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The consistency and the general common solution to some quaternion matrix equations

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Abstract

In this paper, we establish some necessary and sufficient conditions for the solvability to a system of five quaternion matrix equations in terms of the Moore–Penrose inverse and the rank of a matrix, and give an expression of the general solution to the system when it is consistent. As an application, we investigate an \(\eta \)-Hermicity solution of a system. Moreover, we present a numerical example to illustrate the main results of this paper.

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References

  1. Assefa, D., Mansinha, L., Tiampo, K.F.: Local quaternion Fourier transform and color image texture analysis. Signal Process. 90, 1825–1835 (2010)

    MATH  Google Scholar 

  2. Ben-Israel, A., Greville, T.N.E.: Generalized Inverses: Theory and Application. Wiley, New York (1974)

    MATH  Google Scholar 

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

    MATH  Google Scholar 

  4. Castelan, E.B., da Silva, V.G.: On the solution of a Sylvester equation appearing in descriptor systems control theory. Syst. Control Lett. 54, 109–117 (2005)

    MathSciNet  MATH  Google Scholar 

  5. Chen, X.Y., Wang, Q.W.: The \(\eta \)-(anti-)Hermitian solution to a constrained Sylvester-type generalized commutative quaternion matrix equation. Banach J. Math. Anal. 17, No. 3, Paper No. 40 (2023). https://doi.org/10.1007/s43037-023-00262-5

  6. Dmytryshyn, A., Kagstrom, B.: Coupled Sylvester-type matrix equations and block diagonalization. SIAM J. Matrix Anal. Appl. 36, 580–593 (2015)

    MathSciNet  MATH  Google Scholar 

  7. Fan, X., Li, Y., Sun, J., Zhao, J.: Solving quaternion linear system \(AXB= E\) based on semi-tensor product of quaternion matrices. Banach J. Math. Anal. 17(2), 25 (2023)

    MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  9. 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)

    MathSciNet  MATH  Google Scholar 

  10. 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 

  11. He, Z.H., Chen, C., Wang, X.X.: A simultaneous decomposition for three quaternion tensors with applications in color video signal processing. Anal. Appl. Singap. 19(3), 529–549 (2021)

    MathSciNet  MATH  Google Scholar 

  12. Huang, G.X., Yin, F., Guo, K.: An iterative method for the skew-symmetric solution and the optimal approximate solution of the matrix equation \(AXB = C\). J. Comput. Appl. Math. 212, 231–244 (2008)

    MathSciNet  MATH  Google Scholar 

  13. Kyrchei, I.: Explicit representation formulas for the minimum norm least squares solutions of some quaternion matrix equations. Linear Algebra Appl. 438(1), 136–152 (2013)

    MathSciNet  MATH  Google Scholar 

  14. Kyrchei, I.: Determinantal representations of the Drazin inverse over the quaternion skew field with applications to some matrix equations. Appl. Math. Comput. 238, 193–207 (2014)

    MathSciNet  MATH  Google Scholar 

  15. Kyrchei, I.: Determinantal representations of solutions to systems of two-sided quaternion matrix equations. Linear Multilinear Algebra 69(4), 648–672 (2021)

    MathSciNet  MATH  Google Scholar 

  16. Li, F.L., Hu, X.Y., Zhang, L.: The generalized reflexive solution for a class of matrix equations \((AX = B, XC = D)\). Acta Math. Sci. 28(1), 185–193 (2008)

    MathSciNet  MATH  Google Scholar 

  17. Liu, L.S., Wang, Q.W.: The reducible solution to a system of matrix equations over the Hamilton quaternion algebra. Filomat 37(9), 2731–2742 (2023)

    MathSciNet  Google Scholar 

  18. Liu, L.S., Wang, Q.W., Chen, J.F., Xie, Y.Z.: An exact solution to a quaternion matrix equation with an application. Symmetry 14(2), 375 (2022)

    Google Scholar 

  19. Liu, L.S., Wang, Q.W., Mehany, M.S.: A Sylvester-type matrix equation over the Hamilton quaternions with an application. Mathematics 10(10), 1758 (2022)

    Google Scholar 

  20. Li, T., Wang, Q.W., Zhang, X.F.: A modified conjugate residual method and nearest Kronecker product preconditioner for the generalized coupled Sylvester tensor equations. Mathematics 10(10), 1730 (2022)

