Taiwanese Journal of Mathematics

A System of Coupled Two-sided Sylvester-type Tensor Equations over the Quaternion Algebra

Qing-Wen Wang and Xiao Wang

Advance publication

This article is in its final form and can be cited using the date of online publication and the DOI.

Full-text: Open access

Abstract

We establish some necessary and sufficient conditions for the solvability to a system of a pair of coupled two-sided Sylvester-type tensor equations over the quaternion algebra. We also give an expression of the general solution to the system when it is solvable. As applications, we derive some solvability conditions and expressions of the $\eta$-Hermitian solutions to some systems of coupled two-sided Sylvester-type quaternion tensor equations. Moreover, we provide an example to illustrate the main results of this paper.

Article information

Source
Taiwanese J. Math., Advance publication (2020), 18 pages.

Dates
First available in Project Euclid: 29 May 2020

Permanent link to this document
https://projecteuclid.org/euclid.twjm/1590717619

Digital Object Identifier
doi:10.11650/tjm/200504

Subjects
Primary: 11R52: Quaternion and other division algebras: arithmetic, zeta functions 15A09: Matrix inversion, generalized inverses 15A24: Matrix equations and identities 15A69: Multilinear algebra, tensor products 15B33: Matrices over special rings (quaternions, finite fields, etc.)

Keywords
tensor Einstein product tensor equation quaternion algebra Moore-Penrose inverse $\eta$-Hermitian tensor

Citation

Wang, Qing-Wen; Wang, Xiao. A System of Coupled Two-sided Sylvester-type Tensor Equations over the Quaternion Algebra. Taiwanese J. Math., advance publication, 29 May 2020. doi:10.11650/tjm/200504. https://projecteuclid.org/euclid.twjm/1590717619


