MATRICES OF HYBRID NUMBERS

Year 2023, , 1 - 15, 15.10.2023

Çağla Ramis , Yasin Yazlik

https://doi.org/10.33773/jum.1332081

Abstract

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.

Keywords

Complex numbers, Dual numbers, Hyberbolic numbers, Hybrid numbers, Hypercomplex numbers, Complex matrices, Hybrid matrices

References

• 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).