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Euler and De Moivre’s Formulas for Fundamental Matrices of Commutative Quaternions

Year 2020, , 98 - 107, 15.10.2020
https://doi.org/10.36890/iejg.768821

Abstract

In this study, Euler and De Moivre's formulas for fundamental matrices of commutative quaternions are obtained. Simple and effective methods are provided to find the powers and roots of these matrices with the aid of De Moivre's formula obtained from the fundamental matrices of commutative quaternions. Moreover, our results are supported by pseudo-codes and some examples.
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References

  • Segre, C.: \emph{The real representations of complex elements and extension to bicomplex}. Systems. Math. Ann., 40, 413, (1892).
  • Pei, S. C., Chang, J. H., Ding, J. J.: \emph{Commutative reduced biquaternions and their fourier transform for signal and image processing applications}. IEEE Trans. on Signal Proces., 52(7), 2012-2031, (2004).
  • Catoni, F., Boccaletti, D., Cannata, R., Catoni, V., Nichelatti, E., Zampetti, P.: \emph{The mathematics of minkowski space-time with an introduction to commutative hypercomplex numbers}. Birkhauser Verlag AG, Berlin, (2008).
  • Pei S., Chang J., Ding J., Chen M.: \emph{Eigenvalues and singular value decompositions of reduced biquaternion matrices}. IEEE Trans. Circ. Syst. I., 55(9), 1549-8328, (2008).
  • Isokawa, T., Nishimura, H., Matsui, N.: \emph{Commutative quaternion and multistate hopfield neural networks}. In Proc. Int. Joint Conf. Neural Netw., Barcelona, Spain, 1281-1286, (2010).
  • Kosal H. H., Akyigit M., Tosun M.: \emph{Consimilarity of commutative quaternion matrices}. Miskolc Math. Notes, 16(2), 965-977, (2015).
  • Yuan, S. F., Tian,Y., Li, M. Z.: \emph{On Hermitian solutions of the reduced biquaternion matrix equation $\left( AXB,\text{ }CXD \right)=\left( E,G \right)$}. Linear Multilinear Algebra, 1-19 (2018). DOI: 10.1080/03081087.2018.1543383.
  • Kosal H. H., Tosun M.: \emph{Universal similarity factorization equalities for commutative quaternions and their matrices}. Linear Multilinear Algebra, 67(5), 926-938, (2019).
  • Kosal H. H.: \emph{Least-squares solutions of the reduced biquaternion matrix equation $AX=B$ and their applications in colour image restoration.} J. Modern Opt., 66(18), 1802-1810, (2019).
  • Zhang D., Guo Z., Wang G., Jiang T.: \emph{Algebraic techniques for least squares problems in commutative quaternionic theory}. Math Meth Appl Sci., 43, 3513-3523, (2020).
  • Catoni, F., Cannata, R., Zampetti, P.: \emph{An introduction to commutative quaternions}. Adv. Appl. Clifford Algebras, 16, 1-28, (2006).
Year 2020, , 98 - 107, 15.10.2020
https://doi.org/10.36890/iejg.768821

