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Explicit formulas for exponential of 2×2 split-complex matrices

Year 2022, Volume: 71 Issue: 2, 518 - 532, 30.06.2022
https://doi.org/10.31801/cfsuasmas.991894

Abstract

Split-complex (hyperbolic) numbers are ordered pairs of real numbers, written in the form $x+jy$ with $j^{2}=-1$, used to describe the geometry of the Lorentzian plane. Since a null split-complex number does not have an inverse, some methods to calculate the exponential of complex matrices are not valid for split-complex matrices. In this paper, we examined the exponential of a $2x2$ split-complex matrix in three cases : $i:~\Delta=0,~ii:~\Delta\neq0$ and $\Delta$ is not null split-complex number, $iii:~\Delta\neq0$ and $\Delta$ is a null split-complex number where $\Delta=(trA)^{2}-4detA$.

References

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  • Baker, A., Matrix Groups: An Introduction to Lie Group Theory, Springer Science & Business Media, 2012. doi: 10.1007/978-1-4471-0183-3
  • Bernstein, D. S., Orthogonal matrices and the matrix exponential, SIAM Review, 32(4) (1990), 673. doi: 10.1137/1032130
  • Bernstein, D. S., So, W., Some explicit formulas for the matrix exponential, IEEE Transactions on Automatic Control, 38(8) (1993), 1228-1232. doi: 10.1109/9.233156
  • Borota, N. A., Flores, E., Osler, T. J., Spacetime numbers the easy way, Mathematics and Computer Education, 34(2) (2000), 159.
  • Catoni, F., Boccaletti, D., Cannata, R., Catoni, V., Zampetti, P., Geometry of Minkowski Space-time, Springer Science & Business Media, 2011. doi: 10.1007/978-3-642-17977-8
  • Erdogdu, M., Özdemir, M., On exponential of split quaternionic matrices, Applied Mathematics and Computation, 315 (2017), 468-476. doi: 10.1016/j.amc.2017.08.007
  • Erdogdu, M., Özdemir, M., Matrices over hyperbolic split quaternions, Filomat, 30(4) (2016), 913-920. doi: 10.2298/FIL1604913E
  • Ersoy, S., Akyigit, M., One-parameter homothetic motion in the hyperbolic plane and Euler-Savary formula, Advances in Applied Clifford Algebras, 21(2) (2011), 297-313. doi: 10.1007/s00006-010-0255-3
  • Fjelstad, P., Extending special relativity via the perplex numbers, American Journal of Physics, 54(5) (1986), 416-422. doi: 10.1119/1.14605
  • Amorim, R. G. G. D., Santos, W. C. D., Carvalho, L. B., Massa, I. R., A physical approach of perplex numbers, Revista Brasileira de Ensino de F´ısica, 40(3) (2018). doi: 10.1590/1806-9126-RBEF-2017-0356
  • Gürses, N., Sentürk, G. Y., Yüce, S., A study on dual-generalized complex and hyperbolic generalized complex numbers, Gazi University Journal of Science, 34(1) (2021), 180-194.
  • Harkin, A. A., Harkin, J. B., Geometry of generalized complex numbers, Mathematics Magazine, 77(2) (2004), 118-129. doi: 10.1080/0025570X.2004.11953236
  • Kisil, V. V., Geometry of Mobius Transformations: Elliptic, Parabolic and Hyperbolic Actions of SL2(R). World Scientific, 2012.
  • Laksov, D., Diagonalization of matrices over rings, Journal of Algebra, 376 (2013), 123-138. doi: 10.1016/j.jalgebra.2012.10.029
  • Leonard, I. E., The matrix exponential, SIAM review, 38(3) (1996), 507-512. doi: 10.1137/S0036144595286488
  • Machen, C., The exponential of a quaternionic matrix, Rose-Hulman Undergraduate Mathematics Journal, 12(2) (2011), 3.
  • McDonald Bernard, R., Lynn McDonald, et al., Linear algebra over commutative rings, volume 87, Courier Corporation, 1984.
  • Poodiack, R. D., LeClair, K. J., Fundamental theorems of algebra for the perplexes, The College Mathematics Journal, 40(5) (2009), 322-336. doi: 10.4169/074683409X475643
  • Özyurt, G., Alagöz, Y., On hyperbolic split quaternions and hyperbolic split quaternion matrices, Advances in Applied Clifford Algebras, 28(5) (2018), 1-11. doi: 10.1007/s00006-018-0907-2
  • Richter, R. B., Wardlaw, W. P., Diagonalization over commutative rings, The American Mathematical Monthly, 97(3) (1990), 223-227. doi: 10.1080/00029890.1990.11995580
  • Rooney, J., On the three types of complex number and planar transformations, Environment and Planning B: Planning and Design, 5(1) (1978), 89-99. doi: 10.1068/b050089
  • Sobczyk, G., The hyperbolic number plane, The College Mathematics Journal, 26(4) (1995), 268-280. doi: 10.1080/07468342.1995.11973712
  • Sobczyk, G., Complex and Hyperbolic Numbers, In New Foundations in Mathematics (pp. 23-42). Birkh¨auser, Boston. doi: 10.1007/978-0-8176-8385-6_2
  • Sporn, H., Pythagorean triples, complex numbers, and perplex numbers, The College Mathematics Journal, 48(2) (2017), 115-122. doi: 10.4169/college.math.j.48.2.115
  • Tapp, T., Matrix groups for undergraduates, student math, 2005.
  • Ulrych, S., Representations of Clifford algebras with hyperbolic numbers, Advances in Applied Clifford Algebras, 18(1) (2008), 93-114. doi: 10.1007/s00006-007-0057-4
  • Yaglom, I. M., A Simple Non-Euclidean Geometry and Its Physical Basis: An Elementary Account of Galilean Geometry and the Galilean Pinciple of Relativity, Springer Science & Business Media, 2012. doi:10.1007/978-1-4612-6135-3
  • Zhang, F., Matrix Theory: Basic Results and Techniques, Springer Science & Business Media, 2011.
Year 2022, Volume: 71 Issue: 2, 518 - 532, 30.06.2022
https://doi.org/10.31801/cfsuasmas.991894

