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Explicit determinantal representation formulas for the solution of the two-sided restricted quaternionic matrix equation

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Abstract

Weighted singular value decomposition of a quaternion matrix and with its help determinantal representations of the quaternion weighted Moore–Penrose inverse have been derived recently by the author. In this paper, using these determinantal representations, explicit determinantal representation formulas for the solution of the restricted quaternionic matrix equations, \(\mathbf{A}{} \mathbf{X}{} \mathbf{B}=\mathbf{D}\), and consequently, \(\mathbf{A}{} \mathbf{X}=\mathbf{D}\) and \(\mathbf{X}{} \mathbf{B}=\mathbf{D}\) are obtained within the framework of the theory of column–row determinants. We consider all possible cases depending on weighted matrices.

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References

  1. Penrose, R.A.: Generalized inverse for matrices. Proc. Camb. Philos. Soc. 51, 406–413 (1955)

    Article  MATH  Google Scholar 

  2. Ben-Israel, A., Grenville, T.N.E.: Generalized Inverses: Theory and Applications. Springer, Berlin (2002)

    Google Scholar 

  3. Prasad, K.M., Bapat, R.B.: A note of the Khatri inverse, Sankhya: Indian. J. Stat. 54, 291–295 (1992)

    MATH  Google Scholar 

  4. Zhang, F.: Quaternions and matrices of quaternions. Linear Algebra Appl. 251, 21–57 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  5. Kyrchei, I.I.: Determinantal representation of the Moore–Penrose inverse matrix over the quaternion skew field. J. Math. Sci. 180(1), 23–33 (2012)

    Article  MathSciNet  Google Scholar 

  6. Van Loan, C.F.: Generalizing the singular value decomposition. SIAM J. Numer. Anal. 13, 76–83 (1976)

    Article  MathSciNet  MATH  Google Scholar 

  7. Galba, E.F.: Weighted singular decomposition and weighted pseudoinversion of matrices. Ukr. Math. J. 48(10), 1618–1622 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  8. Kyrchei, I.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 

  9. Stanimirovic’, P., Stankovic’, M.: Determinantal representation of weighted Moore–Penrose inverse. Mat. Vesnik 46, 41–50 (1994)

    MathSciNet  MATH  Google Scholar 

  10. Liu, X., Yu, Y., Wang, H.: Determinantal representation of weighted generalized inverses. Appl. Math. Comput. 218(7), 3110–3121 (2011)

    MathSciNet  Google Scholar 

  11. Liu, X., Zhu, G., Zhou, G., Yu, Y.: An analog of the adjugate matrix for the outer inverse \({{\bf A}}_{T,S}^{(2)}\). Math. Probl. Eng. 2012, 591256 (2012)

  12. Kyrchei, I.I.: Analogs of the adjoint matrix for generalized inverses and corresponding Cramer rules. Linear Multilinear Algebra 56(4), 453–469 (2008)

    Article  MathSciNet  MATH  Google Scholar 

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

    Google Scholar 

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

    MathSciNet  Google Scholar 

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

    Google Scholar 

  16. Song, G., Wang, Q., Chang, H.: 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 

  17. Song, G.J.: Determinantal representation of the generalized inverses over the quaternion skew field with applications. Appl. Math. Comput. 219, 656–667 (2012)

    MathSciNet  MATH  Google Scholar 

  18. Liao, A.P., Bai, Z.Z.: Least-squares solution of \(\text{ AXB } = \text{ D }\) over symmetric positive semidefinite matrices X. J. Comput. Math. 21(2), 175–182 (2003)

    MathSciNet  MATH  Google Scholar 

  19. Liu, Y.H.: Ranks of least squares solutions of the matrix equation \(\text{ AXB } = \text{ C }\). Comput. Math. Appl. 55, 1270–1278 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  20. Ling, S., Wei, M., Jia, Z.: Perturbation analysis for the matrix least squares problem \(\text{ AXB }=\text{ C }\). J. Comput. Appl. Math. 273, 150–159 (2015)

    Article  MathSciNet  Google Scholar 

  21. Kyrchei, I.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 

  22. Kyrchei, I.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 

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

    MathSciNet  MATH  Google Scholar 

  24. Kyrchei, I.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 (2016)

  25. Kyrchei, I.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 

  26. Kleyn, A., Kyrchei, I.: Relation of row–column determinants with quasideterminants of matrices over a quaternion algebra. In: Kyrchei, I.I. (ed.) Advanced Linear Algebra Research, pp. 299–324. Nova Science Publishers, New York (2015)

    Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  28. Song, G.J.: Bott-Duffin inverse over the quaternion skew field with applications. J. Appl. Math. Comput. 41, 377–392 (2013)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  30. Chen, L.: Inverse matrix and properties of double determinant over quaternion field. Sci. China Ser. A 34, 528–540 (1991)

    MathSciNet  MATH  Google Scholar 

  31. Jiang, T.: Cramer rule for quaternionic linear equations in quaternionic quantum theory. Rep. Math. Phys. 57, 463–468 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  32. Gelfand, I., Gelfand, S., Retakh, V., Wilson, R.L.: Quasideterminants. Adv. Math. 193, 56–141 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  33. Huang, L., So, W.: On left eigenvalues of a quaternionic matrix. Linear Algebra Appl. 323, 105–116 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  34. So, W.: Quaternionic left eigenvalue problem. Southeast Asian Bull. Math. 29, 555–565 (2005)

    MathSciNet  MATH  Google Scholar 

  35. Wood, R.M.W.: Quaternionic eigenvalues. Bull. Lond. Math. Soc. 17, 137–138 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  36. Brenner, J.L.: Matrices of quaternions. Pac. J. Math. 1, 329–335 (1951)

    Article  MathSciNet  MATH  Google Scholar 

  37. Macías-Virgós, E., Pereira-Sáez, M.J.: A topological approach to left eigenvalues of quaternionic matrices. Linear Multilinear Algebra 62(2), 139–158 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  38. Baker, A.: Right eigenvalues for quaternionic matrices: a topological approach. Linear Algebra Appl. 286, 303–309 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  39. Dray, T., Manogue, C.A.: The octonionic eigenvalue problem. Adv. Appl. Clifford Algebras 8(2), 341–364 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  40. Farenick, D.R., Pidkowich, B.A.F.: The spectral theorem in quaternions. Linear Algebra Appl. 371, 75–102 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  41. Farid, F.O., Wang, Q.W., Zhang, F.: On the eigenvalues of quaternion matrices. Linear Multilinear Algebra 59(4), 451–473 (2011)

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

The author would like to thank to the editor and two anonymous referees for providing many useful comments and suggestions, which greatly improved the original manuscript.

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  1. Pidstrygach Institute for Applied Problems of Mechanics and Mathematics of NAS of Ukraine, Lviv, Ukraine

    Ivan I. Kyrchei

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  1. Ivan I. Kyrchei
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Kyrchei, I.I. Explicit determinantal representation formulas for the solution of the two-sided restricted quaternionic matrix equation. J. Appl. Math. Comput. 58, 335–365 (2018). https://doi.org/10.1007/s12190-017-1148-6

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