Abstract

References

  • Segre, C.: \emph{The real representations of complex elements and extension to bicomplex}. Systems. Math. Ann., 40, 413, (1892).
  • Pei, S. C., Chang, J. H., Ding, J. J.: \emph{Commutative reduced biquaternions and their fourier transform for signal and image processing applications}. IEEE Trans. on Signal Proces., 52(7), 2012-2031, (2004).
  • Catoni, F., Boccaletti, D., Cannata, R., Catoni, V., Nichelatti, E., Zampetti, P.: \emph{The mathematics of minkowski space-time with an introduction to commutative hypercomplex numbers}. Birkhauser Verlag AG, Berlin, (2008).
  • Pei S., Chang J., Ding J., Chen M.: \emph{Eigenvalues and singular value decompositions of reduced biquaternion matrices}. IEEE Trans. Circ. Syst. I., 55(9), 1549-8328, (2008).
  • Isokawa, T., Nishimura, H., Matsui, N.: \emph{Commutative quaternion and multistate hopfield neural networks}. In Proc. Int. Joint Conf. Neural Netw., Barcelona, Spain, 1281-1286, (2010).
  • Kosal H. H., Akyigit M., Tosun M.: \emph{Consimilarity of commutative quaternion matrices}. Miskolc Math. Notes, 16(2), 965-977, (2015).
  • Yuan, S. F., Tian,Y., Li, M. Z.: \emph{On Hermitian solutions of the reduced biquaternion matrix equation $\left( AXB,\text{ }CXD \right)=\left( E,G \right)$}. Linear Multilinear Algebra, 1-19 (2018). DOI: 10.1080/03081087.2018.1543383.
  • Kosal H. H., Tosun M.: \emph{Universal similarity factorization equalities for commutative quaternions and their matrices}. Linear Multilinear Algebra, 67(5), 926-938, (2019).
  • Kosal H. H.: \emph{Least-squares solutions of the reduced biquaternion matrix equation $AX=B$ and their applications in colour image restoration.} J. Modern Opt., 66(18), 1802-1810, (2019).
  • Zhang D., Guo Z., Wang G., Jiang T.: \emph{Algebraic techniques for least squares problems in commutative quaternionic theory}. Math Meth Appl Sci., 43, 3513-3523, (2020).
  • Catoni, F., Cannata, R., Zampetti, P.: \emph{An introduction to commutative quaternions}. Adv. Appl. Clifford Algebras, 16, 1-28, (2006).
There are 11 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Research Article
Authors

Hidayet Hüda Kösal 0000-0002-4083-462X

Tuçe Bilgili 0000-0002-9554-6502

Publication Date October 15, 2020
Acceptance Date October 1, 2020
Published in Issue Year 2020

Cite

APA Kösal, H. H., & Bilgili, T. (2020). Euler and De Moivre’s Formulas for Fundamental Matrices of Commutative Quaternions. International Electronic Journal of Geometry, 13(2), 98-107. https://doi.org/10.36890/iejg.768821
AMA Kösal HH, Bilgili T. Euler and De Moivre’s Formulas for Fundamental Matrices of Commutative Quaternions. Int. Electron. J. Geom. October 2020;13(2):98-107. doi:10.36890/iejg.768821
Chicago Kösal, Hidayet Hüda, and Tuçe Bilgili. “Euler and De Moivre’s Formulas for Fundamental Matrices of Commutative Quaternions”. International Electronic Journal of Geometry 13, no. 2 (October 2020): 98-107. https://doi.org/10.36890/iejg.768821.
EndNote Kösal HH, Bilgili T (October 1, 2020) Euler and De Moivre’s Formulas for Fundamental Matrices of Commutative Quaternions. International Electronic Journal of Geometry 13 2 98–107.
IEEE H. H. Kösal and T. Bilgili, “Euler and De Moivre’s Formulas for Fundamental Matrices of Commutative Quaternions”, Int. Electron. J. Geom., vol. 13, no. 2, pp. 98–107, 2020, doi: 10.36890/iejg.768821.
ISNAD Kösal, Hidayet Hüda - Bilgili, Tuçe. “Euler and De Moivre’s Formulas for Fundamental Matrices of Commutative Quaternions”. International Electronic Journal of Geometry 13/2 (October 2020), 98-107. https://doi.org/10.36890/iejg.768821.
JAMA Kösal HH, Bilgili T. Euler and De Moivre’s Formulas for Fundamental Matrices of Commutative Quaternions. Int. Electron. J. Geom. 2020;13:98–107.
MLA Kösal, Hidayet Hüda and Tuçe Bilgili. “Euler and De Moivre’s Formulas for Fundamental Matrices of Commutative Quaternions”. International Electronic Journal of Geometry, vol. 13, no. 2, 2020, pp. 98-107, doi:10.36890/iejg.768821.
Vancouver Kösal HH, Bilgili T. Euler and De Moivre’s Formulas for Fundamental Matrices of Commutative Quaternions. Int. Electron. J. Geom. 2020;13(2):98-107.