Research Article
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New insight into quaternions and their matrices

Year 2023, Volume: 72 Issue: 1, 43 - 58, 30.03.2023

Gülsüm Yeliz Şentürk Nurten Gürses Salim Yüce

https://doi.org/10.31801/cfsuasmas.1074557

Abstract

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.

Keywords

Generalized quaternion generalized complex number matrix representation elliptic number.

References

  • Hamilton, W.R., Elements of Quaternions, Chelsea Pub. Com., New York, 1969.
  • Hamilton, W.R., Lectures on Quaternions, Hodges and Smith, Dublin, 1853.
  • 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), xxvxxxvi, 1844–1850.
  • 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
  • Cockle, J., On a new imaginary in algebra, Philosophical Magazine, London-Dublin-Edinburgh, 34(3) (1849), 37–47. https://doi.org/10.1080/14786444908646169
  • 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
  • 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
  • Alagöz, Y., Oral, K.H., Yüce, S., Split quaternion matrices, Miskolc Math. Notes, 13(2) (2012), 223–232. https://doi.org/10.18514/MMN.2012.364
  • Pottmann, H., Wallner, J., Computational Line Geometry, Springer, Berlin, 2000. https://doi.org/10.1007/978-3-642-04018-4
  • Rosenfeld, B., Geometry of Lie Groups, Springer, New York, 1997. https://doi.org/10.1007/978-1-4757-5325-7
  • Savin, D., Flaut, C., Ciobanu, C., Some properties of the symbol algebras, Carpathian J. Math., 25(2) (2009), 239–245.
  • Catoni, F., Cannata, R., Catoni, V., Zampetti, P., An introduction to commutative quaternions, Adv. Appl. Clifford Algebr., 16(1) (2006), 1–28. https://doi.org/10.1007/s00006-006-0002-y
  • Catoni, F., Cannata, R., Zampetti, P., An introduction to constant curvature spaces in the commutative (Segre) quaternion geometry, Adv. Appl. Clifford Algebr., 16(1) (2006), 85–101. https://doi.org/10.1007/s00006-006-0010-y
  • 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
  • Edmonds, J.D., Jr., Relativistic Reality: A Modern View, World Scientific, Singapore, 1997. https://doi.org/10.1142/3272
  • Ercan, Z., Y¨uce, S., On properties of the dual quaternions, Eur. J. Pure Appl. Math., 4(2) (2011), 142–146.
  • Majernik, V., Quaternion formulation of the Galilean space-time transformation, Acta Phy. Slovaca, 56 (2006), 9–14.
  • 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
  • Kantor, I., Solodovnikov, A., Hypercomplex Numbers, Springer-Verlag, New York, 1989.
  • Catoni, F., Boccaletti, D., Cannata, R., Catoni, V., Nichelatti, E., Zampetti, P., The Mathematics of Minkowski Space-Time and an Introduction to Commutative Hypercomplex Numbers, Birkhauser Verlag, Basel, 2008. https://doi.org/10.1007/978-3-7643-8614-6
  • 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
  • Catoni, F., Cannata, R., Catoni, V., Zampetti, P., N-dimensional geometries generated by hypercomplex numbers, Adv. Appl. Clifford Algebr. 15(1) (2005), 1–25. https://doi.org/10.1007/s00006-005-0001-4
  • 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
  • Veldsman, S., Generalized complex numbers over near-fields, Quaest. Math., 42(2) (2019), 181–200. https://doi.org/10.2989/16073606.2018.1442884
  • Clifford, W.K., Mathematical papers (ed. R. Tucker), Chelsea Pub. Co., Bronx, NY, 1968.