    Google Scholar 

  21. Li, T., Wang, Q.W., Zhang, X.F.: Gradient based iterative methods for solving symmetric tensor equations. Numer. Linear Algebra Appl. 29(2), e2414 (2022)

    MathSciNet  MATH  Google Scholar 

  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. Liu, X., He, Z.H.: On the split quaternion matrix equation AX= B. Banach J. Math. Anal. 14(1), 228–248 (2020)

    MathSciNet  MATH  Google Scholar 

  24. Liu, X., Zhang, Y.: Consistency of split quaternion matrix equations \(AX^{*}-XB=CY+D\) and \(X-AX^{*}B=CY+D\). Adv. Appl. Clifford Algebras 64, 1–20 (2019)

    MATH  Google Scholar 

  25. Marsaglia, G., Styan, G.P.H.: Equalities and inequalities for ranks of matrices. Linear Multilinear Algebra 2(3), 269–292 (1974)

    MathSciNet  MATH  Google Scholar 

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

    Google Scholar 

  27. Nie, X.R., Wang, Q.W., Zhang, Y.: A system of matrix equations over the quaternion algebra with applications. Algebra Colloq. 24(2), 233–253 (2017)

    MathSciNet  MATH  Google Scholar 

  28. Peng, Z.Y.: The centro-symmetric solutions of linear matrix equation \(AXB = C\) and its optimal approximation. Chin. J. Eng. Math. 6, 60–64 (2003)

    Google Scholar 

  29. Qi, L.Q., Luo, Z.Y., Wang, Q.W., Zhang, X.Z.: Quaternion matrix optimization: motivation and analysis. J. Optim. Theory Appl. 193(1–3), 621–648 (2022)

    MathSciNet  MATH  Google Scholar 

  30. Ren, B.Y., Wang, Q.W., Chen, X.Y.: The \(\eta \)-anti-Hermitian solution to a system of constrained matrix equations over the generalized segre quaternion algebra. Symmetry 15(3), 592 (2023)

    Google Scholar 

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

    MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  33. Villareal, E.R.L., Vargas, J.A.R., Hemerly, E.M.: Static output feedback stabilization using invariant subspaces and Sylvester equations. TEMA Tend. Mat. Apl. Comput. 10(1), 99–110 (2009)

    MathSciNet  MATH  Google Scholar 

  34. Wang, Q.W.: A system of matrix equations and a linear matrix equation over arbitrary regular rings with identity. Linear Algebra Appl. 384, 43–54 (2004)

    MathSciNet  MATH  Google Scholar 

  35. Wang, Q.W.: A system of four matrix equations over von Neumann regular rings and its applications. Acta Math. Sin. 21(2), 323–334 (2005)

    MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  37. Wang, Q.W., He, Z.H.: Some matrix equations with applications. Linear Multilinear Algebra 60(11–12), 1327–1353 (2012)

    MathSciNet  MATH  Google Scholar 

  38. Wang, Q.W., Lv, R.Y., Zhang, Y.: The least-squares solution with the least norm to a system of tensor equations over the quaternion algebra. Linear Multilinear Algebra 70(10), 1942–1962 (2022)

    MathSciNet  MATH  Google Scholar 

  39. Xie, L.M., Wang, Q.W.: A system of matrix equations over the commutative quaternion ring. Filomat 37(1), 97–106 (2023)

    MathSciNet  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  42. Xu, Y.F., Wang, Q.W., Liu, L.S., Mehany, M.S.: A constrained system of matrix equations. Comput. Appl. Math. 41(4), 166 (2022)

    MathSciNet  MATH  Google Scholar 

  43. Zhang, Y.N., Jiang, D.C., Wang, J.: A recurrent neural network for solving Sylvester equation with time-varying coefficients. IEEE Trans. Neural Netw. 13(5), 1053–1063 (2002)

    Google Scholar 

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Acknowledgements

This research was supported by the grant from the National Natural Science Foundation of China (11971294).

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Authors and Affiliations

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

    Xi-Le Xu & Qing-Wen Wang

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  1. Xi-Le Xu
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  2. Qing-Wen Wang
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Correspondence to Qing-Wen Wang.

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Communicated by Dragana Cvetkovic Ilic.

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Xu, XL., Wang, QW. The consistency and the general common solution to some quaternion matrix equations. Ann. Funct. Anal. 14, 53 (2023). https://doi.org/10.1007/s43034-023-00276-y

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