The polar form of hyperbolic commutative quaternions

doi:10.3969/j.issn.1000-5641.201911001

Abstract

Abstract: Firstly, this paper presents the e₁-e₂ representation and matrix representation of hyperbolic commutative quaternions. Secondly, the polar form theorem of hyperbolic commutative quaternion is presented; the existence and uniqueness of the respective polar form are proven, and a series of properties for the hyperbolic commutative quaternion polar form are obtained. Lastly, the relationship between the polar form, e₁-e₂ representation and matrix representation of hyperbolic commutative quaternions are discussed. Hence, the paper provides a theoretical basis for further research on the application of hyperbolic commutative quaternions.

Key words: hyperbolic commutative quaternion, polar form, matrix representation, norm, determinant

CLC Number:

O153.3

KONG Xiangqiang. The polar form of hyperbolic commutative quaternions[J]. Journal of East China Normal University(Natural Science), 2020, 2020(1): 16-23.

References

[1] POGORUY A A, RODRIGUEZ-DAGNINO R M. Some algebraic and analytical properties of coquaternion algebra[J]. Adv Appl Clifford Algebras, 2010, 20(1):79-84. DOI:10.1007/s00006-008-0142-3.
[2] ALAGOZ Y, ORAL K H, YUCE S. Split quaternion matrices[J]. Miskolc Math Notes, 2012, 13(2):223-232. DOI:10.18514/MMN.2012.364.
[3] OZDEMIR M. The roots of split quaternion[J]. Appl Math Lett, 2009, 22(2):258-263. DOI:10.1016/j.aml.2008.03.020.
[4] KULA L, YAYLI Y. Split quaternions and rotations in semi Euclidean space

\scriptsize $ {{\bm E}_2^4} $ \normalsize [J]. J Korean Math Soc, 2007, 44(6):1313-1327. DOI:10.4134/JKMS.2007.44.6.1313.
[5] ERDOGDU M, OZDEMIR M. On complex split quaternion matrices[J]. Adv Appl Clifford Algebras, 2013, 23(3):625-638. DOI:10.1007/s00006-013-0399-z.
[6] ERDOGDU M, OZDEMIR M. On eigenvalues of split quaternion matrices[J]. Adv Appl Clifford Algebras, 2013, 23(3):615-623. DOI:10.1007/s00006-013-0391-7.
[7] ERDOGDU M, OZDEMIR M. On exponential of split quaternionic matrices[J]. Applied Mathematics and Computation, 2017, 315:468-476. DOI:10.1016/j.amc.2017.08.007.
[8] ATASOY A, ATA E, YAYLI Y, et al. A new polar representation for split and dual split quaternions[J]. Adv Appl Clifford Algebras, 2017, 27(3):2307-2319. DOI:10.1007/s00006-017-0797-8.
[9] OZDEMIR M. Introduction to hybrid numbers[J]. Adv Appl Clifford Algebras, 2018, 28:11. DOI:10.1007/s00006-018-0833-3.
[10] SEGRE C. The real representations of complex elements and extension to bicomplex systems[J]. Mathematische Annalen, 1892, 40:413-467. DOI:10.1007/BF01443559.
[11] CATONI F, CANNATA R, ZAMPETTI P. An introduction to commutative quaternions[J]. Adv Appl Clifford Algebras, 2006, 16(1):1-28. DOI:10.1007/s00006-006-0002-y.
[12] KOSAL H H, TOSUN M. Commutative quaternion matrices[J]. Adv Appl Clifford Algebras, 2014, 24(3):769-779. DOI:10.1007/s00006-014-0449-1.
[13] PEI S C, CHANG J H, DING J J. Commutative reduced biquaternions and their Fourier transform for signal and image processing applications[J]. IEEE Transactions on Signal Processing, 2004, 52(7):2012-2031. DOI:10.1109/TSP.2004.828901.
[14] CATONI F, CANNATA R, NICHELATTI E, et al. Commutative hypercomplex numbers and functions of hypercomplex variable:A matrix study[J]. Adv Appl Clifford Algebras, 2005, 15(2):183-212. DOI:10.1007/s00006-005-0011-2.
[15] PINOTSIS D A. Segre quaternions, spectral analysis and a four-dimensional Laplace equation[M]//Progress in Analysis and its Applications. Singapore:World Scientific Press, 2010:240-246.
[16] ROONEY J. On the three types of complex number and planar transformations[J]. Environment and Planning, Series B, 1978, 5(1):89-99. DOI:10.1068/b050089.