New insight into quaternions and their matrices

YÜCE, SALİM; Saçlı, Gülsüm Yeliz; GÜRSES, NURTEN

doi:10.31801/cfsuasmas.1074557

New insight into quaternions and their matrices

Gülsüm Yeliz ŞENTÜRK, (İstanbul Gelişim Üniversitesi, Mühendislik Mimarlık Fakültesi, Bilgisayar Mühendisliği Bölümü, İstanbul, Türkiye)

Nurten GÜRSES, (Yıldız Teknik Üniversitesi, Fen-Edebiyat Fakültesi, Matematik Bölümü, İstanbul, T̈ürkiye)

Salim YÜCE (Yıldız Teknik Üniversitesi, Fen-Edebiyat Fakültesi, Matematik Bölümü, İstanbul, T̈ürkiye)

Communications Faculty of Sciences University of Ankara Series A1: Mathematics and Statistics

5 2

Yıl: 2023 Cilt: 72 Sayı: 1 Sayfa Aralığı: 43 - 58 Metin Dili: İngilizce DOI: 10.31801/cfsuasmas.1074557 İndeks Tarihi: 25-05-2023

New insight into quaternions and their matrices

Öz:

This paper aims to bring together quaternions and generalized complex numbers. Generalized quaternions with generalized complex number components are expressed and their algebraic structures are examined. Several matrix representations and computational results are introduced. An alternative approach for a generalized quaternion matrix with elliptic number entries has been developed as a crucial part.

Anahtar Kelime: Generalized quaternion generalized complex number matrix representation elliptic number.

