Journal of East China Normal University(Natural Science) >

2022 , Vol. 2022 >Issue 6: 8 - 16

DOI: https://doi.org/10.3969/j.issn.1000-5641.2022.06.002

Mathematics

De Moivre’s theorem for a matrix representation of hyperbolic split quaternions

  • Xiangqiang KONG
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  • School of Mathematics and Statistics, Heze University, Heze, Shandong　274015, China

Received date: 2021-01-22

  Online published: 2022-11-22

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Abstract

In this paper, de Moivre’s theorem for a matrix representation of a class of hyperbolic split quaternions is studied. Firstly, the study of hyperbolic split quaternions is transformed into the study of a matrix representation of hyperbolic split quaternions. Secondly, by using the polar representation of a hyperbolic split quaternion, the three forms of de Moivre’s theorem for a matrix representation of the hyperbolic split quaternion are obtained, and Euler’s formula is extended. Thirdly, the root-finding formula of the hyperbolic split quaternion matrix representation equation is obtained. Finally, the validity of the conclusions is verified with some examples.

Key words： hyperbolic split quaternion; matrix representation; de Moivre’s theorem; Euler’s formula

Cite this article

Xiangqiang KONG . De Moivre’s theorem for a matrix representation of hyperbolic split quaternions[J]. Journal of East China Normal University(Natural Science), 2022 , 2022(6) : 8 -16 . DOI: 10.3969/j.issn.1000-5641.2022.06.002

References

1 ERDO?DU M, ?ZDEMIR M. On complex split quaternion matrices. Advances in Applied Clifford Algebras, 2013, 23 (3): 625- 638.
2 POGORUY A A, RODRIGUEZ-DAGNINO M R. Some algebraic and analytical properties of coquaternion algebra. Advances in Applied Clifford Algebras, 2010, 20 (1): 79- 84.
3 ERDO?DU M, ?ZDEMIR M. On eigenvalues of split quaternion matrices. Advances in Applied Clifford Algebras, 2013, 23 (3): 615- 623.
4 孔祥强. 双曲型交换四元数的极表示. 华东师范大学学报(自然科学版), 2020, (1): 16- 23.
5 ?ZDEMIR M, ERGIN A A. Rotations with unit timelike quaternions in Minkowski 3-space. Journal of Geometry and Physics, 2006, 56 (2): 322- 336.
6 孔祥强. 抛物型交换四元数矩阵实表示的性质及应用. 四川师范大学学报(自然科学版), 2019, 42 (6): 789- 793.
7 SCHARLER D F, SIEGELE L, SCHR?CKER H P. Quadratic split quaternion polynomials: Factorization and geometry. Advances in Applied Clifford Algebras, 2020, 30, 11.
8 BEKAR M, YAYLI Y. Involutions in split semi-quaternions. Mathematical Methods in the Applied Sciciences, 2018, 41 (12): 4491- 4505.
9 INALCIK A. Similarity and semi-similarity relations on the degenerate quaternions, pseudodegenerate quaternions and doubly degenerate quaternions. Advances in Applied Clifford Algebras, 2017, 27 (2): 1329- 1341.
10 BEKAR M, YAYLI Y. Involutions in dual split-quaternions. Advances in Applied Clifford Algebras, 2016, 26 (2): 553- 571.
11 ATASOY A, ATA E, YAYLI Y, et al. A new polar representation for split and dual split quaternions. Advances in Applied Clifford Algebras, 2017, 27 (3): 2307- 2319.
12 WANG M H, YUE L L, LIU Q H. Elementary transformation and its applications for split quaternion matrices. Advances in Applied Clifford Algebras, 2020, 30, 1.
13 KISIL V V. Induced representations and hypercomplex numbers. Advances in Applied Clifford Algebras, 2013, 23 (2): 417- 440.
14 ?ZDEMIR M. Introduction to hybrid numbers. Advances in Applied Clifford Algebras, 2018, 28, 11.
15 ERDO?DU M, ?ZDEMIR M. Matrices over hyperbolic split quaternions. Filomat, 2016, 30 (4): 913- 920.
16 ?ZYURT G, ALAG?Z Y. On hyperbolic split quaternions and hyperbolic split quaternion matrices. Advances in Applied Clifford Algebras, 2018, 28, 88.
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