  • Fjelstad, P., Extending special relativity via the perplex numbers, Amer. J. Phys., 54(5) (1986), 416–422. https://doi.org/10.1119/1.14605
  • Sobczyk, G., The hyperbolic number plane, College Math. J., 26(4) (1995), 268–280. https://doi.org/10.2307/2687027
  • Yaglom, I. M., A Simple Non-Euclidean Geometry and Its Physical Basis, Springer-Verlag, New York, 1979.
  • Yaglom, I. M., Complex Numbers in Geometry, Academic Press, New York, 1968.
  • Pennestri, 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
  • Study, E., Geometrie Der Dynamen, Mathematiker Deutschland Publisher, Leibzig, 1903.
  • Messelmi, F., Generalized numbers and their holomorphic functions, Int. J. Open Problems Complex Analysis, 7(1) (2015), 35–47.
  • Zhang, F., Quaternions and matrices of quaternions, Linear Algebra Appl., 251 (1997), 21–57. https://doi.org/10.1016/0024-3795(95)00543-9
  • 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
  • Özen, 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
  • Tian, Y., Biquaternions and their complex matrix representations, Beitr. Algebra Geom. instead of Beitrage zur Algebra und Geometrie/Contributions to Algebra and Geometry, 54(2) (2013), 575–592. https://doi.org/10.1007/s13366-012-0113-7
  • Alagöz, Y., Özyurt, G., Some properties of complex quaternion and complex split quaternion matrices, Miskolc Math. Notes, 20(1) (2019), 45–58. https://doi.org/10.18514/MMN.2019.2550
  • Alagöz, Y., Özyurt, 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
  • Bekar, M., Yaylı, Y., A study on complexified semi-quaternions, International Conference on Operators in Morrey-type Spaces and Applications (OMTSA), 115 (2017).
  • Arizmendi, G., Perez-de la Rosa, Y.M.A., Some extensions of quaternions and symmetries of simply connected space forms, arXiv preprint, arXiv:1906.11370 (2019). https://doi.org/10.48550/arXiv.1906.11370
  • 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
  • Bekar, M., Dual-quasi elliptic planar motion, Mathematical Sciences and Applications ENotes, 4(1) (2016), 136–143. https://doi.org/10.36753/mathenot.421422
  • Erdoğdu, M., Özdemir, M., Matrices over hyperbolic split quaternions, Filomat, 30(4) (2016), 913–920. https://doi.org/10.2298/FIL1604913E
  • 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.
  • Kotelnikov, A. P., Screw calculus and some applications to geometry and mechanics, Annal. Imp. Univ. Kazan, (1895).
  • 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
  • McAulay, A., Octonions: a Development of Clifford’s Biquaternions, 1898.
  • Alagöz, Y., Özyurt, 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
  • Erdoğdu, M., Özdemir, M., On complex split quaternion matrices, Adv. Appl. Clifford Algebr., 23(3)(2013), 625–638. https://doi.org/10.1007/s00006-013-0399-z
  • Özen, K. E., Tosun, M., Elliptic matrix representations of elliptic biquaternions and their applications, Int. Electron. J. Geom., 11(2) (2018), 96–103. https://doi.org/10.36890/iejg.545136
  • Jafari, M., Matrices of generalized dual quaternions, Konuralp J. Math., 3(2) (2015), 110–121.
  • Jafari, M., The algebraic structure of dual semi-quaternions, J. Selçuk Univ. Natur. Appl. Sci., 5(3) (2016), 15–24.
  • Erdoğdu, M., Özdemir, M., Real matrix representations of complex split quaternions with applications, Math. Methods Appl. Sci. 43(12) (2020), 7227–7238. https://doi.org/10.1002/mma.6461
  • 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

