A third-order nilpotent augmentation of the dual-type octonion framework was obtained by replacing the first-order dual layer with a higher-order infinitesimal structure. Algebraic closure and coefficient-wise product expansions were established, enabling all subsequent derivations to be performed in a finite form. A comprehensive invertibility condition was established by limiting the problem to the leading component, and a closed-form inverse was constructed directly up to the second order, producing a computation-ready formula for reciprocal evaluation. The structure of zero divisors and nilpotent elements was characterized, showing how degeneracy and non-invertibility spread via higher-order perturbation layers. The nilpotent constraint was used to establish finite exponential and power expansions, resulting in identities of the Euler and De Moivre types that were obtained for the third-order scenario and explicit formulations for repeated products. A block triangular left-multiplication matrix representation was constructed, from which the determinant and spectral consequences were obtained with the inversion formulas. The resulting framework provides an algebraic tool for explicit computation within dual-type octonionic models and enables higher-order perturbation encoding.
Citation: V. James, B. Sivakumar, V. Rajkumar. Higher-order dual extensions of dual-type octonions with explicit inverses and block matrix representations[J]. AIMS Mathematics, 2026, 11(5): 15037-15055. doi: 10.3934/math.2026619
A third-order nilpotent augmentation of the dual-type octonion framework was obtained by replacing the first-order dual layer with a higher-order infinitesimal structure. Algebraic closure and coefficient-wise product expansions were established, enabling all subsequent derivations to be performed in a finite form. A comprehensive invertibility condition was established by limiting the problem to the leading component, and a closed-form inverse was constructed directly up to the second order, producing a computation-ready formula for reciprocal evaluation. The structure of zero divisors and nilpotent elements was characterized, showing how degeneracy and non-invertibility spread via higher-order perturbation layers. The nilpotent constraint was used to establish finite exponential and power expansions, resulting in identities of the Euler and De Moivre types that were obtained for the third-order scenario and explicit formulations for repeated products. A block triangular left-multiplication matrix representation was constructed, from which the determinant and spectral consequences were obtained with the inversion formulas. The resulting framework provides an algebraic tool for explicit computation within dual-type octonionic models and enables higher-order perturbation encoding.
| [1] | J. B. Kuipers, Quaternions and rotation sequences, A primer with applications to orbits, aerospace, and virtual reality, Princeton University Press, 1999. https://doi.org/10.1515/9780691211701 |
| [2] |
A. Rehman, M. Z. Ur Rahman, A. Ghaffar, C. Martin-Barreiro, C. Castro, V. Leiva, et al., Systems of quaternionic linear matrix equations: Solution, computation, algorithm, and applications, AIMS Math., 9 (2024), 26371–26402. https://doi.org/10.3934/math.20241284 doi: 10.3934/math.20241284
|
| [3] |
Z. H. He, M. L. Deng, J. J. Pan, M. K. Ng, Diagonalization of two quaternion tensors with applications, Appl. Math. Model., 148 (2025), 116278. https://doi.org/10.1016/j.apm.2025.116278 doi: 10.1016/j.apm.2025.116278
|
| [4] |
