Sedenionic matrices and their properties

Year 2024, Volume: 14 Issue: 3, 721 - 734, 15.09.2024

İsmail Gökhan Gürsoy , Özcan Bektaş

https://doi.org/10.17714/gumusfenbil.1415410

Abstract

In this article, the matrix algebra is well-known concept in mathematics, has been extended to sedenionic-coefficient matrices using sedenions, which have many applications in recent years. Subsequently, sedenionic-coefficient matrices have been obtained in their real, complex, quaternionic and octonionic forms. Based on these definitions, the arithmetic operations of addition, multiplication, conjugation, transpose and conjugate transpose for sedenionic matrices and their complex, quaternionic and octonionic variations have been established and their algebraic properties scrutinized. Lastly, vector space over real and complex numbers and module structure over quaternions of sedenionic matrices has been searched.

Keywords

Octonion, Sedenion, Sedenionic matrices, Quaternion

Ethical Statement

The authors of this manuscript declare that the materials and methods used in study to do not require ethical committee approval and/or legal-specific permission.

Thanks

The authors sincerely thank the editors of the journal who devoted their attention to the paper and also the referees who contributed their valuable time to the study.

Sedeniyonik matrisler ve özellikleri

Abstract

Bu makalede matematikte iyi bilinen bir konu olan matris cebiri, son yıllarda pek çok uygulamalaya sahip sedeniyonlar kullanılarak sedeniyonik katsayılı matrislere genişletilmiştir. Daha sonra sedeniyonik katsayılı matrislerin, reel, kompleks, kuaterniyonik ve oktoniyonik katsayılı versiyonları elde edilmiştir. Bu tanımlar doğrultusunda, sedeniyonik matrislerin ve diğer varyasyonlarının toplama, çarpma, eşlenik, tranpoz ve eşlenik transpoz işlemleri tanımlanmış ve cebirsel özellikleri incelenmiştir. Son olarak sedeniyonik matrislerin reel ve kompleks sayılar üzerindeki vektör uzayları ve kuaterniyonlar üzerindeki modül yapısı araştırılmıştır.

Keywords

Oktoniyon, Sedeniyon, Sedeniyonik matrisler, Kuaterniyon

Ethical Statement

Bu yazının yazarları, çalışmada kullanılan materyal ve yöntemlerin etik kurul onayı ve/veya yasal izin gerektirmediğini beyan eder.

Thanks

Yazarlar, makaleye ilgi gösteren dergi editörlerine ve çalışmaya değerli zamanlarını ayıran hakemlere içtenlikle teşekkür ederler.

References

• Bektas, O., (2021). Some properties and special matrix representations of C,H,O coefficient sedenion numbers. Bitlis Eren University Journal of Science, 10 (4), 1416-1425. https://doi.org/10.17798/bitlisfen.959454
• Bektas, O., Senyurt, S., Gür Mazlum, S., (2023) Some properties of fuzzy sedenion numbers and fuzzy sedenion valued series. Maejo International Journal of Science Technology, 17(03), 239-251.
• Bilgici, G., Tokeser, U., & Unal, Z., (2017). Fibonacci and lucas sedenions. Journal of Integer sequence, 20(1), 8.
• Carmody, K., (1988). Circular and hyperbolic quaternions, octonions, and sedenions. Applied Mathematics and Computation, 84(1), 27-47. https://doi.org/10.1016/S0096-3003(96)00051-3
• Catarino, P., (2016). The modified pell and the modified k-pell quaternions and octonions. Advances in Applied Clifford Algebras, 26(2), 577-590. https://doi.org.10.1007/s00006-015-0611-4
• Catarino, P., (2019). k-Pell, k-pell lucas and modified k-pell sedenions. Asian-European Journal of Mathematics, 12(2), 1-10. https://doi.org/10.1142/S1793557119500189
• Cawagas, R. E., (2004). On the structure and zero divisors of the cayley-dickson sedenion algebra. Discussiones Mathematicae General Algebra and Application, 24(2), 251-265.
• Chan, K.C., & Dokovic, D.Z., (2018). Conjugacy class of subalgebras of the real sedenions. Canadian Mathematical Bulletin, 49(4), 492–507. https://doi.org/10.4153/CMB-2006-048-6
• Cimen, C., (2019). On the dual jacobsthal and dual jacobsthal lucas sedenions. Erzincan University Journal of Science and Technology, 12(3), 1759-1766. https://doi.org/10.18185/erzifbed.539189
• Devi, B.M., & Devibala, S., (2021). On mersenne and mersenne-lucas sedenions. Advances and Application in Mathematical Science, 21(1), 383-392.
• Dickson, L. E., (1919). On quaternions and their generalization and the history of eight square theorem. Annals of Mathematics, 20(3), 155-171.
• Imaeda, K., & Imaeda, M., (2000). Sedenions algebra and analysis. Applied Mathematics Computation, 115(2-3), 77-88. https://doi.org/10.1016/S0096-3003(99)00140-X
• Ipek, A., & Cimen, C., (2016a). On pell quaternions and pell lucas quaternions. Advances in Applied Clifford Algebras, 26, 39-51. https://doi.org/10.1007/s00006-015-0571-8
• Ipek, A., & Cimen, C., (2016b). On jacobsthal and jacobstal-lucas octonions. Mediterranean Journal of Mathematics, 14(2), 1-13. DOI:10.1007/s00009-017-0873-2
• Ipek, A., & Cimen, C., (2017). On jacobsthal and the jacobsthal-lucas sedenions and several identities involving these numbers. Mathematica Aaeterna, 7(4), 447-454.
• Ipek, A., Cimen, C., & Cimen, E., (2019). On horadam sedenions. Journal of Science and Arts, 4(49), 887-889.
• Kizilates, C., & Kirlak, S., (2022). New generalization of fibonacci and lucas type sedenions. Journal of Discrete Mathematical Science and Cryptography, 26(8), 2217-2228. https://doi.org/10.1080/09720529.2022.2036405
• Moreno, G., (1997). The zero divisors of the cayley-dickson algebras over the real numbers. https://doi.org/10.48550/arXiv.q-alg/9710013
• Soykan, Y., (2019). Tribonacci and tribonacci-lucas sedenions. Mathematics, 7(1), 74. https://doi.org/10.3390/math7010074
• Soykan, Y., Okumuş, I., & Tasdemir, E., (2020). On generalized tribonacci sedenions. Sarajevo Journal of Mathematics, 16(29), 103-122. https://doi.org/10.48550/arXiv.1901.05312
• Sumer, D., (2022). Reformulation of multifluid plasma equations in terms of sedenion, [Master Thesis, Eskişehir Technical University Institute of Science].
• Tasyurdu, Y., & Akpinar, A., (2020). Perrin octonions and perrin sedenions. Konuralp Journal of Mathematics, 8(2), 384-390.