Abstract
Singular value decomposition plays a prominent role in the theoretical study and numerical calculation of a quaternion matrix in applied sciences. This paper, by means of a complex representation of a quaternion matrix, studies the algorithm for the singular value decomposition of a quaternion matrix, and derives a complex structure-preserving algorithm for the singular value decomposition of a quaternion matrix. This paper also gives two examples to demonstrate the effectiveness of the algorithm.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Data availability
All data generated or analyzed during this study are included in this published article (and its supplementary information files).
References
Adler, S.: Quaternionic Quantum Mechanics and Quantum Fields. Oxford University Press, New York (1995)
Bihan, N.L., Sangwine, S.: Quaternion principal component analysis of color images. IEEE Int. Conf. Image Process 1, 809–812 (2003)
Davies, A.J., McKellar, B.H.: Non-relativistic quaternionic quantum mechanics. Phys. Rev. A 40, 4209–4214 (1989)
Cohn, P.M.: Skew Field Constructions. Cambridge University Press, Cambridge (1977)
Bihan, N.L., Mars, J.: Singular value decomposition ofquaternion matrices: a new tool for vector-sensor signal processing. Signal Process 84, 1177–1199 (2004)
Jiang, T., Jiang, Z., Ling, S.: An algebraic method for quaternion and complex Least squares coneigen-problem in quantum mechanics. Appl. Math. Comput 249, 222–228 (2014)
Jiang, T., Ling, S., Cheng, X.: An algebraic technique for total least squares problem in quaternionic quantum theory. Appl. Math. Lett 52, 58–63 (2016)
Ling, S., Jia, Z., Jiang, T.: LSQR algorithm with structured preconditioner for the least squares problem in quaternionic quantum theory. Comput. Math. Appl 73(10), 2208–2220 (2017)
Ling, S., Jia, Z., Lu, X., Yang, B.: Matrix LSQR algorithm for structured solutions to quaternionic least squares problem. Comput. Math. Appl 77(3), 830–845 (2019)
G. Wang, D. Zhang, V.I. Vasiliev, T. Jiang, A complex structure-preserving algorithm for the full rank decomposition of quaternion matrices and its applications, Numer. Algor, 2022,1-21
Zhang, F.: Quaternions and matrices of quaternions. Linear Algebra Appl 251, 21–57 (1997)
Sangwine, S.J., Bihan, N.L.: Quaternion singualar value decompsition based on bidiagonalization to a real or complex matrix using quaternion householder transformations. Appl. Math. Comput. 182, 727–738 (2006)
S.J. Sangwine, N.L. Bihan, Quaternion toolbox for matlab, http://qtfm.sourceforge.net/
Jia, Z., Wei, M., Ling, S.: A new structure-preserving method for quaternion Hermitian eigenvalue problems. Comput. Appl. Math. 239, 12–24 (2013)
Wang, M., Ma, W.: A structure-preserving algorithm for the quaternion Cholesky decomposition. Appl. Math. Comput. 223, 354–361 (2013)
Li, Y., Wei, M., Zhang, F., Zhao, J.: A fast structure-preserving method for computing the singular value decomposition of quaternion matrices. Appl. Math. Comput. 235, 157–167 (2014)
Li, Y., Wei, M., Zhang, F., Zhao, J.: A real structure-preserving method for the quaternion LU decomposition, revisited. Calcolo 54(4), 1553–1563 (2017)
Jia, Z., Ng, M.K., Song, G.: Lanczos method for large-scale quaternion singular value decomposition. Numer. Algor. 82, 699–717 (2019)
Li, Y., Wei, M., Zhang, F., Zhao, J.: Real structure-preserving algorithms of Householder based transformations for quaternion matrices. Comput. Appl. Math. 305, 82–91 (2016)
Ma, R., Jia, Z., Bai, Z.: A structure-preserving Jacobi algorithm for quaternion Hermitian eigenvalue problems. Comput. Math. Appl. 75(3), 809–820 (2018)
M. Wei, Y. Li, F. Zhang, J. Zhao, Quaternion Matrix Computations, Nova Science Publishers, 2018
Jia, Z., Wei, M., Zhao, M., Chen, Y.: A new real structure-preserving quaternion QR algorithm. J. Comput. Appl. Math. 343, 26–48 (2018)
Golub, G.H., Van Loan, C.F.: Matrix Computations, 3rd edn. The Johns Hopkins University Press, Baltimore, MD (1996)
Unversity of Granada. (Oct. 22, 2012). Computer Vision Group. CVG-UGR Image Data Base. [Online]. Available: http://decsai.ugr.es/cvg/dbimagenes/c512.php
Acknowledgements
We are grateful to the editor and the anonymous referees for providing many useful comments and suggestions, which greatly improve the performance of the original paper.
Funding
This paper is supported by the Shandong Natural Science Foundation(ZR201709250116), the Russian Science Foundation grant (23-71-30013), and the Chinese Government Scholarship (CSC NO. 202108370086, NO. 202008370340).
Author information
Authors and Affiliations
Contributions
Dong Zhang performed the data analyses and wrote the manuscript; Gang Wang and Tongsong Jiang contributed significantly to analysis and manuscript preparation; Chuan Jiang contributed to the conception of the study. All authors reviewed the manuscript.
Corresponding authors
Ethics declarations
Ethics approval
Not Applicable.
Human and animal ethics
Not Applicable.
Consent to participate
Not Applicable.
Consent for publication
Not Applicable.
Conflict of interest
The authors declare no competing interests.
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) 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
Zhang, D., Jiang, T., Jiang, C. et al. A complex structure-preserving algorithm for computing the singular value decomposition of a quaternion matrix and its applications. Numer Algor 95, 267–283 (2024). https://doi.org/10.1007/s11075-023-01571-4
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11075-023-01571-4