Skip to main content
SpringerLink
Log in

Determinantal Representations of Solutions to Systems of Quaternion Matrix Equations

  • Published:
Advances in Applied Clifford Algebras Aims and scope Submit manuscript

Abstract

Within the framework of the theory of quaternion column–row determinants and using determinantal representations of the Moore–Penrose inverse previously obtained by the author, we get explicit determinantal representation formulas of solutions (analogs of Cramer’s rule) to the systems of quaternion matrix equations \( \mathbf{A}_{1}{} \mathbf{X}=\mathbf{C}_{1}\), \( \mathbf{X}\mathbf{B}_{2}=\mathbf{C}_{2} \), and \( \mathbf{A}_{1}{} \mathbf{X}=\mathbf{C}_{1}\), \(\mathbf{A}_{2} \mathbf{X}=\mathbf{C}_{2} \).

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. Bapat, R.B., Bhaskara, K.P.S., Prasad, K.M.: Generalized inverses over integral domains. Linear Algebra Appl. 140, 181–196 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  2. Buxton, J.N., Churchouse, R.F., Tayler, A.B.: Matrices Methods and Applications. Clarendon Press, Oxford (1990)

    Google Scholar 

  3. Farid, F.O., Nie, X.R., Wang, Q.W.: On the solutions of two systems of quaternion matrix equations. Linear Multilinear Algebra (2017). https://doi.org/10.1080/03081087.2017.1395388

    Google Scholar 

  4. Gabriel, R.: Das verallgemeinerte inverse eineer matrix, deren elemente einem beliebigen Körper angehören. J. Reine Angew Math. 234, 107–122 (1967)

    MATH  Google Scholar 

  5. Hajarian, M.: Recent developments in iterative algorithms for solving linear matrix equations. In: Kyrchei, I. (ed.) Advances in Linear Algebra Research, pp. 239–286. Nova Science Publishers, New York (2015)

    Google Scholar 

  6. He, Z.H., Wang, Q.W.: A real quaternion matrix equation with applications. Linear Multilinear Algebra 61(6), 725–740 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  7. 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  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  9. 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  Google Scholar 

  10. 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  MATH  Google Scholar 

  11. Jiang, T., Chen, L.: Algebraic algorithms for least squares problem in quaternionic quantum theory. Comput. Phys. Commun. 176, 481–485 (2007)

    Article  ADS  MATH  Google Scholar 

  12. Kyrchei, I.: Cramer’s rule for quaternion systems of linear equations. Fundam. Prikl. Mat. 13(4), 67–94 (2007)

    MathSciNet  Google Scholar 

  13. Kyrchei, I.: Analogs of the adjoint matrix for generalized inverses and corresponding Cramer rules. Linear Multilinear Algebra 56(9), 453–469 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  14. Kyrchei, I.: Determinantal representations of the Moore–Penrose inverse over the quaternion skew field and corresponding Cramer’s rules. Linear Multilinear Algebra 59(4), 413–431 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  15. Kyrchei, I.: The theory of the column and row determinants in a quaternion linear algebra. In: Baswell, Albert R. (ed.) Advances in Mathematics Research 15, pp. 301–359. Nova Science Publishers, New York (2012)

    Google Scholar 

  16. 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)

    Article  MathSciNet  MATH  Google Scholar 

  17. 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 

  18. Kyrchei, I.: Cramer’s rule for generalized inverse solutions. In: Kyrchei, I. (ed.) Advances in Linear Algebra Research, pp. 79–132. Nova Science Publishers, New York (2015)

    Google Scholar 

  19. Kyrchei, I.: The column and row immanants of matrices over a split quaternion algebra. Adv. Appl. Clifford Algebras 25, 611–619 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  20. Kyrchei, I.: Determinantal representations of the \(W\)-weighted Drazin inverse over the quaternion skew field. Appl. Math. Comput. 264, 453–465 (2015)

    MathSciNet  Google Scholar 

  21. Kyrchei, I.: Explicit determinantal representation formulas of \(W\)-weighted Drazin inverse solutions of some matrix equations over the quaternion skew field. Math. Probl. Eng. 2016(8673809), 1–13 (2016)

    Article  MathSciNet  Google Scholar 

  22. Kyrchei, I.: Cramer’s rules for some hermitian coquaternionic matrix equations. Adv. Appl. Clifford Algebras 27(3), 2509–2529 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  23. Kyrchei, I.: Determinantal representations of the Drazin and \(W\)-weighted Drazin inverses over the quaternion skew field with applications. In: Griffin, S. (ed.) Quaternions: Theory and Applications, pp. 201–275. Nova Science Publishers, New York (2017)

