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A System of Coupled Quaternion Matrix Equations with Seven Unknowns and Its Applications

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Abstract

In this paper, a system of coupled quaternion matrix equations with seven unknowns

$$\begin{aligned} A_{i}X_{i}+Y_{i}B_{i}+C_{i}Z_{i}D_{i}+F_{i}WG_{i}=E_{i} \end{aligned}$$

is considered, where \(A_{i},B_{i},C_{i},D_{i},F_{i},G_{i}\) and \(E_{i}\) are given matrices, \(X_{i},Y_{i},Z_{i}\) and W are unknowns \((i=1,2)\). Some practical necessary and sufficient conditions for the existence of a solution to this system in terms of ranks and Moore–Penrose inverses are provided. The general solution to the system is given when the solvability conditions are satisfied. Applications that are discussed include the solvability conditions and general solutions to some quaternion matrix equations involving \(\phi \)-Hermicity. Some examples are given to illustrate the main results.

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References

  1. Aghamollaei, Gh, Rahjoo, M.: On quaternionic numerical ranges with respect to nonstandard involutions. Linear Algebra Appl. 540, 11–25 (2018)

    Article  MathSciNet  Google Scholar 

  2. Dmytryshyn, A., Futorny, V., Klymchuk, T., Sergeichuk, V.V.: Generalization of Roth’s solvability criteria to systems of matrix equations. Linear Algebra Appl. 527, 294–302 (2017)

    Article  MathSciNet  Google Scholar 

  3. Futorny, V., Klymchuk, T., Sergeichuk, V.V.: Roth’s solvability criteria for the matrix equations \(AX-\widehat{X}B=C\) and \(X-A\widehat{X}B=C\) over the skew field of quaternions with an involutive automorphism \(q\rightarrow \hat{q}\). Linear Algebra Appl. 510, 246–258 (2016)

    Article  MathSciNet  Google Scholar 

  4. He, Z.H.: Structure, properties and applications of some simultaneous decompositions for quaternion matrices involving \(\phi \)-skew-Hermicity. Adv. Appl. Clifford Algebras 29, 6 (2019)

    Article  MathSciNet  Google Scholar 

  5. He, Z.H.: The general solution to a system of coupled Sylvester-type quaternion tensor equations involving \(\eta \)-Hermicity. Bull. Iran. Math. Soc. (2019). https://doi.org/10.1007/s41980-019-00205-7

    Article  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  9. He, Z.H., Wang, Q.W.: A real quaternion matrix equation with with applications. Linear Multilinear Algebra 61, 725–740 (2013)

    Article  MathSciNet  Google Scholar 

  10. He, Z.H., Wang, Q.W.: The \(\eta \)-bihermitian solution to a system of real quaternion matrix equations. Linear Multilinear Algebra 62, 1509–1528 (2014)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  12. He, Z.H., Wang, Q.W., Zhang, Y.: Simultaneous decomposition of quaternion matrices involving \(\eta \)-Hermicity with applications. Appl. Math. Comput. 298, 13–35 (2017)

    MathSciNet  MATH  Google Scholar 

  13. He, Z.H., Wang, Q.W.: The general solutions to some systems of matrix equations. Linear Multilinear Algebra 63(10), 2017–2032 (2015)

    Article  MathSciNet  Google Scholar 

  14. Hungerford, T.W.: Algebra. Spring, New York (1980)

    Book  Google Scholar 

  15. Jiang, T.S., Wei, M.S.: On a solution of the quaternion matrix equation \(X-A\widetilde{X}B=C\) and its application. Acta Math. Sin. 21(3), 483–490 (2005)

    Article  MathSciNet  Google Scholar 

  16. Jia, Z., Wei, M., Zhao, M.X., Chen, Y.: A new real structure-preserving quaternion QR algorithm. J. Comput. Appl. Math. 343, 26–48 (2018)

    Article  MathSciNet  Google Scholar 

  17. Kyrchei, I.: Explicit determinantal representation formulas for the solution of the two-sided restricted quaternionic matrix equation. J. Appl. Math. Comput. 58(1–2), 335–365 (2018)

    Article  MathSciNet  Google Scholar 

  18. Kyrchei, I.: Determinantal representations of solutions and Hermitian solutions to some system of two-sided quaternion matrix equations. J. Math. 12 (2018) (Article ID 6294672)

  19. Kyrchei, I.: Determinantal representations of solutions to systems of quaternion matrix equations. Adv. Appl. Clifford Algebras 28, 23 (2018)

    Article  MathSciNet  Google Scholar 

  20. Kyrchei, I.: Cramer’s rules for Sylvester quaternion matrix equation and its special cases. Adv. Appl. Clifford Algebras 28, 90 (2018)

