# Consimilarity and quaternion matrix equations AX −^X B = C, X − A^X B = C

Special Matrices (2014)

• Volume: 2, Issue: 1, page 180-186, electronic only
• ISSN: 2300-7451

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## Abstract

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L. Huang [Consimilarity of quaternion matrices and complex matrices, Linear Algebra Appl. 331 (2001) 21–30] gave a canonical form of a quaternion matrix with respect to consimilarity transformations A ↦ ˜S−1AS in which S is a nonsingular quaternion matrix and h = a + bi + cj + dk ↦ ˜h := a − bi + cj − dk (a, b, c, d ∈ ℝ). We give an analogous canonical form of a quaternion matrix with respect to consimilarity transformations A ↦^S−1AS in which h ↦ ^h is an arbitrary involutive automorphism of the skew field of quaternions. We apply the obtained canonical form to the quaternion matrix equations AX −^X B = C and X − A^X B = C.

## How to cite

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Tatiana Klimchuk, and Vladimir V. Sergeichuk. "Consimilarity and quaternion matrix equations AX −^X B = C, X − A^X B = C." Special Matrices 2.1 (2014): 180-186, electronic only. <http://eudml.org/doc/269173>.

@article{TatianaKlimchuk2014,
abstract = {L. Huang [Consimilarity of quaternion matrices and complex matrices, Linear Algebra Appl. 331 (2001) 21–30] gave a canonical form of a quaternion matrix with respect to consimilarity transformations A ↦ ˜S−1AS in which S is a nonsingular quaternion matrix and h = a + bi + cj + dk ↦ ˜h := a − bi + cj − dk (a, b, c, d ∈ ℝ). We give an analogous canonical form of a quaternion matrix with respect to consimilarity transformations A ↦^S−1AS in which h ↦ ^h is an arbitrary involutive automorphism of the skew field of quaternions. We apply the obtained canonical form to the quaternion matrix equations AX −^X B = C and X − A^X B = C.},
author = {Tatiana Klimchuk, Vladimir V. Sergeichuk},
journal = {Special Matrices},
keywords = {Quaternion matrices; Consimilarity; Matrix equations; quaternion matrices; consimilarity; matrix equations; canonical form},
language = {eng},
number = {1},
pages = {180-186, electronic only},
title = {Consimilarity and quaternion matrix equations AX −^X B = C, X − A^X B = C},
url = {http://eudml.org/doc/269173},
volume = {2},
year = {2014},
}

TY - JOUR
AU - Tatiana Klimchuk
TI - Consimilarity and quaternion matrix equations AX −^X B = C, X − A^X B = C
JO - Special Matrices
PY - 2014
VL - 2
IS - 1
SP - 180
EP - 186, electronic only
AB - L. Huang [Consimilarity of quaternion matrices and complex matrices, Linear Algebra Appl. 331 (2001) 21–30] gave a canonical form of a quaternion matrix with respect to consimilarity transformations A ↦ ˜S−1AS in which S is a nonsingular quaternion matrix and h = a + bi + cj + dk ↦ ˜h := a − bi + cj − dk (a, b, c, d ∈ ℝ). We give an analogous canonical form of a quaternion matrix with respect to consimilarity transformations A ↦^S−1AS in which h ↦ ^h is an arbitrary involutive automorphism of the skew field of quaternions. We apply the obtained canonical form to the quaternion matrix equations AX −^X B = C and X − A^X B = C.
LA - eng
KW - Quaternion matrices; Consimilarity; Matrix equations; quaternion matrices; consimilarity; matrix equations; canonical form
UR - http://eudml.org/doc/269173
ER -

## References

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1. [1] J.H. Bevis, F.J. Hall, R.E. Hartwig, Consimilarity and thematrix equation A¯X −XB = C, Current Trends inMatrix Theory (Auburn, Ala., 1986), North-Holland, New York, 1987, pp. 51–64.
2. [2] H. Dym, Linear Algebra in Action, American Mathematical Society, 2007. Zbl1113.15001
3. [3] F.R. Gantmacher, The Theory of Matrices, Vol. 1, Chelsea, New York, 1959. Zbl0085.01001
4. [4] Y.P. Hong, R.A. Horn, A canonical form for matrices under consimilarity, Linear Algebra Appl. 102 (1988) 143–168. Zbl0657.15008
5. [5] R.A. Horn, C.R. Johnson, Matrix Analysis, 2nd ed., Cambridge University Press, Cambridge, 2013. Zbl1267.15001
6. [6] R.A. Horn, C.R. Johnson, Topics in Matrix Analysis, Cambridge University Press, Cambridge, 1991. Zbl0729.15001
7. [7] L. Huang, Consimilarity of quaternion matrices and complex matrices, Linear Algebra Appl. 331 (2001) 21–30. Zbl0982.15019
8. [8] N. Jacobson, The Theory of Rings, American Mathematical Society, New York, 1943. Zbl0060.07302
9. [9] T. Jiang, M. Wei, On solutions of the matrix equations X − AXB = C and X − A¯X B = C, Linear Algebra Appl. 367 (2003) 225–233.
10. [10] T.S. Jiang, M.S. Wei, On a solution of the quaternion matrix equation X − A˜X B = C and its application, Acta Math. Sin. 21 (2005) 483–490. Zbl1083.15019
11. [11] A.W. Knapp, Advanced Algebra, Birkhaüser, 2007.
12. [12] P. Lancaster, Explicit solutions of linear matrix equations, SIAM Review 12 (1970) 544–566. [WoS][Crossref] Zbl0209.06502
13. [13] P. Lancaster, L. Lerer, Factored forms for solutions of AX − XB = C and X − AXB = C in companion matrices, Linear Algebra Appl. 62 (1984) 19–49.
14. [14] P. Lancaster, M. Tismenetsky, The Theory of Matrices with Applications, 2nd ed., Academic Press, 1985. Zbl0558.15001
15. [15] C. Song, G. Chen, On solutions of matrix equation XF − AX = C and XF − A˜X = C over quaternion field, J. Appl. Math. Comput. 37 (2011) 57–68.
16. [16] C.Q. Song, G.L. Chen, Q.B. Liu, Explicit solutions to the quaternion matrix equations X − AXF = C and X − A˜XF = C, Int. J. Comput. Math. 89 (2012) 890–900.
17. [17] C. Song, J. Feng, X.Wang, J. Zhao, A real representation method for solving Yakubovich-j-conjugate quaternionmatrix equation, Abstr. Appl. Anal. 2014, Art. ID 285086, 9 pp.
18. [18] M.-F. Vignéras, Arithmétique des Algèbres de Quaternions, Springer, Berlin, 1980. Zbl0422.12008
19. [19] N.A. Wiegmann, Some theorems on matrices with real quaternion elements, Canad. J. Math. 7 (1955) 191–201. Zbl0064.01604
20. [20] S.F. Yuan, A.P. Liao, Least squares solution of the quaternion matrix equation X − A^XB = C with the least norm, Linear Multilinear Algebra 59 (2011) 985–998. [WoS] Zbl1228.65072
21. [21] F. Zhang, Quaternions and matrices of quaternions, Linear Algebra Appl. 251 (1997) 21–57. Zbl0873.15008

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