Z. H. He, Y. Z. Xu, Q. W. Wang, X. X. Wang, Solving a system of quaternion matrix equations by using PSVD for multiple matrices with applications, J. Comput. Appl. Math., 476 (2026), 117072. https://doi.org/10.1016/j.cam.2025.117072 doi: 10.1016/j.cam.2025.117072
|
| [5] |
J. C. Baez, The octonions, Bull. Amer. Math. Soc., 39 (2002), 145–205. https://doi.org/10.1090/S0273-0979-01-00934-X doi: 10.1090/S0273-0979-01-00934-X
|
| [6] | G. M. Dixon, Division algebras: octonions, quaternions, complex numbers and the algebraic design of physics, Mathematics and Its Applications, Vol. 290, Springer, 1994. https://doi.org/10.1007/978-1-4757-2315-1 |
| [7] |
S. De Leo, K. Abdel-Khalek, Octonionic Dirac equation, Prog. Theor. Phys., 96 (1996), 833–845. https://doi.org/10.1143/PTP.96.833 doi: 10.1143/PTP.96.833
|
| [8] | H. Pottmann, J. Wallner, Computational line geometry, Mathematics and Visualization, Springer, 2001. https://doi.org/10.1007/978-3-642-04018-4 |
| [9] |
G. R. Veldkamp, On the use of dual numbers, vectors and matrices in instantaneous, spatial kinematics, Mech. Mach. Theory, 11 (1976), 141–156. https://doi.org/10.1016/0094-114X(76)90006-9 doi: 10.1016/0094-114X(76)90006-9
|
| [10] |
H. H. Cheng, Programming with dual numbers and its applications in mechanisms design, Eng. Comput., 10 (1994), 212–229. https://doi.org/10.1007/BF01202367 doi: 10.1007/BF01202367
|
| [11] |
Y. L. Gu, J. Luh, Dual-number transformation and its applications to robotics, IEEE J. Rob. Autom., 3 (1987), 615–623. https://doi.org/10.1109/JRA.1987.1087138 doi: 10.1109/JRA.1987.1087138
|
| [12] |
K. Daniilidis, Hand-Eye calibration using dual quaternions, Int. J. Rob. Res., 18 (1999), 286–298. https://doi.org/10.1177/02783649922066213 doi: 10.1177/02783649922066213
|
| [13] | J. R. Dooley, J. M. McCarthy, Spatial rigid body dynamics using dual quaternion components, Proceedings. 1991 IEEE International Conference on Robotics and Automation, 1 (1991), 90–95. https://doi.org/10.1109/ROBOT.1991.131559 |
| [14] |
E. Pennestrì, R. Stefanelli, Linear algebra and numerical algorithms using dual numbers, Multibody Syst. Dyn., 18 (2007), 323–344. https://doi.org/10.1007/s11044-007-9088-9 doi: 10.1007/s11044-007-9088-9
|
| [15] |
B. C. Chanyal, Dual octonion electrodynamics with the massive field of dyons, J. Math. Phys., 57 (2016), 033503. https://doi.org/10.1063/1.4943594 doi: 10.1063/1.4943594
|
| [16] |
S. Halıcı, On dual Fibonacci octonions, Adv. Appl. Clifford Algebras, 25 (2015), 905–914. https://doi.org/10.1007/s00006-015-0550-0 doi: 10.1007/s00006-015-0550-0
|
| [17] |
N. Kılıç, On dual Horadam octonions, Notes Number Theory Discrete Math., 25 (2019), 137–149. https://doi.org/10.7546/nntdm.2019.25.1.137-149 doi: 10.7546/nntdm.2019.25.1.137-149
|
| [18] |
A. Dağdeviren, F. Kürüz, On dual type octonions and their properties, Turk. J. Math. Comput. Sci., 17 (2025), 93–101. https://doi.org/10.47000/tjmcs.1590392 doi: 10.47000/tjmcs.1590392
|
| [19] |
Y. Yan, D. Cheng, J. Feng, H. Li, J. Yue, Survey on applications of algebraic state space theory of logical systems to finite state machines, Sci. China Inf. Sci., 66 (2023), 111201. https://doi.org/10.1007/s11432-022-3538-4 doi: 10.1007/s11432-022-3538-4
|
| [20] | D. Condurache, Higher-order kinematics in dual Lie algebra, In: F. Bulnes, Advances on tensor analysis and their applications, kinematics, IntechOpen, 2020. https://doi.org/10.5772/intechopen.91779 |
V. James, B. Sivakumar, V. Rajkumar. Higher-order dual extensions of dual-type octonions with explicit inverses and block matrix representations[J]. AIMS Mathematics, 2026, 11(5): 15037-15055. doi: 10.3934/math.2026619
DownLoad: