Skip to main content
SpringerLink
Log in

A System of Sylvester-like Quaternion Tensor Equations with an Application

  • Research Article
  • Published:
Frontiers of Mathematics Aims and scope Submit manuscript

Abstract

This paper establishes the solvability conditions and an expression of the exact solution to a system of three Sylvester-like quaternion tensor equations in four variables. Based on a comprehensive analysis of the general solution and the solvability conditions associated with the system, necessary and sufficient conditions are deduced to a system of Sylvester-like tensor equations, including the unknowns as η-Hermitian quaternion tensors. Ultimately, we design an algorithm to compute the general solution, even a numerical example to illustrate the essential findings of this paper.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Che M.L., Wei Y.M., Theory and Computation of Complex Tensors and Its Applications. Singapore: Springer, 2020

    Book  Google Scholar 

  2. Chen Y., Wang Q.W., Xie L.M., Dual quaternion matrix equation AXB = C with applications. Symmetry, 2024, 16(3): 287

    Article  Google Scholar 

  3. De Lathauwer L., Castaing J., Tensor-based techniques for the blind separation of DS-CDMA signals. Signal Process, 2007, 87(2): 322–336

    Article  Google Scholar 

  4. De Leo S., Scolarici G., Right eigenvalue equation in quaternionic quantum mechanics. J. Phys. A, 2000, 33(15): 2971–2995

    Article  ADS  MathSciNet  Google Scholar 

  5. Dehghan M., Hajarian M., Analysis of an iterative algorithm to solve the generalized coupled Sylvester matrix equations. Appl. Math. Model., 2011, 35(7): 3285–3300

    Article  MathSciNet  Google Scholar 

  6. Ding F., Chen T.W., Gradient based iterative algorithms for solving a class of matrix equations. IEEE Trans. Automat. Control, 2005, 50(8): 1216–1221

    Article  MathSciNet  Google Scholar 

  7. Ding W.Y., Wei Y.M., Generalized tensor eigenvalue problems. SIAM J. Matrix Anal. Appl., 2015, 36(3): 1073–1099

    Article  MathSciNet  Google Scholar 

  8. Ding W.Y., Wei Y.M., Theory and Computation of Tensors. Multi-dimensional Arrays, London: Elsevier/Academic Press, 2016

    Google Scholar 

  9. Duan G.R., Zhou B., Solution to the second-order Sylvester matrix equation MVF2 + DVF + KV = BW. IEEE Trans. Automat. Control, 2006, 51(5): 805–809

    Article  MathSciNet  Google Scholar 

  10. Einstein A., The foundation of the general theory of relativity. Ann. Phys., 1916, 49(7): 769–822

    Article  CAS  Google Scholar 

  11. Fernandez J.M., Schneeberger W.A., Quaternionic computing. 2003, arXiv:quant-ph/0307017

  12. Guan Y., Chu D.L., Numerical computation for orthogonal low-rank approximation of tensors. SIAM J. Matrix Anal. Appl., 2019, 40(3): 1047–1065

    Article  MathSciNet  Google Scholar 

  13. Guan Y., Chu M.T., Chu D.L., Convergence analysis of an SVD-based algorithm for the best rank-1 tensor approximation. Linear Algebra Appl., 2018, 555: 53–69

    Article  MathSciNet  Google Scholar 

  14. Guan Y., Chu M.T., Chu D.L., SVD-based algorithms for the best rank-1 approximation of a symmetric tensor. SIAM J. Matrix Anal. Appl., 2018, 39(3): 1095–1115

    Article  MathSciNet  Google Scholar 

  15. Hajarian M., Computing symmetric solutions of general Sylvester matrix equations via Lanczos version of biconjugate residual algorithm. Comput. Math. Appl., 2018, 76(4): 686–700

    Article  MathSciNet  Google Scholar 

  16. Hamilton W.R., Elements of Quaternions, Second Edition. London: Longmans, Green & Co., 1899

    Google Scholar 

  17. He Z.H., Structure, properties and applications of some simultaneous decompositions for quaternion matrices involving ϕ-skew-Hermicity. Adv. Appl. Clifford Algebr., 2019, 29 (1): Paper No. 6, 31 pp.

