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The BMV-conjecture over quaternions and octonions

Abstract

This paper investigates generalizations of the BMV-conjecture for quaternionic and octonionic matrices. For quaternions the correctness of the formulation is shown as well as its equivalence to the original conjecture for complex matrices. General properties of octonions and Hermitian matrices over them are examined for the BMV-conjecture formulation over octonions.

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References

  1. 1.

    D. Bessis, P. Moussa, and M. Villani, “Monotonic converging variational approximations to the functional integrals in quantum statistical mechanics,” J. Math. Phys., 16, 2318–2325 (1975).

    MathSciNet  Article  Google Scholar 

  2. 2.

    S. Burgdorf, “Sums of Hermitian squares as an approach to the BMV conjecture,” Linear and Multilinear Algebra, 59, No. 1, 1–9 (2011).

    MathSciNet  MATH  Article  Google Scholar 

  3. 3.

    M. Fannes and D. Petz, “On the function e H+itK,” Int. J. Math. Math. Sci., 29, 389–394 (2001).

    MathSciNet  Article  Google Scholar 

  4. 4.

    M. Fannes and D. Petz, “Perturbation of Wigner matrices and a conjecture,” Proc. Am. Math. Soc., 131, 1981–1988 (2003).

    MathSciNet  MATH  Article  Google Scholar 

  5. 5.

    D. Hägele, “Proof of the case p ≤ 7 of the Lieb–Seiringer formulation of the Bessis–Moussa–Villani conjecture,” J. Stat. Phys., 127, No. 6, 1167–1171 (2007).

    MathSciNet  MATH  Article  Google Scholar 

  6. 6.

    C. J. Hillar, “Advances on the Bessis–Moussa–Villani trace conjecture,” Linear Algebra Appl., 426, No. 1, 130–142 (2007).

    MathSciNet  MATH  Article  Google Scholar 

  7. 7.

    C. J. Hillar and C. R. Johnson, “Eigenvalues of words in two positive definite letters,” SIAM J. Matrix Anal. Appl., 23, No. 4, 916–928 (2002).

    MathSciNet  MATH  Article  Google Scholar 

  8. 8.

    R. A. Horn and C. R. Johnson, Matrix Analysis, Cambridge Univ. Press, New York (1985).

    MATH  Book  Google Scholar 

  9. 9.

    I. Klep and M. Schweighofer, “Sums of Hermitian squares and the BMV conjecture,” J. Stat. Phys., 133, 739–760 (2008).

    MathSciNet  MATH  Article  Google Scholar 

  10. 10.

    E. H. Lieb and R. Seiringer, “Equivalent forms of the Bessis–Moussa–Villani conjecture,” J. Stat. Phys., 115, 185–190 (2004).

    MathSciNet  MATH  Article  Google Scholar 

  11. 11.

    J. P. Ward, Quaternions and Cayley Numbers: Algebra and Applications, Kluwer Academic, Boston (1997).

    MATH  Book  Google Scholar 

  12. 12.

    F. Zhang, “Quaternions and matrices of quaternions,” Linear Algebra Appl., 251, 21–57 (1997).

    MathSciNet  MATH  Article  Google Scholar 

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Correspondence to A. S. Smirnov.

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Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 17, No. 6, pp. 185–222, 2011/12.

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Smirnov, A.S. The BMV-conjecture over quaternions and octonions. J Math Sci 193, 775–801 (2013). https://doi.org/10.1007/s10958-013-1497-0

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Keywords

  • Diagonal Entry
  • Complex Matrice
  • Hermitian Matrix
  • Hermitian Matrice
  • Inverse Element