  MATH  Google Scholar 

  13. Z. Lin, Notes on n-D polynomial matrix factorizations. Multidimens. Syst. Signal Process. 10, 379–393 (1999)

    Article  MATH  Google Scholar 

  14. Z. Lin, N.K. Bose, A generalization of Serre’s conjecture and related issues. Linear Algebra Appl. 338, 125–138 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  15. Z. Lin, J. Ying, L. Xu, Factorizations for nD polynomial matrices. Circuits Syst. Signal Process. 20(6), 601–618 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  16. Z. Lin, L. Xu, H. Fan, On minor prime factorization for n-D polynomial matrices. IEEE Trans. Circuits Syst. II: Express Briefs 52(9), 568–571 (2005)

    Article  Google Scholar 

  17. Z. Lin, L. Xu, N.K. Bose, A tutorial on Gröbner bases with applications in signals and systems. IEEE Trans. Circuits Syst. I 55(1), 445–461 (2008)

    Article  MathSciNet  Google Scholar 

  18. J. Liu, M. Wang, Notes on factor prime factorizations for n-D polynomial matrices. Multidimens. Syst. Signal Process. 21(1), 87–97 (2010)

    Article  MathSciNet  Google Scholar 

  19. N.E. Mastorakis, N.J. Theodorou, S.G. Tzafestas, A general factorization method for multivariate polynomials, Multidimens. Syst. Signal Process. 5, 151–178 (1994)

    MATH  MathSciNet  Google Scholar 

  20. J.F. Pommaret, Solving Bose conjecture on linear multidimensional systems, in Proc. Eur. Control Conf., (2001), pp. 1853–1855

    Google Scholar 

  21. V. Srinivas, A generalized Serre problem. J. Algebra 278(2), 621–627 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  22. M. Wang, D. Feng, On Lin–Bose problem. Linear Algebra Appl. 390, 279–285 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  23. M. Wang, C.P. Kwong, On multivariate polynomial matrix factorization problems. Math. Control Signals Syst. 17(4), 297–311 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  24. M. Wang, On factor prime factorization for n-D polynomial matrices. IEEE Trans. Circuits Syst. I, Reg. Papers 54(6), 1398–1405 (2007)

    Article  Google Scholar 

  25. M. Wang, Remarks on n-D polynomial matrix factorization problems. IEEE Trans. Circuits Syst. II: Express Briefs 55(1), 61–64 (2008)

    Article  Google Scholar 

  26. D.C. Youla, G. Gnavi, Notes on n-dimensional system theory. IEEE Trans. Circuits Syst. CAS-26(2), 105–111 (1979)

    Article  MathSciNet  Google Scholar 

  27. D.C. Youla, P.F. Pickel, The Quillen–Suslin theorem and the structure of n-dimensional elementary polynomial matrices. IEEE Trans. Circuits Syst. CAS-31(6), 513–518 (1984)

    Article  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. College of Mathematics and Computation, Hunan Science and Technology University, Xiangtan, Hunan, 411201, China

    Jinwang Liu & Dongmei Li

  2. State Key Lab of Information Security, Institute of Software, Chinese Academy of Sciences, Beijing, China

    Mingsheng Wang

Authors
  1. Jinwang Liu
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  2. Dongmei Li
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  3. Mingsheng Wang
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Correspondence to Mingsheng Wang.

Additional information

Liu was supported in part by the Natural Science Foundation of China under Grant (10771058), and Wang was partially supported by the Natural Science Foundation of China under Grant (60970134), and the 973 project under Grant (2007CB311201).

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Liu, J., Li, D. & Wang, M. On General Factorizations for n-D Polynomial Matrices. Circuits Syst Signal Process 30, 553–566 (2011). https://doi.org/10.1007/s00034-010-9229-x

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