  18. He Z.H., Agudelo O.M., Wang Q.W., Moor D.B., Two-sided coupled generalized Sylvester matrix equations solving using a simultaneous decomposition for fifteen matrices. Linear Algebra Appl., 2016, 496: 549–593

    Article  MathSciNet  Google Scholar 

  19. He Z.H., Liu J., Tam T.-Y., The general ϕ-Hermitian solution to mixed pairs of quaternion matrix Sylvester equations. Electron. J. Linear Algebra, 2017, 32: 475–499

    Article  MathSciNet  Google Scholar 

  20. He Z.H., Navasca C., Wang Q.-W., Tensor decompositions and tensor equations over quaternion algebra. 2017, arXiv:1710.07552

  21. He Z.H., Wang Q.W., A system of periodic discrete-time coupled Sylvester quaternion matrix equations. Algebra Colloq., 2017, 24(1): 169–180

    Article  MathSciNet  Google Scholar 

  22. He Z.H., Wang Q.W., Zhang Y., A system of quaternary coupled Sylvester-type real quaternion matrix equations. Automatica J. IFAC, 2018, 87: 25–31

    Article  MathSciNet  Google Scholar 

  23. He Z.H., Wang Q.W., Zhang Y., A simultaneous decomposition for seven matrices with applications. J. Comput. Appl. Math., 2019, 349: 93–113

    Article  MathSciNet  Google Scholar 

  24. Horn R.A., Zhang F.Z., A generalization of the complex Autonne–Takagi factorization to quaternion matrices. Linear Multilinear Algebra, 2012, 60(11–12): 1239–1244

    Article  MathSciNet  Google Scholar 

  25. Le Bihan N., Mars J., Singular value decomposition of quaternion matrices: a new tool for vector-sensor signal processing. Signal Process., 2004, 84(7): 1177–1199

    Article  Google Scholar 

  26. Li B.W., Tian S., Sun Y.S., Hu Z.M., Schur-decomposition for 3D matrix equations and its application in solving radiative discrete ordinates equations discretized by Chebyshev collocation spectral method. J. Comput. Phys., 2010, 229(4): 1198–1212

    Article  ADS  MathSciNet  CAS  Google Scholar 

  27. Mehany M.S., Wang Q.W., Three symmetrical systems of coupled Sylvester-like quaternion matrix equations. Symmetry, 2022, 14(3): 550

    Article  ADS  Google Scholar 

  28. Qi L.Q., Eigenvalues of a real supersymmetric tensor. J. Symbolic Comput., 2005, 40(6): 1302–1324

    Article  MathSciNet  Google Scholar 

  29. Qin J., Wang Q.W., Solving a system of two-sided Sylvester-like quaternion tensor equations. Comput. Appl. Math., 2023, 42 (5): Paper No. 232, 30 pp.

  30. Rabanser S., Shchur O., Günnemann S., Introduction to tensor decompositions and their applications in machine learning. 2017, arXiv:1711.10781

  31. Rodman L., Topics in Quaternion Linear Algebra. Princeton Series in Applied Mathematics, Princeton, NJ: Princeton University Press, 2014

    Book  Google Scholar 

  32. Shahzad A., Jones B.L., Kerrigan E.C., Constantinides G.A., An efficient algorithm for the solution of a coupled Sylvester equation appearing in descriptor systems. Automatica J. IFAC, 2011, 47(1): 244–248

    Article  MathSciNet  Google Scholar 

  33. Simoncini V., Computational methods for linear matrix equations. SIAM Rev., 2016, 58(3): 377–441

    Article  MathSciNet  Google Scholar 

  34. Sun L.Z., Zheng B.D., Bu C.J., Wei Y.M., Moore–Penrose inverse of tensors via Einstein product. Linear Multilinear Algebra, 2016, 64: 686–698

    Article  MathSciNet  Google Scholar 

  35. Sylvester J., Sur l’equation en matrices px = xq. C. R. Acad. Sci. Paris, 1884, 99: 67–71, 115–116

    Google Scholar 

  36. Took C.C., Mandic D.P., Quaternion-valued stochastic gradient-based adaptive IIR filtering. IEEE Trans. Signal Process., 2010, 58(7): 3895–3901

