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Simultaneous equivalence of square matrices

Part of the Lecture Notes in Mathematics book series (LNM,volume 1404)

Abstract

Recent results of M. Maruyama on vector bundles over the projective plane give new information on the problem of classifying m-tuples of n by n matrices upto simultaneous equivalence.

As an application we also give a linearization procedure for partial differential equations of the form \(\sum\limits_{i + j + k = n} {a_{ijk} \tfrac{{\partial ^n \psi }}{{\partial x^i \partial y^j \partial z^k }}} = c^n \psi\).

AMS-classification

  • (primary) 15 A 21
  • (secondary) 16 A 64
  • 14 F 05
  • Key Words
  • Simultaneous equivalence
  • vector bundles
  • representation theory
  • partial differential equations

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References

  1. W. Barth: Moduli of vector bundles on the projective plane, Invent. Math. 42 (1977) 63–91.

    CrossRef  MathSciNet  MATH  Google Scholar 

  2. W. Barth: Some properties of stable rank 2 vector bundles on n , Math. Ann. 226 (1966) 125–150.

    CrossRef  MathSciNet  Google Scholar 

  3. W. Barth, K. Hulek: Monads and moduli of vector bundles, Manuscripta Math. 25 (1978) 323–347.

    CrossRef  MathSciNet  MATH  Google Scholar 

  4. S. Bloch: Torsion algebraic cycles, K 2 and the Brauer group of functionfields, Bull. AMS 80 (1974).

    Google Scholar 

  5. L. Childs: Linearization of n-ic forms and generalized Clifford algebras., Lin. Mult. Alg. 5 (1978) 267–278.

    CrossRef  MathSciNet  MATH  Google Scholar 

  6. Clemens: Scrapbook on complex curve theory.

    Google Scholar 

  7. J.L Colliot-Thélène, J. Sansuc: Principal homogeneous spaces under flasque tori with applications to various problems, J. of Algebra, to appear.

    Google Scholar 

  8. E. Formanek: The center of the ring of 3 by 3 generic matrices, Lin. Mult. Alg. 7 (1979) 203–212.

    CrossRef  MathSciNet  MATH  Google Scholar 

  9. E. Formanek: The center of the ring of 4 by 4 generic matrices, J. Algebra 62 (1980) 304–319.

    CrossRef  MathSciNet  MATH  Google Scholar 

  10. K. Hulek: On the classification of stable rank r vectorbundles over the projective plane, Birkhaüser PM 7 (1980) 113–144.

    MathSciNet  Google Scholar 

  11. V. Kač: Root systems, representations of quivers and invariant theory, LNM 996, 74–108.

    Google Scholar 

  12. S. Kleiman: Les théorèmes de finitude pour le foncteur de Picard, LNM 225, 616–666.

    Google Scholar 

  13. L. Le Bruyn: Some remarks on rational matrix invariants, to appear.

    Google Scholar 

  14. M. Maruyama: Vector bundles on ℙ 2 and torsion sheaves on the dual plane, Proceedings Tata 1984.

    Google Scholar 

  15. M. Maruyama: The equations of plane curves and the moduli spaces of vector bundles on ℙ 2

    Google Scholar 

  16. M. Maruyama: Stable rationality of some moduli spaces of vector bundles on ℙ 2, LNM 1194, 80–89.

    Google Scholar 

  17. C. Procesi: Rings with polynomial identities, Marcel (1973).

    Google Scholar 

  18. C. Procesi: Invariant theory of n by n matrices, Adv. Math. 19 (1976) 306–381.

    CrossRef  MathSciNet  MATH  Google Scholar 

  19. C. Procesi: Relazioni tra geometrica algebrica ed algebra non commutative, Boll. Un. Math. Ital. (5) 18-A (1981) 1–10.

    MathSciNet  MATH  Google Scholar 

  20. D. Saltman: Retract rational fields and cyclic Galois extensions, Israel J. Math. 46 (1983).

    Google Scholar 

  21. D. Saltman: The Brauer group and the center of generic matrices, J. Alg. (1986).

    Google Scholar 

  22. M. Van den Bergh: Linearization of binary and ternary forms, to appear.

    Google Scholar 

  23. M. Van den Bergh: Center of generic division algebras, to appear.

    Google Scholar 

  24. W. Van der Kallen: The Merkurjev-Suslin Theorem, in: Integral Representations and applications; Springer LNM (1986).

    Google Scholar 

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© 1989 Springer-Verlag

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Le Bruyn, L. (1989). Simultaneous equivalence of square matrices. In: Malliavin, MP. (eds) Séminaire d’Algèbre Paul Dubreil et Marie-Paul Malliavin. Lecture Notes in Mathematics, vol 1404. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0084074

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  • DOI: https://doi.org/10.1007/BFb0084074

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-51812-9

  • Online ISBN: 978-3-540-46814-1

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