ON SOLUTIONS OF QUATERNION MATRIX EQUATIONS XF-AX = BY AND XF-AX = BY

doi:10.1016/S0252-9602(12)60153-2

Abstract

Abstract:

In this paper, the quaternion matrix equations XF − AX = BY and XF −AX= BY are investigated. For convenience, they were called generalized Sylvester-quaternion matrix equation and generalized Sylvester-j-conjugate quaternion matrix equa-tion, which include the Sylvester matrix equation and Lyapunov matrix equation as special cases. By applying of Kronecker map and complex representation of a quaternion matrix, the sufficient conditions to compute the solution can be given and the expressions of the explicit solutions to the above two quaternion matrix equations XF − AX = BY and XF − AX = BY are also obtained. By the established expressions, it is easy to compute
the solution of the quaternion matrix equation in the above two forms. In addition, two practical algorithms for these two quaternion matrix equations are give. One is complex representation matrix method and the other is a direct algorithm by the given expression. Furthermore, two illustrative examples are proposed to show the efficiency of the given method.

Key words: Kronecker map, explicit solution, generalized Sylvester-quaternion matrix equation, complex representation method

CLC Number:

15A06

SONG Cai-Qin, CHEN Guan-Liang, WANG Xiao-Dong. ON SOLUTIONS OF QUATERNION MATRIX EQUATIONS XF -AX = BY AND XF-AX = BY[J].Acta mathematica scientia,Series B, 2012, 32(5): 1967-1982.

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References

[1] Davies A J, Mckellar B H. Observability of quaternionic quantum mechanics. Physical Review A, 1992, 46: 3671–3675

[2] Sangwine S J, Ell T A, Moxey C E. Vetor phase correlation. Electron Letter, 2001, 37(25): 1513–1515

[3] Bihan N L, Sangwine S J. Color image decomposition using quaternion singular value decomposition. IEEE International Conference on Visual Information Engineering of Quaternion (VIE), UK: Guidford, 2003.

[4] Davies A J. Quaternionic Dirac equation. Physical Review D, 1990, 41: 2628–2630

[5] Adler S L. Quaternionic quantum field theory. Comm Math Phys, 1986, 104: 611–656

[6] Farenick D R, Pidkowich B A F. The spectral theorem in quaternions. Linear Algebra Appl, 2003, 371: 75–102

[7] Moxey C E, Sangwine S J, Ell T A. Hypercomplex correlation techniques for vector imagines. IEEE Trans Signal Process, 2003, 51(7): 1177–1199

[8] Adler S L. Scattering and decay theory for quaternionic quantum mechanics and stucture of induced T nonconservation. Physical Review D, 1988, 37: 3654–3662

[9] Clive C T, Danilo P M, Zhang F Z. On the unitary diagonalisation of a special class of quaternion matrices. Appl Math Letters, 2001, 24: 1806–1809

[10] Joe M, Chi S W. Charaterization of Wishart-Laplasce diatributions via Jordan algabra homomorphisms. Linear Algebra Appl, 2010, 432: 1578–1594

[11] Tu B X. Generalization of Schur theorem over quaternion field. Math Annal Analise 1998, 2: 131–138

[12] Chen L X. Generalization of Cayley-Hamilton theorem over quaternion field. Sci Bull, 1991, 17: 1291–1293

[13] Wu J L. The generalization of Wieland-Hoffman theorems on quaternion field. Math Res Exp, 1995, 15: 121–124

[14] Wu A G, Duan G R. On solution to the generalized Sylvester matrix equation AV + BW = EV F. IET Control Theory and Applications, 2007, 1(1): 402–408

[15] Kyrchiei I I. Cramer’s rule for quaternionic system of linear equations. J Math Sci, 2008, 155(6): 839–858

[16] Zhang F Z. Quaternions and matrices of quaternions. Linear Algebra Appl, 1997, 251: 21–57

[17] Wu J L. Distribution and estimation for eigenvalues of real quaternion matrices. Comput Math Appl, 2008, 55: 1998–2004

[18] Wang Q W, Yu S W, Xie W. Extreme ranks of real matrices in solution of the quaternion matrix equation AXB = C with applications. Algebra Colloquium, 2010, 17(2): 345–360

[19] Wang Q W, Zhang H S, Yu S W. On the real and pure imaginary solutions to the quaternion matrix equation AXB + CY D = E. Elect J Linear Algebra, 2008, 17: 343–358

[20] Wang Q W, van der Woude J W, Yu S W. An equivalence canonical form of a matrix triplet over an arbitrary division ring with applications. Science China Mathematics, 2011, 54(5): 907–924

[21] Wang Q W, Van der Woude J W, Chang H X. A system of real quaternion matrix equations with appli-cations. Linear Algebra Appl, 2009, 431: 2291–2303

[22] Wang Q W, Zhang H S, Song G J. A new solvable condition for a pair of generalized Sylvester equations. Elect J Linear Algebra, 2009, 18: 289–301

[23] Wang Q W, Yu S W, Zhang Q. The real solution to a system of quaternion matrix equations with applications. Comm Algebra, 2009, 37(6): 2060–2079

[24] Wang Q W, Li C K. Ranks and the least-norm of the general solution to a system of quaternion matrix equations. Linear Algebra Appl, 2009, 430: 1626–1640

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

[26] Wang Q W. Bisymmetric and centrosymmetric solutions to a systems of real quaternion matrix equations. Comput Math Appl, 2005, 49: 641–650

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

[28] Jiang T S, Wei M S. On solution of the quaternion matrix equation X − AeXB = C and its application. Acta Math Sin English Ser, 2005, 21: 483–490

[29] Huang L P. The matrix equation AXB −CXD = E over the quaternion field. Linear Algebra Appl, 1996, 234: 197–208

[30] Feng L G, Cheng W. Estimates of the double determinants of quaternion matrices. Acta Math Sci, 2010, 30B(4): 1189–1198

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

ON SOLUTIONS OF QUATERNION MATRIX EQUATIONS XF -AX = BY AND XF-AX = BY

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