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Trace Identities for Skew-Symmetric Matrices

Received: 10 May 2016     Accepted: 14 June 2016     Published: 29 June 2016
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Abstract

We derive an expression for the product of the Pfaffians of two skew-symmetric matrices A and B as a sum of products of the traces of powers of AB and an expression for the inverse matrix A-1, or equivalently B-1, as a finite-order polynomial of AB with coefficients depending on the traces of powers of AB.

Published in Mathematics and Computer Science (Volume 1, Issue 2)
DOI 10.11648/j.mcs.20160102.11
Page(s) 21-28
Creative Commons

This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.

Copyright

Copyright © The Author(s), 2016. Published by Science Publishing Group

Keywords

Characteristic Polynomial, Cayley-Hamilton Theorem, Skew-Symmetric Matrix, Determinant, Pfaffian

References
[1] N. Bourbaki, Elements of mathematics, 2. Linear and multilinear algebra (Addison-Wesley, 1973).
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[3] L. A. Kondratyuk, M. I. Krivoruchenko, Superconducting quark matter in SU (2) color group, Z. Phys. A 344, 99 (1992).
[4] L. S. Brown, Quantum Field Theory (Cambridge University Press, 1994).
[5] Q.-L. Hu, Z.-C. Gao, and Y. S. Chen, Matrix elements of one-body and two-body operators between arbitrary HFB multi-quasiparticle states, Phys. Lett. B 734, 162 (2014).
[6] Yang Sun, Projection techniques to approach the nuclear many-body problem, Phys. Scr. 91, 043005 (2016).
[7] Wei Li, Heping Zhang, Dimer statistics of honeycomb lattices on Klein bottle, Möbius strip and cylinder, Physica A 391, 3833 (2012).
[8] G. H. Hardy and S. Ramanujan, Asymptotic formulae in combinatory analysis, Proc. London Math. Soc. 17, 75 (1918).
[9] Ya. V. Uspensky, Asymptotic expressions of numerical functions occurring in problems concerning the partition of numbers into summands, Bull. Acad. Sci. de Russie 14, 199 (1920).
[10] M. I. Krivoruchenko, Recurrence relations for the number of solutions of a class of Diophantine equations, Rom. J. Phys. 58, 1408 (2013).
[11] E. T. Bell, Partition polynomials, Ann. Math. 29, 38 (1928).
[12] Th. Voronov, Pfaffian, in: Concise Encyclopedia of Supersymmetry and Noncommutative Structures in Mathematics and Physics, Eds. S. Duplij, W. Siegel, J. Bagger (Springer, Berlin, N. Y. 2005), p. 298.
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Cite This Article
  • APA Style

    M. I. Krivoruchenko. (2016). Trace Identities for Skew-Symmetric Matrices. Mathematics and Computer Science, 1(2), 21-28. https://doi.org/10.11648/j.mcs.20160102.11

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    ACS Style

    M. I. Krivoruchenko. Trace Identities for Skew-Symmetric Matrices. Math. Comput. Sci. 2016, 1(2), 21-28. doi: 10.11648/j.mcs.20160102.11

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    AMA Style

    M. I. Krivoruchenko. Trace Identities for Skew-Symmetric Matrices. Math Comput Sci. 2016;1(2):21-28. doi: 10.11648/j.mcs.20160102.11

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  • @article{10.11648/j.mcs.20160102.11,
      author = {M. I. Krivoruchenko},
      title = {Trace Identities for Skew-Symmetric Matrices},
      journal = {Mathematics and Computer Science},
      volume = {1},
      number = {2},
      pages = {21-28},
      doi = {10.11648/j.mcs.20160102.11},
      url = {https://doi.org/10.11648/j.mcs.20160102.11},
      eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.mcs.20160102.11},
      abstract = {We derive an expression for the product of the Pfaffians of two skew-symmetric matrices A and B as a sum of products of the traces of powers of AB and an expression for the inverse matrix A-1, or equivalently B-1, as a finite-order polynomial of AB with coefficients depending on the traces of powers of AB.},
     year = {2016}
    }
    

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    AB  - We derive an expression for the product of the Pfaffians of two skew-symmetric matrices A and B as a sum of products of the traces of powers of AB and an expression for the inverse matrix A-1, or equivalently B-1, as a finite-order polynomial of AB with coefficients depending on the traces of powers of AB.
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Author Information
  • Theoretical Physics Division, Institute for Theoretical and Experimental Physics, Moscow, Russia

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