Year 2023, Volume: 72 Issue: 1, 43 - 58, 30.03.2023

Gülsüm Yeliz Şentürk Nurten Gürses Salim Yüce

https://doi.org/10.31801/cfsuasmas.1074557

Abstract

References

  • Hamilton, W.R., Elements of Quaternions, Chelsea Pub. Com., New York, 1969.
  • Hamilton, W.R., Lectures on Quaternions, Hodges and Smith, Dublin, 1853.
  • 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), xxvxxxvi, 1844–1850.
  • 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
  • Cockle, J., On a new imaginary in algebra, Philosophical Magazine, London-Dublin-Edinburgh, 34(3) (1849), 37–47. https://doi.org/10.1080/14786444908646169
  • 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
  • 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
  • Alagöz, Y., Oral, K.H., Yüce, S., Split quaternion matrices, Miskolc Math. Notes, 13(2) (2012), 223–232. https://doi.org/10.18514/MMN.2012.364
  • Pottmann, H., Wallner, J., Computational Line Geometry, Springer, Berlin, 2000. https://doi.org/10.1007/978-3-642-04018-4
  • Rosenfeld, B., Geometry of Lie Groups, Springer, New York, 1997. https://doi.org/10.1007/978-1-4757-5325-7
  • Savin, D., Flaut, C., Ciobanu, C., Some properties of the symbol algebras, Carpathian J. Math., 25(2) (2009), 239–245.
  • Catoni, F., Cannata, R., Catoni, V., Zampetti, P., An introduction to commutative quaternions, Adv. Appl. Clifford Algebr., 16(1) (2006), 1–28. https://doi.org/10.1007/s00006-006-0002-y
  • Catoni, F., Cannata, R., Zampetti, P., An introduction to constant curvature spaces in the commutative (Segre) quaternion geometry, Adv. Appl. Clifford Algebr., 16(1) (2006), 85–101. https://doi.org/10.1007/s00006-006-0010-y
  • 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
  • Edmonds, J.D., Jr., Relativistic Reality: A Modern View, World Scientific, Singapore, 1997. https://doi.org/10.1142/3272
  • Ercan, Z., Y¨uce, S., On properties of the dual quaternions, Eur. J. Pure Appl. Math., 4(2) (2011), 142–146.
  • Majernik, V., Quaternion formulation of the Galilean space-time transformation, Acta Phy. Slovaca, 56 (2006), 9–14.
  • 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
  • Kantor, I., Solodovnikov, A., Hypercomplex Numbers, Springer-Verlag, New York, 1989.
  • Catoni, F., Boccaletti, D., Cannata, R., Catoni, V., Nichelatti, E., Zampetti, P., The Mathematics of Minkowski Space-Time and an Introduction to Commutative Hypercomplex Numbers, Birkhauser Verlag, Basel, 2008. https://doi.org/10.1007/978-3-7643-8614-6
  • 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
  • Catoni, F., Cannata, R., Catoni, V., Zampetti, P., N-dimensional geometries generated by hypercomplex numbers, Adv. Appl. Clifford Algebr. 15(1) (2005), 1–25. https://doi.org/10.1007/s00006-005-0001-4
  • 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
  • Veldsman, S., Generalized complex numbers over near-fields, Quaest. Math., 42(2) (2019), 181–200. https://doi.org/10.2989/16073606.2018.1442884
  • Clifford, W.K., Mathematical papers (ed. R. Tucker), Chelsea Pub. Co., Bronx, NY, 1968.
  • Fjelstad, P., Extending special relativity via the perplex numbers, Amer. J. Phys., 54(5) (1986), 416–422. https://doi.org/10.1119/1.14605
  • Sobczyk, G., The hyperbolic number plane, College Math. J., 26(4) (1995), 268–280. https://doi.org/10.2307/2687027
  • Yaglom, I. M., A Simple Non-Euclidean Geometry and Its Physical Basis, Springer-Verlag, New York, 1979.
  • Yaglom, I. M., Complex Numbers in Geometry, Academic Press, New York, 1968.
  • Pennestri, 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
  • Study, E., Geometrie Der Dynamen, Mathematiker Deutschland Publisher, Leibzig, 1903.
  • Messelmi, F., Generalized numbers and their holomorphic functions, Int. J. Open Problems Complex Analysis, 7(1) (2015), 35–47.
  • Zhang, F., Quaternions and matrices of quaternions, Linear Algebra Appl., 251 (1997), 21–57. https://doi.org/10.1016/0024-3795(95)00543-9
  • 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
  • Özen, 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
  • Tian, Y., Biquaternions and their complex matrix representations, Beitr. Algebra Geom. instead of Beitrage zur Algebra und Geometrie/Contributions to Algebra and Geometry, 54(2) (2013), 575–592. https://doi.org/10.1007/s13366-012-0113-7
  • Alagöz, Y., Özyurt, G., Some properties of complex quaternion and complex split quaternion matrices, Miskolc Math. Notes, 20(1) (2019), 45–58. https://doi.org/10.18514/MMN.2019.2550
  • Alagöz, Y., Özyurt, 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
  • Bekar, M., Yaylı, Y., A study on complexified semi-quaternions, International Conference on Operators in Morrey-type Spaces and Applications (OMTSA), 115 (2017).
  • Arizmendi, G., Perez-de la Rosa, Y.M.A., Some extensions of quaternions and symmetries of simply connected space forms, arXiv preprint, arXiv:1906.11370 (2019). https://doi.org/10.48550/arXiv.1906.11370
  • 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
  • Bekar, M., Dual-quasi elliptic planar motion, Mathematical Sciences and Applications ENotes, 4(1) (2016), 136–143. https://doi.org/10.36753/mathenot.421422
  • Erdoğdu, M., Özdemir, M., Matrices over hyperbolic split quaternions, Filomat, 30(4) (2016), 913–920. https://doi.org/10.2298/FIL1604913E
  • 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.
  • Kotelnikov, A. P., Screw calculus and some applications to geometry and mechanics, Annal. Imp. Univ. Kazan, (1895).
  • 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
  • McAulay, A., Octonions: a Development of Clifford’s Biquaternions, 1898.
  • Alagöz, Y., Özyurt, 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
  • Erdoğdu, M., Özdemir, M., On complex split quaternion matrices, Adv. Appl. Clifford Algebr., 23(3)(2013), 625–638. https://doi.org/10.1007/s00006-013-0399-z
  • Özen, K. E., Tosun, M., Elliptic matrix representations of elliptic biquaternions and their applications, Int. Electron. J. Geom., 11(2) (2018), 96–103. https://doi.org/10.36890/iejg.545136
  • Jafari, M., Matrices of generalized dual quaternions, Konuralp J. Math., 3(2) (2015), 110–121.
  • Jafari, M., The algebraic structure of dual semi-quaternions, J. Selçuk Univ. Natur. Appl. Sci., 5(3) (2016), 15–24.
  • Erdoğdu, M., Özdemir, M., Real matrix representations of complex split quaternions with applications, Math. Methods Appl. Sci. 43(12) (2020), 7227–7238. https://doi.org/10.1002/mma.6461
  • 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
There are 54 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Research Articles
Authors