Abstract

References

  • Ablamowicz, R., Matrix exponential via Clifford algebras, Journal of Nonlinear Mathematical Physics, 5(3) (1998), 294-313. doi: 10.2991/jnmp.1998.5.3.5
  • Baker, A., Matrix Groups: An Introduction to Lie Group Theory, Springer Science & Business Media, 2012. doi: 10.1007/978-1-4471-0183-3
  • Bernstein, D. S., Orthogonal matrices and the matrix exponential, SIAM Review, 32(4) (1990), 673. doi: 10.1137/1032130
  • Bernstein, D. S., So, W., Some explicit formulas for the matrix exponential, IEEE Transactions on Automatic Control, 38(8) (1993), 1228-1232. doi: 10.1109/9.233156
  • Borota, N. A., Flores, E., Osler, T. J., Spacetime numbers the easy way, Mathematics and Computer Education, 34(2) (2000), 159.
  • Catoni, F., Boccaletti, D., Cannata, R., Catoni, V., Zampetti, P., Geometry of Minkowski Space-time, Springer Science & Business Media, 2011. doi: 10.1007/978-3-642-17977-8
  • Erdogdu, M., Özdemir, M., On exponential of split quaternionic matrices, Applied Mathematics and Computation, 315 (2017), 468-476. doi: 10.1016/j.amc.2017.08.007
  • Erdogdu, M., Özdemir, M., Matrices over hyperbolic split quaternions, Filomat, 30(4) (2016), 913-920. doi: 10.2298/FIL1604913E
  • Ersoy, S., Akyigit, M., One-parameter homothetic motion in the hyperbolic plane and Euler-Savary formula, Advances in Applied Clifford Algebras, 21(2) (2011), 297-313. doi: 10.1007/s00006-010-0255-3
  • Fjelstad, P., Extending special relativity via the perplex numbers, American Journal of Physics, 54(5) (1986), 416-422. doi: 10.1119/1.14605
  • Amorim, R. G. G. D., Santos, W. C. D., Carvalho, L. B., Massa, I. R., A physical approach of perplex numbers, Revista Brasileira de Ensino de F´ısica, 40(3) (2018). doi: 10.1590/1806-9126-RBEF-2017-0356
  • Gürses, N., Sentürk, G. Y., Yüce, S., A study on dual-generalized complex and hyperbolic generalized complex numbers, Gazi University Journal of Science, 34(1) (2021), 180-194.
  • Harkin, A. A., Harkin, J. B., Geometry of generalized complex numbers, Mathematics Magazine, 77(2) (2004), 118-129. doi: 10.1080/0025570X.2004.11953236
  • Kisil, V. V., Geometry of Mobius Transformations: Elliptic, Parabolic and Hyperbolic Actions of SL2(R). World Scientific, 2012.
  • Laksov, D., Diagonalization of matrices over rings, Journal of Algebra, 376 (2013), 123-138. doi: 10.1016/j.jalgebra.2012.10.029
  • Leonard, I. E., The matrix exponential, SIAM review, 38(3) (1996), 507-512. doi: 10.1137/S0036144595286488
  • Machen, C., The exponential of a quaternionic matrix, Rose-Hulman Undergraduate Mathematics Journal, 12(2) (2011), 3.
  • McDonald Bernard, R., Lynn McDonald, et al., Linear algebra over commutative rings, volume 87, Courier Corporation, 1984.
  • Poodiack, R. D., LeClair, K. J., Fundamental theorems of algebra for the perplexes, The College Mathematics Journal, 40(5) (2009), 322-336. doi: 10.4169/074683409X475643
  • Özyurt, G., Alagöz, Y., On hyperbolic split quaternions and hyperbolic split quaternion matrices, Advances in Applied Clifford Algebras, 28(5) (2018), 1-11. doi: 10.1007/s00006-018-0907-2
  • Richter, R. B., Wardlaw, W. P., Diagonalization over commutative rings, The American Mathematical Monthly, 97(3) (1990), 223-227. doi: 10.1080/00029890.1990.11995580
  • Rooney, J., On the three types of complex number and planar transformations, Environment and Planning B: Planning and Design, 5(1) (1978), 89-99. doi: 10.1068/b050089
  • Sobczyk, G., The hyperbolic number plane, The College Mathematics Journal, 26(4) (1995), 268-280. doi: 10.1080/07468342.1995.11973712
  • Sobczyk, G., Complex and Hyperbolic Numbers, In New Foundations in Mathematics (pp. 23-42). Birkh¨auser, Boston. doi: 10.1007/978-0-8176-8385-6_2
  • Sporn, H., Pythagorean triples, complex numbers, and perplex numbers, The College Mathematics Journal, 48(2) (2017), 115-122. doi: 10.4169/college.math.j.48.2.115
  • Tapp, T., Matrix groups for undergraduates, student math, 2005.
  • Ulrych, S., Representations of Clifford algebras with hyperbolic numbers, Advances in Applied Clifford Algebras, 18(1) (2008), 93-114. doi: 10.1007/s00006-007-0057-4
  • Yaglom, I. M., A Simple Non-Euclidean Geometry and Its Physical Basis: An Elementary Account of Galilean Geometry and the Galilean Pinciple of Relativity, Springer Science & Business Media, 2012. doi:10.1007/978-1-4612-6135-3
  • Zhang, F., Matrix Theory: Basic Results and Techniques, Springer Science & Business Media, 2011.
There are 29 citations in total.