    Article  MathSciNet  Google Scholar 

  21. Kyrchei, I.: Determinantal representations of general and (skew-)Hermitian solutions to the generalized Sylvester-Type quaternion matrix equation. Abstr. Appl. Anal. 14 (Article ID 5926832) (2019)

  22. Liu, X.: The \(\eta \)-anti-Hermitian solution to some classic matrix equations. Appl. Math. Comput. 320, 264–270 (2018)

    MathSciNet  MATH  Google Scholar 

  23. Marsaglia, G., Styan, G.P.H.: Equalities and inequalities for ranks of matrices. Linear Multilinear Algebra 2, 269–292 (1974)

    Article  MathSciNet  Google Scholar 

  24. Rodman, L.: Topics in Quaternion Linear Algebra. Princeton University Press, Princeton (2014)

    Book  Google Scholar 

  25. Rehman, A., Wang, Q.W., He, Z.H.: Solution to a system of real quaternion matrix equations encompassing \(\eta \)-Hermicity. Appl. Math. Comput. 265, 945–957 (2015)

    MathSciNet  MATH  Google Scholar 

  26. Song, C., Chen, G., Zhang, X.: An iterative solution to coupled quaternion matrix equations. Filomat 26(4), 809–826 (2012)

    Article  MathSciNet  Google Scholar 

  27. Song, G.J., Wang, Q.W.: Condensed Cramer rule for some restricted quaternion linear equations. Appl. Math. Comput. 218(7), 3110–3121 (2011)

    MathSciNet  MATH  Google Scholar 

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

    Article  Google Scholar 

  29. Took, C.C., Mandic, D.P.: The quaternion LMS algorithm for adaptive filtering of hypercomplex real world processes. IEEE Trans. Signal Process. 57, 1316–1327 (2009)

    Article  MathSciNet  ADS  Google Scholar 

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

    Article  MathSciNet  ADS  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  32. Wang, Q.W., He, Z.H.: Some matrix equations with applications. Linear Multilinear Algebra 60, 1327–1353 (2012)

    Article  MathSciNet  Google Scholar 

  33. Wang, Q.W., Jiang, J.: Extreme ranks of (skew-)Hermitian solutions to a quaternion matrix equation. Elctron. J. Linear Algebra. 20, 552–573 (2010)

    MathSciNet  MATH  Google Scholar 

  34. Wang, Q.W., Chang, H.X., Lin, C.Y.: P-(skew)symmetric common solutions to a pair of quaternion matrix equations. Appl. Math. Comput. 195, 721–732 (2008)

    MathSciNet  MATH  Google Scholar 

  35. Wang, Q.W.: Bisymmetric and centrosymmetric solutions to system of real quaternion matrix equations. Comput. Math. Appl. 49, 641–650 (2005)

    Article  MathSciNet  Google Scholar 

  36. Wang, Q.W., van der Woude, J.W., Chang, H.X.: A system of real quaternion matrix equations with applications. Linear Algebra Appl. 431, 2291–2303 (2009)

    Article  MathSciNet  Google Scholar 

  37. Wei, M.S., Li, Y., Zhang, F., Zhao, J.: Quaternion Matrix Computations. Nova Science Publishers, New York (2018)

    Google Scholar 

  38. Yuan, S.F., Wang, Q.W., Yu, Y.B., Tian, Y.: On Hermitian solutions of the split quaternion matrix equation \(AXB+CXD=E\). Adv. Appl. Clifford Algebras 27(2), 1–18 (2017)

    MathSciNet  MATH  Google Scholar 

  39. Zhang, X.: A system of generalized Sylvester quaternion matrix equations and its applications. Appl. Math. Comput. 273, 74–81 (2016)

    MathSciNet  MATH  Google Scholar 

  40. Zhang, Y., Wang, R.H.: The exact solution of a system of quaternion matrix equations involving \(\eta \)-Hermicity. Appl. Math. Comput. 222, 201–209 (2013)

    MathSciNet  MATH  Google Scholar 

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Acknowledgements

The author would like to thank the anonymous referees for their valuable suggestions.

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  1. Department of Mathematics, Shanghai University, Shanghai, 200444, People’s Republic of China

    Zhuo-Heng He

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Correspondence to Zhuo-Heng He.

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Communicated by Pierre-Philippe Dechant

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This research was supported by the National Natural Science Foundation of China (Grant no. 11801354).

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He, ZH. A System of Coupled Quaternion Matrix Equations with Seven Unknowns and Its Applications. Adv. Appl. Clifford Algebras 29, 38 (2019). https://doi.org/10.1007/s00006-019-0955-2

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