Export citation

References

  • A. D. Barbour and S. Utev, Solving the Stein equation in compound Poisson approximation, Adv. in Appl. Probab. 30 (1998), no. 2, 449–475.
  • C. Bauckhage, Robust tensor classifiers for color object recognition, in: Image Analysis and Recognition, 352–363, Lecture Notes in Computer Science 4633, 2007.
  • S. Brahma and B. Datta, An optimization approach for minimum norm and robust partial quadratic eigenvalue assignment problems for vibrating structures, J. Sound Vibration 324 (2009), no. 3-5, 471–489.
  • Z. Chen and L. Lu, A projection method and Kronecker product preconditioner for solving Sylvester tensor equations, Sci. China Math. 55 (2012), no. 6, 1281–1292.
  • M. Dehghan and M. Hajarian, Analysis of an iterative algorithm to solve the generalized coupled Sylvester matrix equations, Appl. Math. Model. 35 (2011), no. 7, 3285–3300.
  • L. De Lathlauwer, A survey of tensor methods, 2009 IEEE International Symposium on Circuits and Systems, (2009), 2773–2776.
  • S. De Leo and G. Scolarici, Right eigenvalue equation in quaternionic quantum mechanics, J. Phys. A 33 (2000), no. 15, 2971–2995.
  • A. Einstein, The foundation of the general theory of relativity, in: The Collected Papers of Albert Einstein, Vol. 6: The Berlin years: writings, 1914–1917, 146–200, Princeton University Press, 1997.
  • J. M. Fernandez and W. A. Schneeberger, Quaternionic computing.v2, (2004).
  • L. Grasedyck, Existence and computation of low Kronecker-rank approximations for large linear systems of tensor product structure, Computing 72 (2004), no. 3-4, 247–265.
  • Y. Guan and D. Chu, Numerical computation for orthogonal low-rank approximation of tensors, SIAM J. Matrix Anal. Appl. 40 (2019), no. 3, 1047–1065.
  • Y. Guan, M. T. Chu and D. Chu, Convergence analysis of an SVD-based algorithm for the best rank-1 tensor approximation, Linear Algebra Appl. 555 (2018), 53–69.
  • ––––, SVD-based algorithms for the best rank-1 approximation of a symmetric tensor, SIAM J. Matrix Anal. Appl. 39 (2018), no. 3, 1095–1115.
  • W. R. Hamilton, Elements of Quaternions, Longmans Green and Co London, 1866.
  • Z.-H. He, The general solution to a system of coupled Sylvester-type quaternion tensor equations involving $\eta$-Hermicity, Bull. Iranian Math. Soc. 45 (2019), no. 5, 1407–1430.
  • Z.-H. He, O. M. Agudelo, Q.-W. Wang and B. De Moor, Two-sided coupled generalized Sylvester matrix equations solving using a simultaneous decomposition for fifteen matrices, Linear Algebra Appl. 496 (2016), 549–593.
  • Z.-H. He, C. Navasca and Q.-W. Wang, Tensor decompositions and tensor equations over quaternion algebra, arXiv:1710.07552.
  • Z.-H. He and Q.-W. Wang, A real quaternion matrix equation with applications, Linear Multilinear Algebra 61 (2013), no. 6, 725–740.
  • Z.-H. He, Q.-W. Wang and Y. Zhang, Simultaneous decomposition of quaternion matrices involving $\eta$-Hermicity with applications, Appl. Math. Comput. 298 (2017), 13–35.
  • A. Klein and P. Spreij, On Fisher's information matrix of an ARMAX process and Sylvester's resultant matrices, Linear Algebra Appl. 237/238 (1996), 579–590.
  • T. Levi-Civita, The Absolute Differential Calculus: Calculus of tensors, Dover Phoenix Editions, Dover Publications, Mineola, NY, 2005.
  • B.-W. Li, S. Tian, Y.-S. Sun and Z.-M. Hu, Schur-decomposition for 3D matrix equations and its application in solving radiative discrete ordinates equations discretized by Chebyshev collocation spectral method, J. Comput. Phys. 229 (2010), no. 4, 1198–1212.
  • T. Li, Q.-W. Wang and X.-F. Duan, Numerical algorithms for solving discrete Lyapunov tensor equation, J. Comput. Appl. Math. 370 (2020), 112676, 11 pp.
  • N. Liu, B. Zhang, J. Yan, Z. Chen, W. Liu, F. Bai and L. Chien, Text representation: From vector to tensor, Fifth IEEE International Conference on Data Mining, (2005), 725–728.
  • C. Navasca, L. De Lathauwer and S. Kindermann, Swamp reducing technique for tensor decomposition, 16th European Signal Processing Conference, (2008).
  • L. Qi, H. Chen and Y. Chen, Tensor Eigenvalues and Their Applications, Advances in Mechanics and Mathematics 39, Springer, Singapore, 2018.
  • L. Qi and Z. Luo, Tensor Analysis: Spectral theory and special tensors, Other Titles in Applied Mathematics, SIAM, Philadelphia, 2017.
  • L. Rodman, Topics in Quaternion Linear Algebra, Princeton Series in Applied Mathematics, Princeton University Press, Princeton, NJ, 2014.
  • A. Shashua and T. Hazan, Non-negative tensor factorization with applications to statistics and computer vision, Proceeding of the 22nd international conference on Machine learning, (2005), 792–799.
  • L. Sun, B. Zheng, C. Bu and Y. Wei, Moore-Penrose inverse of tensors via Einstein product, Linear Multilinear Algebra 64 (2016), no. 4, 686–698.
  • C. C. Took and D. P. Mandic, Quaternion-valued stochastic gradient-based adaptive IIR filtering, IEEE Trans. Signal Process. 58 (2010), no. 7, 3895–3901.
  • ––––, Augmented second-order statistics of quaternion random signals, Signal Process. 91 (2011), no. 2, 214–224.
  • A. Varga, Robust pole assignment via Sylvester equation based state feedback parametrization, Proceedings of the 2000 IEEE International Symposium on Computer-Aided Control System Design, Anchorage, Alaska, (2000), 13–18.
  • Q.-W. Wang and Z.-H. He, Solvability conditions and general solution for mixed Sylvester equations, Automatica J. IFAC 49 (2013), no. 9, 2713–2719.
  • Q.-W. Wang, Z.-H. He and Y. Zhang, Constrained two-sided coupled Sylvester-type quaternion matrix equations, Automatica J. IFAC 101 (2019), 207–213.
  • Q.-W. Wang and X. Xu, Iterative algorithms for solving some tensor equations, Linear Multilinear Algebra 67 (2019), no. 7, 1325–1349.
  • Q.-W. Wang, X. Xu and X. Duan, Least squares solution of the quaternion Sylvester tensor equation, Accepted in Linear Multilinear Algebra, 2019.
  • X. Xu and Q.-W. Wang, Extending BiCG and BiCR methods to solve the Stein tensor equation, Comput. Math. Appl. 77 (2019), no. 12, 3117–3127.
  • F. Zhang, Quaternions and matrices of quaternions, Linear Algebra Appl. 251 (1997), 21–57.
  • Y. Zhang, D. Jiang and J. Wang, A recurrent neural network for solving Sylvester equation with time-varying coefficients, IEEE Trans. Neural Networks 13 (2002), no. 5, 1053–1063.