    Google Scholar 

  24. Kyrchei, I.: Weighted singular value decomposition and determinantal representations of the quaternion weighted Moore–Penrose inverse. Appl. Math. Comput. 309, 1–16 (2017)

    Article  MathSciNet  Google Scholar 

  25. Li, Y.T., Wu, W.J.: Symmetric and skew-antisymmetric solutions to systems of real quaternion matrix equations. Comput. Math. Appl. 55, 1142–1147 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  26. Maciejewski, A.A., Klein, C.A.: Obstacle avoidance for kinematically redundant manipulators in dynamically varying environments. Int. J. Robot. Res. 4(3), 109–117 (1985)

    Article  Google Scholar 

  27. Mitra, S.K.: The matrix equations \(AX=C, XB=D\). Linear Algebra Appl. 59, 171–181 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  28. 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)

    Article  MathSciNet  MATH  Google Scholar 

  29. 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  Google Scholar 

  30. Rehman, A., Wang, Q.W., Ali, I., Akram, M., Ahmad, M.O.: A constraint system of generalized Sylvester quaternion matrix equations. Adv. Appl. Clifford Algebras 27(4), 3183–3196 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  31. Song, G.: Characterization of the \(W\)-weighted Drazin inverse over the quaternion skew field with applications. Electron. J. Linear Algebra 26, 1–14 (2013)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  32. Song, G.J., Dong, C.Z.: New results on condensed Cramer’s rule for the general solution to some restricted quaternion matrix equations. J. Appl. Math. Comput. 53, 321–341 (2017)

    Article  MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  34. Song, G.J., Wang, Q.W., Chang, H.X.: Cramer rule for the unique solution of restricted matrix equations over the quaternion skew field. Comput. Math. Appl. 61, 1576–1589 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  35. Stanimirovic’, P.S.: General determinantal representation of pseudoinverses of matrices. Mat. Vesn. 48, 1–9 (1996)

    MathSciNet  MATH  Google Scholar 

  36. Wang, Q.W.: The general solution to a system of real quaternion matrix equations. Comput. Math. Appl. 49, 665–675 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  37. Wang, Q.W., Wu, Z.C., Lin, C.Y.: Extremal ranks of a quaternion matrix expression subject to consistent systems of quaternion matrix equations with applications. Appl. Math. Comput. 182(2), 1755–1764 (2006)

    MathSciNet  MATH  Google Scholar 

  38. Wang, Q.W., van der Woude, J.W., Chang, H.X.: A system of real quaternion matrix equations with applications. Linear Algebra Appl. 431(1), 2291–2303 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  39. Wang, Q.W., Rehman, A., He, Z.H., Zhang, Y.: Constraint generalized Sylvester matrix equations. Automatica 69, 60–64 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  40. Wang, L., Wang, Q.W., He, Z.H.: The common solution of some matrix equations. Algebra Colloq. 23(1), 71–81 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  41. Yuan, Y.: Least-squares solutions to the matrix equations \(AX=B\) and \(XC=D\). Appl. Math. Comput. 216, 3120–3125 (2010)

    MathSciNet  MATH  Google Scholar 

  42. Yuan, S.F., Wang, Q.W.: Two special kinds of least squares solutions for the quaternion matrix equation \(AXB+CXD=E\). Electron. J. Linear Algebra 23, 57–74 (2012)

    MathSciNet  MATH  Google Scholar 

  43. Yuan, S.F., Wang, Q.W., Duan, X.F.: On solutions of the quaternion matrix equation \(AX=B\) and their applications in color image restoration. Appl. Math. Comput. 221, 10–20 (2013)

    MathSciNet  MATH  Google Scholar 

  44. 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(4), 3235–3252 (2017)

    Article  MathSciNet  MATH  Google Scholar 

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

    MathSciNet  Google Scholar 

  46. Zhang, F., Wei, M., Lia, Y., Zhao, J.: Special least squares solutions of the quaternion matrix equation \(AX=B\) with applications. Appl. Math. Comput. 270, 425–433 (2015)

    MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Pidstrygach Institute for Applied Problems of Mechanics and Mathematics NAS of Ukraine, Str. Naukova 3b, Lviv, 79060, Ukraine

    Ivan Kyrchei

Authors
  1. Ivan Kyrchei
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Ivan Kyrchei.

Additional information

Communicated by Rafał Abłamowicz.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Kyrchei, I. Determinantal Representations of Solutions to Systems of Quaternion Matrix Equations. Adv. Appl. Clifford Algebras 28, 23 (2018). https://doi.org/10.1007/s00006-018-0843-1

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s00006-018-0843-1

Mathematics Subject Classification

Keywords

Search

Navigation