Abstract
In this paper, we investigate some properties of dual complex numbers, dual complex vectors, and dual complex matrices. First, based on the magnitude of the dual complex number, we study the Young inequality, the Hölder inequality, and the Minkowski inequality in the setting of dual complex numbers. Second, we define the p-norm of a dual complex vector, which is a nonnegative dual number, and show some related properties. Third, we study the properties of eigenvalues of unitary matrices and unitary triangulation of arbitrary dual complex matrices. In particular, we introduce the operator norm of dual complex matrices induced by the p-norm of dual complex vectors, and give expressions of three important operator norms of dual complex matrices.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Brezov, D.: Factorization and generalized roots of dual complex matrices with Rodrigues’ formula. Adv. Appl. Clifford Algebras 30, 29 (2020)
Cheng, H.H., Thompson, S.: Dual polynomials and complex dual numbers for analysis of spatial mechanisms. In: Proceedings of the ASME Design Engineering Technical Conference and Computers in Engineering Conference, Irvine, California, 18–22 (1996)
Cheng, J., Kim, J., Jiang, Z., Che, W.: Dual quaternion-based graph SLAM. Robot. Auton. Syst. 77, 15–24 (2016)
Clifford, W.K.: Preliminary sketch of bi-quaternions. Proc. Lond. Math. Soc. 4, 381–395 (1873)
Daniilidis, K.: Hand-eye calibration using dual quaternions. Int. J. Robot. Res. 18, 286–298 (1999)
Gunn, C.: On the homogeneous model of Euclidean geomery. In: Dorst, L., Lasenby, J. (eds.) Guide to Geometric Algebra in Practice. Springer, London (2011)
Güngör, M.A., Tetik, Ö.: De-Moivre and Euler formulae for dual complex numbers. Univ. J. Math. Appl. 2, 126–129 (2019)
Matsuda, G., Kaji, S., Ochiai, H.: Anti-commutative dual complex numbers and 2D rigid transformation. In: Anjyo, K. (ed.) Mathematical Progress in Expressive Image Synthesis I: Extended and Selected Results from the Symposium MEIS2013, Mathematics for Industry, pp. 131–138. Springer, Japan (2014)
Pennestrì, E., Stefanelli, R.: Linear algebra and numerical algorithms using dual numbers. Multibody Syst. Dyn. 18, 323–344 (2007)
Qi, L., Alexander, D.M., Chen, Z.M., Ling, C., Luo, Z.: Low rank approximation of dual complex matrices. arXiv:2201.12781v1 (2022)
Qi, L., Ling, C., Yan, H.: Dual quaternions and dual quaternion vectors. Commun. Appl. Math. Comput. 4(4), 1494–1508 (2022). https://doi.org/10.1007/s42967-022-00189-y
Qi, L., Luo, Z.: Eigenvalues and singular values of dual quaternion matrices. arXiv:2111.12211 (2021)
Ulrich, M., Steger, C.: Hand-eye calibration of SCARA robots using dual quaternions. Pattern Recognit. Image Anal. 26, 231–239 (2016)
Wang, X., Yu, C., Lin, Z.: A dual quaternion solution to attitude and position control for rigid body coordination. IEEE Trans. Robot. 28, 1162–1170 (2012)
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflict of Interest
The authors declare that they have no conflict of interest.
Rights and permissions
Springer Nature or its licensor holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Miao, XH., Huang, ZH. Norms of Dual Complex Vectors and Dual Complex Matrices. Commun. Appl. Math. Comput. 5, 1484–1508 (2023). https://doi.org/10.1007/s42967-022-00215-z
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s42967-022-00215-z