Belge Türü: Makale Makale Türü: Araştırma Makalesi Erişim Türü: Erişime Açık

[1] Hamilton, W.R., Elements of Quaternions, Chelsea Pub. Com., New York, 1969.
[2] Hamilton, W.R., Lectures on Quaternions, Hodges and Smith, Dublin, 1853.
[3] Hamilton, W.R., On quaternions; or on a new system of imaginaries in algebra, The London, Edinburgh and Dublin Philosophical Magazine and Journal of Science (3rd Series), xxv- xxxvi, 1844–1850.
[4] Dickson, L.E., On the theory of numbers and generalized quaternions, Amer. J. Math., 46(1) (1924), 1–16. https://doi.org/10.2307/2370658
[5] Cockle, J., On a new imaginary in algebra, Philosophical Magazine, London-Dublin- Edinburgh, 34(3) (1849), 37–47. https://doi.org/10.1080/14786444908646169
[6] Griffiths, L.W., Generalized quaternion algebras and the theory of numbers, Amer. J. Math. 50(2) (1928), 303–314. https://doi.org/10.2307/2371761
[7] Jafari, M., Yaylı, Y., Generalized quaternions and their algebratic properties, Commun. Fac. Sci. Univ. Ank. Ser. A1. Math. Stat., 64(1) (2015), 15–27. https://doi.org/10.1501/Commua1 0000000724
[8] Alag ̈oz, Y., Oral, K.H., Y ̈uce, S., Split quaternion matrices, Miskolc Math. Notes, 13(2) (2012), 223–232. https://doi.org/10.18514/MMN.2012.364
[9] Pottmann, H., Wallner, J., Computational Line Geometry, Springer, Berlin, 2000. https://doi.org/10.1007/978-3-642-04018-4
[10] Rosenfeld, B., Geometry of Lie Groups, Springer, New York, 1997. https://doi.org/10.1007/978-1-4757-5325-7
[11] Savin, D., Flaut, C., Ciobanu, C., Some properties of the symbol algebras, Carpathian J. Math., 25(2) (2009), 239–245.
[12] Catoni, F., Cannata, R., Catoni, V., Zampetti, P., An introduction to commutative quater- nions, Adv. Appl. Clifford Algebr., 16(1) (2006), 1–28. https://doi.org/10.1007/s00006-006- 0002-y
[13] Catoni, F., Cannata, R., Zampetti, P., An introduction to constant curvature spaces in the commutative (Segr ́e) quaternion geometry, Adv. Appl. Clifford Algebr., 16(1) (2006), 85–101. https://doi.org/10.1007/s00006-006-0010-y
[14] Clifford, W.K., Preliminary sketch of bi-quaternions, Proc. Lond. Math. Soc., s1–4(1) (1873), 381–395. https://doi.org/10.1112/plms/s1-4.1.381
[15] Edmonds, J.D., Jr., Relativistic Reality: A Modern View, World Scientific, Singapore, 1997. https://doi.org/10.1142/3272
[16] Ercan, Z., Y ̈uce, S., On properties of the dual quaternions, Eur. J. Pure Appl. Math., 4(2) (2011), 142–146.
[17] Majernik, V., Quaternion formulation of the Galilean space-time transformation, Acta Phy. Slovaca, 56 (2006), 9–14.
[18] Majernik, V., Nagy, M., Quaternionic form of Maxwell’s equations with sources, Lett. Nuovo Cimento, 16 (1976), 165–169. https://doi.org/10.1007/BF02747070
[19] Kantor, I., Solodovnikov, A., Hypercomplex Numbers, Springer-Verlag, New York, 1989.
[20] Catoni, F., Boccaletti, D., Cannata, R., Catoni, V., Nichelatti, E., Zampetti, P., The Mathe- matics of Minkowski Space-Time and an Introduction to Commutative Hypercomplex Num- bers, Birkhauser Verlag, Basel, 2008. https://doi.org/10.1007/978-3-7643-8614-6
[21] Catoni, F., Cannata, R., Catoni, V., Zampetti, P., Two-dimensional hypercomplex numbers and related trigonometries and geometries, Adv. Appl. Clifford Algebr., 14 (2004), 47–68. https://doi.org/10.1007/s00006-004-0008-2
[22] Catoni, F., Cannata, R., Catoni, V., Zampetti, P., N -dimensional geometries gen- erated by hypercomplex numbers, Adv. Appl. Clifford Algebr. 15(1) (2005), 1–25. https://doi.org/10.1007/s00006-005-0001-4
[23] Harkin, A.A., Harkin, J.B., Geometry of generalized complex numbers, Math. Mag., 77(2) (2004), 118–129. https://doi.org/10.1080/0025570X.2004.11953236
[24] Veldsman, S., Generalized complex numbers over near-fields, Quaest. Math., 42(2) (2019), 181–200. https://doi.org/10.2989/16073606.2018.1442884
[25] Clifford, W.K., Mathematical papers (ed. R. Tucker), Chelsea Pub. Co., Bronx, NY, 1968.
[26] Fjelstad, P., Extending special relativity via the perplex numbers, Amer. J. Phys., 54(5) (1986), 416–422. https://doi.org/10.1119/1.14605
[27] Sobczyk, G., The hyperbolic number plane, College Math. J., 26(4) (1995), 268–280. https://doi.org/10.2307/2687027
[28] Yaglom, I. M., A Simple Non-Euclidean Geometry and Its Physical Basis, Springer-Verlag, New York, 1979.
[29] Yaglom, I. M., Complex Numbers in Geometry, Academic Press, New York, 1968.
[30] Pennestr`ı, E., Stefanelli, R., Linear algebra and numerical algorithms using dual numbers, Multibody Syst. Dyn., 18(3) (2007), 323–344. https://doi.org/10.1007/s11044-007-9088-9
[31] Study, E., Geometrie Der Dynamen, Mathematiker Deutschland Publisher, Leibzig, 1903.
[32] Messelmi, F., Generalized numbers and their holomorphic functions, Int. J. Open Problems Complex Analysis, 7(1) (2015), 35–47.
[33] Zhang, F., Quaternions and matrices of quaternions, Linear Algebra Appl., 251 (1997), 21–57. https://doi.org/10.1016/0024-3795(95)00543-9
[34] Flaut, C., Shpakivskyi, V., An efficient method for solving equations in generalized quaternion and octonion algebras, Adv. Appl. Clifford Algebr., 25(2) (2015), 337–350. https://doi.org/10.1007/s00006-014-0493-x
[35] ̈Ozen, K. E., Tosun, M., On the matrix algebra of elliptic biquaternions, Math. Methods Appl. Sci., 43(6) (2020), 2984–2998. https://doi.org/10.1002/mma.6096
[36] Tian, Y., Biquaternions and their complex matrix representations, Beitr. Algebra Geom. instead of Beitr ̈age zur Algebra und Geometrie/Contributions to Algebra and Geometry, 54(2) (2013), 575–592. https://doi.org/10.1007/s13366-012-0113-7
[37] Alag ̈oz, Y., ̈Ozyurt, G., Some properties of complex quaternion and com- plex split quaternion matrices, Miskolc Math. Notes, 20(1) (2019), 45–58. https://doi.org/10.18514/MMN.2019.2550
[38] Alag ̈oz, Y., ̈Ozyurt, G., Linear equations systems of real and complex semi-quaternions, Iran. J. Sci. Technol. Trans. A Sci., 44(5) (2020), 1483–1493. https://doi.org/10.1007/s40995-020- 00956-7
[39] Bekar, M., Yaylı, Y., A study on complexified semi-quaternions, International Conference on Operators in Morrey-type Spaces and Applications (OMTSA), 115 (2017).
[40] Arizmendi, G., P ́erez-de la Rosa, Y.M.A., Some extensions of quaternions and sym- metries of simply connected space forms, arXiv preprint, arXiv:1906.11370 (2019). https://doi.org/10.48550/arXiv.1906.11370
[41] Ata, E., Yıldırım, Y., A different polar representation for generalized and generalized dual quaternions, Adv. Appl. Clifford Algebr., 28(4) (2018), 1–20. https://doi.org/10.1007/s00006- 018-0895-2
[42] Bekar, M., Dual-quasi elliptic planar motion, Mathematical Sciences and Applications E- Notes, 4(1) (2016), 136–143. https://doi.org/10.36753/mathenot.421422
[43] Erdo ̆gdu, M., ̈Ozdemir, M., Matrices over hyperbolic split quaternions, Filomat, 30(4) (2016), 913–920. https://doi.org/10.2298/FIL1604913E
[44] Hamilton, W.R., On the geometrical interpretation of some results obtained by calculation with biquaternions, In Halberstam and Ingram and first published in Proceedings of the Royal Irish Academy, 1853.
[45] Kotelnikov, A. P., Screw calculus and some applications to geometry and mechanics, Annal. Imp. Univ. Kazan, (1895).
[46] Kula, L., Yaylı, Y., Dual split quaternions and screw motion in Minkowski 3-space, Iran. J. Sci. Technol. Trans. A Sci., 30(3) (2006), 245–258. https://doi.org/10.22099/IJSTS.2006.2758
[47] McAulay, A., Octonions: a Development of Clifford’s Biquaternions, 1898.
[48] Alag ̈oz, Y., ̈Ozyurt, G., Real and hyperbolic matrices of split semi quaternions, Adv. Apl. Clifford Algebr., 29 (2019), 29–53. https://doi.org/10.1007/s00006-019-0973-0
[49] Erdo ̆gdu, M., ̈Ozdemir, M., On complex split quaternion matrices, Adv. Appl. Clifford Al- gebr., 23(3)(2013), 625–638. https://doi.org/10.1007/s00006-013-0399-z
[50] ̈Ozen, K. E., Tosun, M., Elliptic matrix representations of elliptic biquater- nions and their applications, Int. Electron. J. Geom., 11(2) (2018), 96–103. https://doi.org/10.36890/iejg.545136
[51] Jafari, M., Matrices of generalized dual quaternions, Konuralp J. Math., 3(2) (2015), 110–121.
[52] Jafari, M., The algebraic structure of dual semi-quaternions, J. Sel ̧cuk Univ. Natur. Appl. Sci., 5(3) (2016), 15–24.
[53] Erdo ̆gdu, M., ̈Ozdemir, M., Real matrix representations of complex split quater- nions with applications, Math. Methods Appl. Sci. 43(12) (2020), 7227–7238. https://doi.org/10.1002/mma.6461
[54] Qi L., Ling C., Yan, H., Dual quaternions and dual quaternion vectors, Commun. Appl. Math. Comput., (2022), 1–15. https://doi.org/10.1007/s42967-022-00189-y