    Article  ADS  MathSciNet  Google Scholar 

  37. Took C.C., Mandic D.P., Augmented second-order statistics of quaternion random signals. Signal Process., 2011, 91(2): 214–224

    Article  Google Scholar 

  38. Took C.C., Mandic D.P., Zhang F.Z., On the unitary diagonalisation of a special class of quaternion matrices. Appl. Math. Lett., 2011, 24(11): 1806–1809

    Article  MathSciNet  Google Scholar 

  39. Wang Q.W., He Z.H., Solvability conditions and general solution for the mixed Sylvester equations. Automatica, 2013, 49(9): 2713–2719

    Article  MathSciNet  Google Scholar 

  40. Wang Q.W., He Z.H., Systems of coupled generalized Sylvester matrix equations. Automatica, 2014, 50(11): 2840–2844

    Article  MathSciNet  Google Scholar 

  41. Wang Q.W., He Z.H., Zhang Y., Constrained two-sided coupled Sylvester-type quaternion matrix equations. Automatica, 2019, 101: 207–213

    Article  MathSciNet  CAS  Google Scholar 

  42. Wang Q.W., Wang X., A system of coupled two-sided Sylvester-type tensor equations over the quaternion algebra. Taiwanese J. Math., 2020, 24(6): 1399–1416

    Article  MathSciNet  CAS  Google Scholar 

  43. Wang Q.W., Wang X., Zhang Y.S., A constraint system of coupled two-sided Sylvesterlike quaternion tensor equations. Comput. Appl. Math., 2020, 39 (4): Paper No. 317, 15 pp.

  44. Wimmer H.K., Consistency of a pair of generalized Sylvester equations. IEEE Trans. Automat. Control, 1994, 39(5): 1014–1016

    Article  MathSciNet  Google Scholar 

  45. Wu A.G., Duan G.R., Zhou B., Solution to generalized Sylvester matrix equations. IEEE Trans. Automat. Control, 2008, 53(3): 811–815

    Article  MathSciNet  Google Scholar 

  46. Xie M.Y., Wang Q.W., Reducible solution to a quaternion tensor equation. Front. Math. China, 2020, 15(5): 1047–1070

    Article  MathSciNet  Google Scholar 

  47. Xie M.Y., Wang Q.W., He Z.H., Mehany M.S., A system of Sylvester-type quaternion matrix equations with ten variables. Acta Math. Sin. (Engl. Ser.), 2022, 38(8): 1399C–1420

    Article  MathSciNet  Google Scholar 

  48. Xu Y.F., Wang Q.W., Liu L.S., Mehany M.S., A constrained system of matrix equations. Comput. Appl. Math., 2022, 41 (4): Paper No. 166, 24 pp.

  49. Zhou B., Li Z.Y., Duan G.R., Wang Y., Weighted least squares solutions to general coupled Sylvester matrix equations. J. Comput. Appl. Math., 2009, 224(2): 759–776

    Article  ADS  MathSciNet  Google Scholar 

Download references

Acknowledgements

This research was supported by the National Natural Science Foundation of China (Nos. 12371023, 11971294). The authors are highly grateful to the referees for their valuable comments and suggestions which led to improvements of this paper.

Author information

Authors and Affiliations

  1. Department of Mathematics and Newtouch Center for Mathematics, Shanghai University, Shanghai, 200444, China

    Mahmoud Saad Mehany, Qingwen Wang & Longsheng Liu

  2. Pure Mathematics and Computer Science Department, Ain Shams University, Cairo, 11566, Egypt

    Mahmoud Saad Mehany

Authors
  1. Mahmoud Saad Mehany
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Qingwen Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Longsheng Liu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Qingwen Wang.

Ethics declarations

Conflict of Interest The authors declare no conflict of interest.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Mehany, M.S., Wang, Q. & Liu, L. A System of Sylvester-like Quaternion Tensor Equations with an Application. Front. Math (2024). https://doi.org/10.1007/s11464-021-0389-8

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s11464-021-0389-8

Keywords

MSC2020

Search

Navigation