Research on the Computation and Properties of the Hyperbolic Complex Determinant

Jie Zhang; Yajing Ma; Kaitong Jin

doi:10.54097/8433ec19

Authors

Jie Zhang
Yajing Ma
Kaitong Jin

DOI:

https://doi.org/10.54097/8433ec19

Keywords:

Hyperbolic Complex, Determinant, Zero Divisor Factorization.

Abstract

In this paper, the calculation results and properties of determinants under hyperbolic complex numbers are studied. The hyperbolic complex number is a commutative ring consisting of two real numbers with zero divisors. This paper overcomes the difficulty that the hyperbolic complex number contains zero divisors, and calculates the decomposition of the determinant of the hyperbolic complex number by its decomposition property. This paper investigates the decomposition operation of hyperbolic complex determinants and proves that any hyperbolic complex determinant can be decomposed into two real determinants through zero factors. Based on this decomposition, the necessary and sufficient conditions for the invertibility of hyperbolic complex determinants are further derived. In addition, the property that the determinant of the product of hyperbolic complex matrices is equal to the product of their respective determinants is discussed. This will lay a foundation for the study of hyperbolic complex numbers in algebraic theory and provide a new direction for their application in physics.

Downloads

Download data is not yet available.

References

[1] Zhao Haiquan, Chen Jinsong, Wang Zhuonan, et.al. Frequency estimation of distributed power systems based on enhanced complex-log-hyperbolic cosine algorithm [J]. Signal processing, 2023, 39 (11): 2071 - 2079.

[2] Chen Zhiyong, Luo An, Huang Xucheng, et.al. A method for assessing the stability of broadband harmonic resonance based on Euler's formula[J]. Processing of the CSEE, 2020, 40 (5): 1509 - 1522.

[3] Kumar D, Pathan M. On a Class of Integrals of Beta Family: Series Representations and Fractional Maps [J]. Mediterranean journal of mathematics, 2024, 21.

[4] Li Anchen. Research on recommendation model based on graph representation learning [D]. Jilin: Jilin University, 2023.

[5] Zhu, Mingmin, Jiang, Jiewei, Gao, Weifeng. A fast ADMM algorithm for sparse precision matrix estimation using lasso penalized D-trace loss [J]. Egyptian informatics journal, 2024, 25.

[6] Cortinovis A, Kressner D. On randomized trace estimates for indefinite matrices with an application to determinants [J]. Found Comput Math 22, 2022, 875 – 903.

[7] Dar I, Rather N, Faiq I. Bounds on the zeros of a quaternionic polynomial using matrix methods [J]. Filoma, 2024, 38: 3001 - 3010.

[8] Siavash A, Shawn C. Shadden. A singular woodbury and pseudo-determinant matrix identities and application to gaussian process regression [J]. Applied Mathematics and Computation, 452, 2023, 128032.

[9] Drescher M, Salmhofer M, Enss T. Bosonic Functional Determinant Approach and its Application to Polaron Spectra [J], 2024: 1 - 10.

[10] Masaki K. Reducibilities of hyperbolic neural networks[J]. Neurocomputing, 378, 2020: 129 - 141.

[11] Blankers V, Rendfrey T, Shukert A, Patrick D. Julia and mandelbrot sets for dynamics over the hyperbolic numbers [M]. Fractal and Fractional, Shipman, Blankers, 2019: 3 (1), 6.

[12] Yang, M, Verma H, Zhang, et.al. Hypformer exploring efficient transformer fully in hyperbolic space [J]. Proceedings of the Acm Sigkdd International Conference on Knowledge Discovery and Data Mining, 2024, 3770 – 3781.

[13] Gamaliel Y, Juan B. Generalized iterated function system on hyperbolic number Plane [J], 2019, Fractals.

[14] A. Calzolari, A. Catellani, M. BuongiornoNardelli, M. Fornari, Hyperbolic Metamaterials with Extreme Mechanical Hardness. Adv. Optical Mater. 2021, 9, 2001904.