Trace Identities for Skew-Symmetric Matrices

M. I. Krivoruchenko

doi:doi:10.11648/j.mcs.20160102.11

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Trace Identities for Skew-Symmetric Matrices

M. I. Krivoruchenko

Published in Mathematics and Computer Science (Volume 1, Issue 2)

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), 2024. 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).
[2]	F. R. Gantmacher, The Theory of Matrices. Vol. 1 and 2. (Chelsea Pub. Co., N. Y. 1959).
[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.
[13]	D. C. Youla, A normal form for a matrix under the unitary congruence group, Canad. J. Math. 13, 694 (1961).

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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},
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  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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Author Information

M. I. Krivoruchenko

Theoretical Physics Division, Institute for Theoretical and Experimental Physics, Moscow, Russia; Department of Nano/Bio, Information and Cognitive Technologies, Moscow Institute of Physics and Technology, Dolgoprudny, Russia; Bogoliubov Laboratory of Theoretical Physics, Joint Institute for Nuclear Research, Dubna, Russia

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Plain Text BibTeX RIS

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

Copy | Download

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

Copy | Download

@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}
}

Copy | Download

TY  - JOUR
T1  - Trace Identities for Skew-Symmetric Matrices
AU  - M. I. Krivoruchenko
Y1  - 2016/06/29
PY  - 2016
N1  - https://doi.org/10.11648/j.mcs.20160102.11
DO  - 10.11648/j.mcs.20160102.11
T2  - Mathematics and Computer Science
JF  - Mathematics and Computer Science
JO  - Mathematics and Computer Science
SP  - 21
EP  - 28
PB  - Science Publishing Group
SN  - 2575-6028
UR  - https://doi.org/10.11648/j.mcs.20160102.11
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.
VL  - 1
IS  - 2
ER  -

Copy | Download