Skip to main content
SpringerLink
Log in

On the Normal Form with Respect to the Semiscalar Equivalence of Polynomial Matrices Over the Field

  • Published:
Journal of Mathematical Sciences Aims and scope Submit manuscript

For a matrix pencil A 0 x − A 1 , where A 0 and A 1 are (n × n) -matrices over an arbitrary field F , and A0 is a nonsingular matrix, we establish the normal form with respect to semiscalar equivalence. We also describe the structure of nonsingular polynomial matrices over the field F , which can be reduced to the established form by the transformations of semiscalar equivalence.

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

Access this article

Log in via an institution

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

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. P. S. Kazimirskii and D. M. Bilonoga, “Semiscalar equivalence of polynomial matrices with pairwise coprime elementary divisors,” Dokl. Akad. Nauk Ukr. SSR, Ser. A, No. 4, 8–9 (1990).

    Google Scholar 

  2. P. S. Kazimirskii and O. M. Mel’nik, “Solution of the problem of semiscalar equivalence of polynomial matrices with pairwise different characteristic roots,” Dokl. Akad. Nauk Ukr. SSR, Ser. A, No. 11, 9–12 (1988).

  3. P. S. Kazimirs’kyi, Decomposition of Polynomial Matrices into Factors [in Ukrainian], Naukova Dumka, Kyiv (1981).

    Google Scholar 

  4. P. S. Kazimirs’kyi and V. M. Petrychkovych, “On the equivalence of polynomial matrices,” in: Theoretical and Applied Problems of Algebra and Differential Equations [in Ukrainian], Naukova Dumka, Kyiv (1977), pp. 61–66.

    Google Scholar 

  5. P. Lancaster, Theory of Matrices [Russian translation], Nauka, Moscow (1982).

    Google Scholar 

  6. V. M. Prokip, “Canonical form with respect to semiscalar equivalence for a matrix pencil with nonsingular first matrix,” Ukr. Mat. Zh., 63, No. 8, 1435–1440 (2011); English translation: Ukr. Math. J., 63, No. 8, 1314–1320 (2012).

    Article  MATH  Google Scholar 

  7. B. Z. Shavarovskii, “On some “tame” and “wild” aspects of the problem of semiscalar equivalence of polynomial matrices,” Mat. Zametki, 76, No. 1, 119–132 (2004); English translation: Math. Notes, 76, No. 1–2, 111–123 (2004).

    Article  MathSciNet  Google Scholar 

  8. B. Z. Shavarovskii, “On semiscalar and quasidiagonal equivalences of matrices,” Ukr. Mat. Zh., 52, No. 10, 1435–1440 (2000); English translation: Ukr. Math. J., 52, No. 10, 1638–1643 (2000).

    Article  MathSciNet  Google Scholar 

  9. J. A. Dias da Silva and T. J. Laffey, “On simultaneous similarity of matrices and related questions,” Linear Algebra Appl., 291, 167–184 (1999).

    Article  MathSciNet  MATH  Google Scholar 

  10. M. Dodig, “Eigenvalues of partially prescribed matrices,” Electron. J. Linear Algebra, 17, 316–332 (2008).

    MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Pidstryhach Institute for Applied Problems of Mechanics and Mathematics, Ukrainian National Academy of Sciences, Lviv, Ukraine

    V. M. Prokip

Authors
  1. V. M. Prokip
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Translated from Matematychni Metody ta Fizyko-Mekhanichni Polya, Vol. 55, No. 3, pp. 21–26, July–September, 2012.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Prokip, V.M. On the Normal Form with Respect to the Semiscalar Equivalence of Polynomial Matrices Over the Field. J Math Sci 194, 149–155 (2013). https://doi.org/10.1007/s10958-013-1515-2

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10958-013-1515-2

Keywords

Search

Navigation