Gülsüm Yeliz Şentürk 0000-0002-8647-1801

Nurten Gürses 0000-0001-8407-854X

Salim Yüce 0000-0002-8296-6495

Publication Date March 30, 2023
Submission Date February 16, 2022
Acceptance Date July 3, 2022
Published in Issue Year 2023 Volume: 72 Issue: 1

Cite

APA Şentürk, G. Y., 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. https://doi.org/10.31801/cfsuasmas.1074557
AMA Şentürk GY, Gürses N, Yüce S. New insight into quaternions and their matrices. Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat. March 2023;72(1):43-58. doi:10.31801/cfsuasmas.1074557
Chicago Şentürk, Gülsüm Yeliz, Nurten Gürses, and Salim Yüce. “New Insight into Quaternions and Their Matrices”. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics 72, no. 1 (March 2023): 43-58. https://doi.org/10.31801/cfsuasmas.1074557.
EndNote Şentürk GY, Gürses N, Yüce S (March 1, 2023) New insight into quaternions and their matrices. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics 72 1 43–58.
IEEE G. Y. Şentürk, N. Gürses, and S. Yüce, “New insight into quaternions and their matrices”, Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat., vol. 72, no. 1, pp. 43–58, 2023, doi: 10.31801/cfsuasmas.1074557.
ISNAD Şentürk, Gülsüm Yeliz et al. “New Insight into Quaternions and Their Matrices”. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics 72/1 (March 2023), 43-58. https://doi.org/10.31801/cfsuasmas.1074557.
JAMA Şentürk GY, Gürses N, Yüce S. New insight into quaternions and their matrices. Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat. 2023;72:43–58.
MLA Şentürk, Gülsüm Yeliz et al. “New Insight into Quaternions and Their Matrices”. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics, vol. 72, no. 1, 2023, pp. 43-58, doi:10.31801/cfsuasmas.1074557.
Vancouver Şentürk GY, Gürses N, Yüce S. New insight into quaternions and their matrices. Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat. 2023;72(1):43-58.

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

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