Matrices with Hyperbolic Number Entries

In this study, firstly, we will present some properties of hyperbolic numbers. Then, we will introduce hyperbolic matrices, which are matrices with hyperbolic number entries. Additionally, we will examine the algebraic properties of these matrices and reveal its difference from other matrix structures such as real, dual, and complex matrices. As a result of comparing the results found in this work with real, dual, and complex matrices, it will be revealed that there are similarities in additive properties and differences in some multiplicative properties. Finally, we will define some special hyperbolic matrices and give their properties and relations with real matrices.

Keywords:

Hyperbolic matrices hyperbolic numbers, special hyperbolic matrices,

PDF

___

Alagöz, Y., Oral, K.H., Yüce, S., Split quaternion matrices, Miskolc Mathematical Notes, 13(2)(2012), 223–232.
Assis, A.K.T., Perplex numbers and quaternions, International Journal of Mathematical Education in Science and Technology, 22(4)(1991), 555–562.
Beauregard, R.A., Suryanarayan E.R., Pythagorean triples: the hyperbolic view, The College Mathematics Journal, 27(3)(1996), 170–181.
Dağdeviren, A., Kürüz, F., Special real and dual matrices with Hadamard product, Journal of Engineering Technology and Applied Sciences, 6(2)(2021), 127–134.
Dağdeviren, A., Lorentz matris carpimi ve dual matrislerin ozellikleri, Master’s Thesis, Yildiz Technical University, 2013.
Fjelstad, P., Extending special relativity via the perplex numbers, American Journal of Physics, 54(5)(1986), 416–422.
Gutin, R., Matrix decompositions over the split-complex numbers, arXiv preprint arXiv:2105.08047, (2021).
Kulyabov, D.S., Korolkova, A.V., Gevorkyan, M.N., Hyperbolic numbers as Einstein numbers, In Journal of Physics: Conference Series, IOP Publishing, 1557(1)(2020), 12–27.
Motter, A.E., Rosa, M.A.F., Hyperbolic calculus, Advances in Applied Clifford Algebras, 8(1)(1998), 109–128.
Petroudi, S.H.J., Pirouz, M., Akbiyik, M., Yilmaz, F., Some special matrices with harmonic numbers, Konuralp Journal of Mathematics, 10(1)(2022), 188–196.
Sobczyk, G., The hyperbolic number plane, The College Mathematics Journal, 26(4)(1995), 268–280.
Stanimirovi´c, P., Nikolov, J., Stanimirovi´c, I., A generalization of Fibonacci and Lucas matrices, Discrete Applied Mathematics, 156(14)(2008), 2606–2619.
Yaglom, I.M., Complex Numbers in Geometry, Academic Press, 2014.