Araştırma Makalesi
Yıl 2023,
Cilt: 6 Sayı: 3-Supplement, 1 - 15, 15.10.2023
Öz
In this study, we investigate the matrices over the new extension of the real numbers in four dimensional space E2^4
called the hybrid numbers. Since the hybrid multiplication is noncommutative, this leads to finding a linear
transformation on the complex field. Thus we characterize the hybrid matrices and examine their algebraic properties with respect to their complex adjoint matrices. Moreover, we define the co-determinant of hybrid matrices which plays an important role to construct the Lie groups.
Anahtar Kelimeler
Complex numbers Dual numbers Hyberbolic numbers Hybrid numbers Hypercomplex numbers Complex matrices Hybrid matrices
Kaynakça
- Y. Alagöz, K. H. Oral, S. Yüce, Split quaternion matrices, Miskolc Mathematical Notes, Vol.13, No.2, pp.223-232 (2012).
- S. A. Billings, Nonlinear system identication: NARMAX methods in the time, frequency, and spatio-temporal domains, John Wiley & Sons, (2013).
- F. Catoni, R. Cannata, V. Catoni, P. Zampetti, Two dimensional hypercomplex number and related trigonometries, Adv. Appl. Cliord Algebras, Vol.14, No.1, pp.47-68 (2004).
- F. Catoni, D. Boccaletti, R. Cannata, V. Catoni, E. Nichelatti, P. Zampetti, The mathematics of Minkowski space-time: with an introduction to commutative hypercomplex numbers, Birkhauser, Basel, (2008).
- Y. Choo, On the generalized bi-periodic Fibonaci and Lucas quaternions, Miskolc Mathematical Notes, Vol.20, No.2, pp.807-821 (2019).
- W.K. Cliord, Preliminary sketch of biquaternions, Proc. Lond. Math. Soc., Vol.1, No.1, pp.381-395 (1871).
- J. Cockle, III. On a new imaginary in algebra, Lond. Edinb. Dublin Philos. Mag., Vol.34, No.226, pp.37-47 (1849).
- N. Cohen, S. De Leo, The quaternionic determinant, The Electronic Journal of Linear Algebra, Vol.7, pp.100-111 (2000).
- G. Dattoli, et al., Hybrid complex numbers: the matrix version, Adv. Appl. Cliord Algebras, Vol.28, pp.1-17 (2018).
- P.A.M. Dirac, The principles of quantum mechanics, 4th edition, Oxford University Press, Oxford, (1958).
- M. Erdoğdu, M. Özdemir, On eigenvalues of split quaternion matrices, Adv. Appl. Cliord Algebras, Vol.23, No.3, pp.615-623 (2013).
- W. R. Hamilton, Elements of quaternions, Longmans, Green, & Company, (1866).
- A.A. Harkin, J.B. Harkin, Geometry of generalized complex numbers, Math. Mag., Vol.77, No.2, pp.118-29 (2004).
- L. Huang, W. So, On left eigenvalues of a quaternionic matrix, Linear algebra and its applications, Vol.323, No.1-3, pp.105-116 (2001).
- T. Jiang, Z. Zhang, Z. Jiang, Algebraic techniques for eigenvalues and eigenvectors of a split quaternion matrix in split quaternionic mechanics, Computer Physics Communications,Vol.229, pp.1-7 (2018).
- V.V. Kisil, Erlangen program at large-1: geometry of invariants, SIGMA Symmetry Integr. Geom. Methods Appl., Vol.6, No.076, pp.45 (2010).
- M. Ozdemir, Introduction to hybrid numbers. Adv. Appl. Cliord Algebras, Vol.28, pp.1-32 (2018).
- M. Ozdemir, Finding n-th roots of a 2 2 real matrix using De Moivre's formula, Adv. Appl. Cliord Algebras, Vol.29, No.1, pp. 2 (2019).
- Ç . Ramis, Y. Yaylı, Dual split quaternions and Chasles' theorem in 3 dimensional Minkowski space E31 , Adv. Appl. Cliord Algebras, Vol.23, pp.951-964 (2013).
- B. A. Rosenfeld, A history of Non-Euclidean geometry, Studies in the History of Mathematics and Physical Sciences, Springer, (1988).
- G. Sobczyk, New foundations in mathematics: the geometric concept of number, Birkhauser, Boston, (2013).
- A. Szynal-Liana, The Horadam hybrid numbers, Discuss. Math. Gen. Algebra Appl., Vol.38, pp.91-98 (2018).
- I.M. Yaglom, Complex Numbers in Geometry, Academic Press, New York, (1968).
- I.M. Yaglom, A Simple Non-Euclidean Geometry and its Physical Basis, Heidelberg Science Library, Springer, New York, (1979).
- Y. Yazlik, S. Kome, C. Kome, Bicomplex generalized kHoradam quaternions, Miskolc Mathematical Notes, Vol.20, No.2, pp.1315-1330 (2019).
- F. Zhang, Quaternions and matrices of quaternions. Linear Algebra Appl., Vol.251, pp.21-57 (1997).