APA	Saçlı G, GÜRSES N, YÜCE S (2023). New insight into quaternions and their matrices. , 43 - 58. 10.31801/cfsuasmas.1074557
Chicago	Saçlı Gülsüm Yeliz,GÜRSES NURTEN,YÜCE SALİM New insight into quaternions and their matrices. (2023): 43 - 58. 10.31801/cfsuasmas.1074557
MLA	Saçlı Gülsüm Yeliz,GÜRSES NURTEN,YÜCE SALİM New insight into quaternions and their matrices. , 2023, ss.43 - 58. 10.31801/cfsuasmas.1074557
AMA	Saçlı G,GÜRSES N,YÜCE S New insight into quaternions and their matrices. . 2023; 43 - 58. 10.31801/cfsuasmas.1074557
Vancouver	Saçlı G,GÜRSES N,YÜCE S New insight into quaternions and their matrices. . 2023; 43 - 58. 10.31801/cfsuasmas.1074557
IEEE	Saçlı G,GÜRSES N,YÜCE S "New insight into quaternions and their matrices." , ss.43 - 58, 2023. 10.31801/cfsuasmas.1074557
ISNAD	Saçlı, Gülsüm Yeliz vd. "New insight into quaternions and their matrices". (2023), 43-58. https://doi.org/10.31801/cfsuasmas.1074557

