Journal of Mathematical Sciences

, Volume 157, Issue 5, pp 697–700 | Cite as

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

Article

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.

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References

  1. 1.
    C. F. Van Loan, “A symplectic method for approximating all the eigenvalues of a Hamiltonian matrix,” Linear Algebra Appl., 16, 233–251 (1984).CrossRefGoogle Scholar
  2. 2.
    M. P. Drazin, “A note on skew-symmetric matrices,” Math. Gazette, 36, 253–255 (1952).MATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    B. D. O. Anderson, “Orthogonal decompositions defined by a pair of two skew-symmetric matrices,” Linear Algebra Appl., 63, 119–132 (1984).CrossRefMathSciNetGoogle Scholar
  4. 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).MATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    D. Z. Djokovic, “On the product of two alternating matrices,” Amer. Math. Monthly, 98, 935–936 (1991).MATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    L. Rodman, “Products of symmetric and skew-symmetric matrices,” Linear Multilinear Algebra, 43, 19–34 (1997).MATHCrossRefMathSciNetGoogle Scholar
  7. 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).CrossRefMathSciNetGoogle Scholar
  8. 8.
    J. R. Bunch, “A note on the stable decomposition of skew-symmetric matrices,” Math. Comput., 38(158), 475–479 (1982).MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media, Inc. 2009

Authors and Affiliations

  1. 1.Moscow State UniversityMoscowRussia
  2. 2.Technische Universität BraunschweigBraunschweigGermany

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