Yıl 2023,
Cilt: 6 Sayı: 3-Supplement, 1 - 15, 15.10.2023
Öz
Kaynakça
- Y. Alagöz, K. H. Oral, S. Yüce, Split quaternion matrices, Miskolc Mathematical Notes, Vol.13, No.2, pp.223-232 (2012).
- S. A. Billings, Nonlinear system identication: NARMAX methods in the time, frequency, and spatio-temporal domains, John Wiley & Sons, (2013).
- F. Catoni, R. Cannata, V. Catoni, P. Zampetti, Two dimensional hypercomplex number and related trigonometries, Adv. Appl. Cliord Algebras, Vol.14, No.1, pp.47-68 (2004).
- F. Catoni, D. Boccaletti, R. Cannata, V. Catoni, E. Nichelatti, P. Zampetti, The mathematics of Minkowski space-time: with an introduction to commutative hypercomplex numbers, Birkhauser, Basel, (2008).
- Y. Choo, On the generalized bi-periodic Fibonaci and Lucas quaternions, Miskolc Mathematical Notes, Vol.20, No.2, pp.807-821 (2019).
- W.K. Cliord, Preliminary sketch of biquaternions, Proc. Lond. Math. Soc., Vol.1, No.1, pp.381-395 (1871).
- J. Cockle, III. On a new imaginary in algebra, Lond. Edinb. Dublin Philos. Mag., Vol.34, No.226, pp.37-47 (1849).
- N. Cohen, S. De Leo, The quaternionic determinant, The Electronic Journal of Linear Algebra, Vol.7, pp.100-111 (2000).
- G. Dattoli, et al., Hybrid complex numbers: the matrix version, Adv. Appl. Cliord Algebras, Vol.28, pp.1-17 (2018).
- P.A.M. Dirac, The principles of quantum mechanics, 4th edition, Oxford University Press, Oxford, (1958).
- M. Erdoğdu, M. Özdemir, On eigenvalues of split quaternion matrices, Adv. Appl. Cliord Algebras, Vol.23, No.3, pp.615-623 (2013).
- W. R. Hamilton, Elements of quaternions, Longmans, Green, & Company, (1866).
- A.A. Harkin, J.B. Harkin, Geometry of generalized complex numbers, Math. Mag., Vol.77, No.2, pp.118-29 (2004).
- L. Huang, W. So, On left eigenvalues of a quaternionic matrix, Linear algebra and its applications, Vol.323, No.1-3, pp.105-116 (2001).
- T. Jiang, Z. Zhang, Z. Jiang, Algebraic techniques for eigenvalues and eigenvectors of a split quaternion matrix in split quaternionic mechanics, Computer Physics Communications,Vol.229, pp.1-7 (2018).
- V.V. Kisil, Erlangen program at large-1: geometry of invariants, SIGMA Symmetry Integr. Geom. Methods Appl., Vol.6, No.076, pp.45 (2010).
- M. Ozdemir, Introduction to hybrid numbers. Adv. Appl. Cliord Algebras, Vol.28, pp.1-32 (2018).
- M. Ozdemir, Finding n-th roots of a 2 2 real matrix using De Moivre's formula, Adv. Appl. Cliord Algebras, Vol.29, No.1, pp. 2 (2019).
- Ç . Ramis, Y. Yaylı, Dual split quaternions and Chasles' theorem in 3 dimensional Minkowski space E31 , Adv. Appl. Cliord Algebras, Vol.23, pp.951-964 (2013).
- B. A. Rosenfeld, A history of Non-Euclidean geometry, Studies in the History of Mathematics and Physical Sciences, Springer, (1988).
- G. Sobczyk, New foundations in mathematics: the geometric concept of number, Birkhauser, Boston, (2013).
- A. Szynal-Liana, The Horadam hybrid numbers, Discuss. Math. Gen. Algebra Appl., Vol.38, pp.91-98 (2018).
- I.M. Yaglom, Complex Numbers in Geometry, Academic Press, New York, (1968).
- I.M. Yaglom, A Simple Non-Euclidean Geometry and its Physical Basis, Heidelberg Science Library, Springer, New York, (1979).
- Y. Yazlik, S. Kome, C. Kome, Bicomplex generalized kHoradam quaternions, Miskolc Mathematical Notes, Vol.20, No.2, pp.1315-1330 (2019).
- F. Zhang, Quaternions and matrices of quaternions. Linear Algebra Appl., Vol.251, pp.21-57 (1997).
Toplam 26 adet kaynakça vardır.
Ayrıntılar
| Birincil Dil | İngilizce |
|---|---|
| Konular | Cebirsel ve Diferansiyel Geometri, Reel ve Kompleks Fonksiyonlar |
| Bölüm | Araştırma Makalesi |
| Yazarlar | |
| Yayımlanma Tarihi | 15 Ekim 2023 |
| Gönderilme Tarihi | 24 Temmuz 2023 |
| Kabul Tarihi | 8 Ekim 2023 |
| Yayımlandığı Sayı | Yıl 2023 Cilt: 6 Sayı: 3-Supplement |