APA	Saçlı G, GÜRSES N, YÜCE S (2023). New insight into quaternions and their matrices. Communications Faculty of Sciences University of Ankara Series A1: Mathematics and Statistics, 72(1), 43 - 58. 10.31801/cfsuasmas.1074557
Chicago	Saçlı Gülsüm Yeliz,GÜRSES NURTEN,YÜCE SALİM New insight into quaternions and their matrices. Communications Faculty of Sciences University of Ankara Series A1: Mathematics and Statistics 72, no.1 (2023): 43 - 58. 10.31801/cfsuasmas.1074557
MLA	Saçlı Gülsüm Yeliz,GÜRSES NURTEN,YÜCE SALİM New insight into quaternions and their matrices. Communications Faculty of Sciences University of Ankara Series A1: Mathematics and Statistics, vol.72, no.1, 2023, ss.43 - 58. 10.31801/cfsuasmas.1074557
AMA	Saçlı G,GÜRSES N,YÜCE S New insight into quaternions and their matrices. Communications Faculty of Sciences University of Ankara Series A1: Mathematics and Statistics. 2023; 72(1): 43 - 58. 10.31801/cfsuasmas.1074557
Vancouver	Saçlı G,GÜRSES N,YÜCE S New insight into quaternions and their matrices. Communications Faculty of Sciences University of Ankara Series A1: Mathematics and Statistics. 2023; 72(1): 43 - 58. 10.31801/cfsuasmas.1074557
IEEE	Saçlı G,GÜRSES N,YÜCE S "New insight into quaternions and their matrices." Communications Faculty of Sciences University of Ankara Series A1: Mathematics and Statistics, 72, ss.43 - 58, 2023. 10.31801/cfsuasmas.1074557
ISNAD	Saçlı, Gülsüm Yeliz vd. "New insight into quaternions and their matrices". Communications Faculty of Sciences University of Ankara Series A1: Mathematics and Statistics 72/1 (2023), 43-58. https://doi.org/10.31801/cfsuasmas.1074557