Skip to main content
SpringerLink
Log in

On the product of two skew-Hamiltonian or two skew-symmetric matrices

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

We show that the product C of two skew-Hamiltonian matrices obeys the Stenzel conditions. If at least one of the factors is nonsingular, then the Stenzel conditions amount to the requirement that every elementary divisor corresponding to a nonzero eigenvalue of C occurs an even number of times. The same properties are valid for the product of two skew-pseudosymmetric matrices. We observe that the method proposed by Van Loan for computing the eigenvalues of real Hamiltonian and skew-Hamiltonian matrices can be extended to complex skew-Hamiltonian matrices. Finally, we show that the computation of the eigenvalues of a product of two skew-symmetric matrices reduces to the computation of the eigenvalues of a similar skew-Hamiltonian matrix. Bibliography: 8 titles.

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. C. F. Van Loan, “A symplectic method for approximating all the eigenvalues of a Hamiltonian matrix,” Linear Algebra Appl., 16, 233–251 (1984).

    Article  Google Scholar 

  2. M. P. Drazin, “A note on skew-symmetric matrices,” Math. Gazette, 36, 253–255 (1952).

    Article  MATH  MathSciNet  Google Scholar 

  3. B. D. O. Anderson, “Orthogonal decompositions defined by a pair of two skew-symmetric matrices,” Linear Algebra Appl., 63, 119–132 (1984).

    Article  MathSciNet  Google Scholar 

  4. R. Gow and T. J. Laffey, “Pairs of alternating forms and products of two skew-symmetric matrices,” Linear Algebra Appl., 63, 119–132 (1984).

    Article  MATH  MathSciNet  Google Scholar 

  5. D. Z. Djokovic, “On the product of two alternating matrices,” Amer. Math. Monthly, 98, 935–936 (1991).

    Article  MATH  MathSciNet  Google Scholar 

  6. L. Rodman, “Products of symmetric and skew-symmetric matrices,” Linear Multilinear Algebra, 43, 19–34 (1997).

    Article  MATH  MathSciNet  Google Scholar 

  7. H. Stenzel, “Über die Darstellbarkeit einer Matrix als Produkt von zwei symmetrischer Matrizen, als Produkt von zwei alternierenden Matrizen und als Produkt von einer symmetrischen und einer alternierenden Matrix,” Math. Z., 15, 1–25 (1922).

    Article  MathSciNet  Google Scholar 

  8. J. R. Bunch, “A note on the stable decomposition of skew-symmetric matrices,” Math. Comput., 38(158), 475–479 (1982).

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Moscow State University, Moscow, Russia

    Kh. D. Ikramov

  2. Technische Universität Braunschweig, Braunschweig, Germany

    H. Fassbender

Authors
  1. Kh. D. Ikramov
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. H. Fassbender
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Kh. D. Ikramov.

Additional information

Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 359, 2008, pp. 45–51.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Ikramov, K.D., Fassbender, H. On the product of two skew-Hamiltonian or two skew-symmetric matrices. J Math Sci 157, 697–700 (2009). https://doi.org/10.1007/s10958-009-9352-z

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10958-009-9352-z

Keywords

Search

Navigation