Details

Primary Language English
Subjects Applied Mathematics
Journal Section Research Articles
Authors

Hasan Çakır 0000-0003-4317-7968

Mustafa Özdemir 0000-0002-1359-4181

Publication Date June 30, 2022
Submission Date September 6, 2021
Acceptance Date January 18, 2022
Published in Issue Year 2022 Volume: 71 Issue: 2

Cite

APA Çakır, H., & Özdemir, M. (2022). Explicit formulas for exponential of 2×2 split-complex matrices. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics, 71(2), 518-532. https://doi.org/10.31801/cfsuasmas.991894
AMA Çakır H, Özdemir M. Explicit formulas for exponential of 2×2 split-complex matrices. Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat. June 2022;71(2):518-532. doi:10.31801/cfsuasmas.991894
Chicago Çakır, Hasan, and Mustafa Özdemir. “Explicit Formulas for Exponential of 2×2 Split-Complex Matrices”. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics 71, no. 2 (June 2022): 518-32. https://doi.org/10.31801/cfsuasmas.991894.
EndNote Çakır H, Özdemir M (June 1, 2022) Explicit formulas for exponential of 2×2 split-complex matrices. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics 71 2 518–532.
IEEE H. Çakır and M. Özdemir, “Explicit formulas for exponential of 2×2 split-complex matrices”, Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat., vol. 71, no. 2, pp. 518–532, 2022, doi: 10.31801/cfsuasmas.991894.
ISNAD Çakır, Hasan - Özdemir, Mustafa. “Explicit Formulas for Exponential of 2×2 Split-Complex Matrices”. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics 71/2 (June 2022), 518-532. https://doi.org/10.31801/cfsuasmas.991894.
JAMA Çakır H, Özdemir M. Explicit formulas for exponential of 2×2 split-complex matrices. Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat. 2022;71:518–532.
MLA Çakır, Hasan and Mustafa Özdemir. “Explicit Formulas for Exponential of 2×2 Split-Complex Matrices”. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics, vol. 71, no. 2, 2022, pp. 518-32, doi:10.31801/cfsuasmas.991894.
Vancouver Çakır H, Özdemir M. Explicit formulas for exponential of 2×2 split-complex matrices. Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat. 2022;71(2):518-32.

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

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