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An elementary proof of the Jordan-Kronecker theorem

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Abstract

This paper presents a proof of the Jordan-Kronecker theorem on the reduction to canonical form of a pair of skew-symmetric bilinear forms on a finite-dimensional linear space over an algebraically closed field.

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References

  1. A. V. Bolsinov and A. A. Oshemkov, “Bi-Hamiltonian structures and singularities of integrable systems,” Regul. Chaotic Dyn. 14(4–5), 431–454 (2009).

    Article  MATH  MathSciNet  Google Scholar 

  2. I. M. Gel’fand and I. S. Zakharevich, “Spectral theory of a pencil of skew-symmetric differential operators of third order on S 1,” Funktsional. Anal. Prilozhen. 23(2), 1–11 (1989) [Functional Anal. Appl. 23 (2), 85–93 (1989)].

    MathSciNet  Google Scholar 

  3. F. J. Turiel, “Classification locale simultanée de deux formes symplectiques compatibles,” Manuscripta Math. 82(3–4), 349–362 (1994).

    Article  MATH  MathSciNet  Google Scholar 

  4. F. J. Turiel, On the Local Theory of Veronese Webs, arXiv: math. DG/1001.3098v1.

  5. F. J. Turiel, Local Product Theorem for Bihamiltonian Structures, arXiv: math. SG/1107.2243v1.

  6. I. S. Zakharevich, Kronecker Webs, Bihamiltonian Structures, and the Method of Argument Translation, arXiv: math. SG/9908034v3.

  7. A. S. Mishchenko and A. T. Fomenko, “Euler equations on finite-dimensional Lie groups,” Izv. Akad. Nauk SSSR Ser. Mat. 42(2), 396–415 (1978) [Math. USSR-Izv. 12, 371–389 (1978)].

    MATH  MathSciNet  Google Scholar 

  8. A. S. Mishchenko and A. T. Fomenko, “Generalized Liouville method of integration of Hamiltonian systems,” Funktsional. Anal. Prilozhen. 12(2), 46–56 (1978) [Functional Anal. Appl. 12 (2), 113–121 (1978)].

    MATH  Google Scholar 

  9. A. T. Fomenko, “On symplectic structures and integrable systems on symmetric spaces,” Mat. Sb. 115(2), 263–280 (1981) [Math. USSR-Sb. 43, 235–250 (1982)].

    MathSciNet  Google Scholar 

  10. F. R. Gantmakher, Theory of Matrices (Nauka, Moscow, 1988) [in Russian].

    MATH  Google Scholar 

  11. R. C. Thompson, “Pencils of complex and real symmetric and skew matrices,” Linear Algebra Appl. 147, 323–371 (1991).

    Article  MATH  MathSciNet  Google Scholar 

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  1. Moscow State University, Moscow, Russia

    I. K. Kozlov

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  1. I. K. Kozlov
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Correspondence to I. K. Kozlov.

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Original Russian Text © I. K. Kozlov, 2013, published in Matematicheskie Zametki, 2013, Vol. 94, No. 6, pp. 857–870.

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Kozlov, I.K. An elementary proof of the Jordan-Kronecker theorem. Math Notes 94, 885–896 (2013). https://doi.org/10.1134/S0001434613110254

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