Skip to main content

Advertisement

Springer Nature Link
Log in

On the equivalence of multivariate polynomial matrices

  • Published:
Multidimensional Systems and Signal Processing Aims and scope Submit manuscript

Abstract

The equivalence of system is an important concept in multidimensional (\(n\)D) system, which is closely related to equivalence of multivariate polynomial matrices. This paper mainly investigates the equivalence of some \(n\)D polynomial matrices, several new results and conditions on the reduction by equivalence of a given \(n\)D polynomial matrix to its Smith form are obtained.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Explore related subjects

Discover the latest articles, news and stories from top researchers in related subjects.

References

  • Bose, N. (1982). Applied multidimensional systems theory. New Work: Van Nostrand Reinhold.

    MATH  Google Scholar 

  • Bose, N., Buchberger, B., & Guiver, J. (2003). Multidimensional systems theory and applications. Dordrecht, The Netherlands: Kluwer.

    Google Scholar 

  • Boudellioua, M. S., & Quadrat, A. (2010). Serre’s reduction of linear function systems. Mathematics in Computer Science, 4(2), 289–312.

    Article  MATH  MathSciNet  Google Scholar 

  • Boudellioua, M. S. (2012). Computation of the Smith form for multivariate polynomial matrices using Maple. American Journal of Computational Mathematics, 2, 21–26.

    Article  Google Scholar 

  • Cluzeau, T., & Quadrat, A. (2013). Isomorphisms and Serre’s reduction of linear systems, Proceedings of nDS13, Erlangen, Germany, 09–11/09/13.

  • Cluzeau, T., & Quadrat, A. (2008). Factoring and decomposing a class of linear functional Systems. Linear Algebra and Its Applications, 428, 324–381.

    Article  MATH  MathSciNet  Google Scholar 

  • Frost, M., & Boudellioua, M. S. (1986). Some further results concerning matrices with elements in a polynomial ring. International Journal of Control, 43(5), 1543–1555.

    Article  MATH  MathSciNet  Google Scholar 

  • Kailath, T. (1980). Linear systems. Englewood Cliffs, NJ: Prentice Hall.

    MATH  Google Scholar 

  • Kung, S., Levy, B., Morf, M., & Kailath, T. (1977). New results in 2-D systems theory: Part II. Proceeding of the IEEE, 65, 945–961.

    Article  Google Scholar 

  • Lee, E., & Zak, S. (1983). Smith form over \(R[z_1, z_2]\). IEEE Transactions on Automatic Control, 28(1), 115–118.

    Article  MATH  MathSciNet  Google Scholar 

  • Lin, Z., Boudellioua, M.S., & Xu, L. (2006). On the equivalence and factorization of multivariate polynomial matrices, Proceeding of the IEEE (pp. 4911–4914).

  • Lin, Z. (1988). On matrix fraction descriptions of multivariable linear n-D systems. IEEE Transactions on Circuits and Systems, 35(10), 1317–1322.

    Article  MATH  MathSciNet  Google Scholar 

  • Lin, Z. (1999). Notes on n-D polynomial matrix factorization. Multidimensional Systems and Signal Process, 10, 379–393.

    Article  MATH  MathSciNet  Google Scholar 

  • Lin, Z., & Bose, N. (2001). A generalization of Serre’s conjecture and some related issues. Linear Algebra and Its Applications, 10, 125–138.

    Article  MATH  MathSciNet  Google Scholar 

  • Lin, Z., Ying, J., & Xu, L. (2001). Factorization for nD polynomial matrices. Circuits, Systems and Signal Processing, 20(6), 601–618.

    Article  MATH  MathSciNet  Google Scholar 

  • Lin, Z., Xu, L., & Fan, H. (2005). On minor prim factorizations for n-D polynomial matrices. IEEE Transactions on Circuits and Systems II: Express Briefs, 52(9), 568–571.

    Article  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  • Liu, J., & Wang, M. (2010). Notes on factor prime factorization for n-D polynomial matrixs. Multidimensional Systems and Signal Processing, 21(1), 87–97.

    Article  MathSciNet  Google Scholar 

  • Liu, J., Li, D., & Wang, M. (2011). On general factorization for n-D polynomial matrixs. Circuits Signal Process, 30, 553–566.

    Article  Google Scholar 

  • Liu, J., & Wang, M. (2013). New results on multivariate polynomial matrix factorizations. Linear Algebra and its Applicationgs, 438, 87–95.

    Article  MATH  MathSciNet  Google Scholar 

  • Liu, J., Li, D., & Zheng, L. (2014). The Lin–Bose problem. IEEE Transactions on Circuits and Systems II, 61, 41–43.

    Article  Google Scholar 

  • Morf, M., Levy, B., & Kung, S. (1977). New results in 2-D systems theory: Part I. Proceeding of the IEEE, 65, 861–872.

    Article  Google Scholar 

  • Rosenbrock, H. H. (1970). State space and multivariable theory. New Work, London: Nelson-Wiley.

    MATH  Google Scholar 

  • Wang, M., & Feng, D. (2004). On Lin–Bose problem. Linear Algebra and Its Applications, 390, 279–285.

    Article  MATH  MathSciNet  Google Scholar 

  • Youla, D., & Gnavi, G. (1979). Notes on n-dimensional system theory. IEEE Transactions on Circuits and Systems, CAS–26(2), 105–111.

    Article  MATH  MathSciNet  Google Scholar 

Download references

Acknowledgments

The authors are thankful to the referees for their carefully reading the article and made numerous helpful suggestions.

Author information

Authors and Affiliations

  1. School of Mathematics and Computations, Hunan University of Science and Technology, Xiangtan, 411201, Hunan, China

    Dongmei Li & Jinwang Liu

  2. School of Mathematics and Statistics, Central South University, Changsha, 410075, China

    Licui Zheng

Authors
  1. Dongmei Li
    View author publications

    You can also search for this author inPubMed Google Scholar

  2. Jinwang Liu
    View author publications

    You can also search for this author inPubMed Google Scholar

  3. Licui Zheng
    View author publications

    You can also search for this author inPubMed Google Scholar

Corresponding author

Correspondence to Jinwang Liu.

Additional information

This research is supported by the National Natural Science Foundation of China (11471108), the Tian Yuan Special Funds of National Natural Science Foundation of China (11426101) and Hunan provincial Natural Science Foundation of China (14JJ6027, 2015JJ2051).

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Li, D., Liu, J. & Zheng, L. On the equivalence of multivariate polynomial matrices. Multidim Syst Sign Process 28, 225–235 (2017). https://doi.org/10.1007/s11045-015-0329-4

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11045-015-0329-4

Keywords