Journal of Applied Mathematics

Least Squares Pure Imaginary Solution and Real Solution of the Quaternion Matrix Equation $AXB+CXD=E$ with the Least Norm

Shi-Fang Yuan

Abstract

Using the Kronecker product of matrices, the Moore-Penrose generalized inverse, and the complex representation of quaternion matrices, we derive the expressions of least squares solution with the least norm, least squares pure imaginary solution with the least norm, and least squares real solution with the least norm of the quaternion matrix equation $AXB+CXD=E$, respectively.

Article information

Source
J. Appl. Math., Volume 2014 (2014), Article ID 857081, 9 pages.

Dates
First available in Project Euclid: 2 March 2015

https://projecteuclid.org/euclid.jam/1425305687

Digital Object Identifier
doi:10.1155/2014/857081

Mathematical Reviews number (MathSciNet)
MR3198408

Citation

Yuan, Shi-Fang. Least Squares Pure Imaginary Solution and Real Solution of the Quaternion Matrix Equation $AXB+CXD=E$ with the Least Norm. J. Appl. Math. 2014 (2014), Article ID 857081, 9 pages. doi:10.1155/2014/857081. https://projecteuclid.org/euclid.jam/1425305687

References

• F. Z. Zhang, “Quaternions and matrices of quaternions,” Linear Algebra and Its Applications, vol. 251, pp. 21–57, 1997.
• Y.-H. Au-Yeung and C.-M. Cheng, “On the pure imaginary quaternionic solutions of the Hurwitz matrix equations,” Linear Algebra and Its Applications, vol. 419, no. 2-3, pp. 630–642, 2006.
• K.-W. E. Chu, “Singular value and generalized singular value decompositions and the solution of linear matrix equations,” Linear Algebra and Its Applications, vol. 88-89, pp. 83–98, 1987.
• M. Dehghan and M. Hajarian, “Finite iterative algorithms for the reflexive and anti-reflexive solutions of the matrix equation ${A}_{1}{X}_{1}{B}_{1}+{A}_{2}{X}_{2}{B}_{2}=C$,” Mathematical and Computer Modelling, vol. 49, no. 9-10, pp. 1937–1959, 2009.
• J. D. Gardiner, A. J. Laub, J. J. Amato, and C. B. Moler, “Solution of the Sylvester matrix equation $AX{B}^{T}+CX{D}^{T}=E$,” ACM Transactions on Mathematical Software, vol. 18, no. 2, pp. 223–231, 1992.
• Z.-H. He and Q.-W. Wang, “A real quaternion matrix equation with applications,” Linear and Multilinear Algebra, vol. 61, no. 6, pp. 725–740, 2013.
• N. Li and Q.-W. Wang, “Iterative algorithm for solving a class of quaternion matrix equation over the generalized $(P,Q)$-reflexive matrices,” Abstract and Applied Analysis, vol. 2013, Article ID 831656, 15 pages, 2013.
• N. Li, Q.-W. Wang, and J. Jiang, “An efficient algorithm for the reflexive solution of the quaternion matrix equation $AXB+C{X}^{H}D=F$,” Journal of Applied Mathematics, vol. 2013, Article ID 217540, 14 pages, 2013.
• Y.-T. Li and W.-J. Wu, “Symmetric and skew-antisymmetric solutions to systems of real quaternion matrix equations,” Computers & Mathematics with Applications, vol. 55, no. 6, pp. 1142–1147, 2008.
• A.-P. Liao and Y. Lei, “Least-squares solution with the minimum-norm for the matrix equation $(AXB,GXH)=(C,D)$,” Computers & Mathematics with Applications, vol. 50, no. 3-4, pp. 539–549, 2005.
• A. Mansour, “Solvability of $AXB-CXD=E$ in the operators algebra $B(H)$,” Lobachevskii Journal of Mathematics, vol. 31, no. 3, pp. 257–261, 2010.
• S. K. Mitra, “Common solutions to a pair of linear matrix equations ${A}_{1}X{B}_{1}={C}_{1}$ and ${A}_{2}X{B}_{2}={C}_{2}$,” Mathematical Proceedings of the Cambridge Philosophical Society, vol. 74, pp. 213–216, 1973.
• A. Navarra, P. L. Odell, and D. M. Young, “A representation of the general common solution to the matrix equations ${A}_{1}X{B}_{1}={C}_{1}$ and ${A}_{2}X{B}_{2}={C}_{2}$ with applications,” Computers & Mathematics with Applications, vol. 41, no. 7-8, pp. 929–935, 2001.
• Y.-X. Peng, X.-Y. Hu, and L. Zhang, “An iterative method for symmetric solutions and optimal approximation solution of the system of matrix equations ${A}_{1}X{B}_{1}={C}_{1}$, ${A}_{2}X{B}_{2}={C}_{2}$,” Applied Mathematics and Computation, vol. 183, no. 2, pp. 1127–1137, 2006.
• Z.-H. Peng, X.-Y. Hu, and L. Zhang, “An efficient algorithm for the least-squares reflexive solution of the matrix equation ${A}_{1}X{B}_{1}={C}_{1}$, ${A}_{2}X{B}_{2}={C}_{2}$,” Applied Mathematics and Computation, vol. 181, no. 2, pp. 988–999, 2006.
• Z.-Y. Peng, “Solutions of symmetry-constrained least-squares problems,” Numerical Linear Algebra with Applications, vol. 15, no. 4, pp. 373–389, 2008.
• J. W. van der Woude, “On the existence of a common solution $X$ to the matrix equations ${A}_{i}X{B}_{j}={C}_{ij},(i,j)\in \Gamma$,” Linear Algebra and Its Applications, vol. 375, pp. 135–145, 2003.
• Y. X. Yuan, “Two classes of best approximation problems of matrices,” Mathematica Numerica Sinica, vol. 23, no. 4, pp. 429–436, 2001.
• Y. X. Yuan, “The minimum norm solutions of two classes of matrix equations,” Numerical Mathematics, vol. 24, no. 2, pp. 127–134, 2002.
• Y. X. Yuan, “The optimal solution of linear matrix equations by using matrix decompositions,” Mathematica Numerica Sinica, vol. 24, no. 2, pp. 165–176, 2002 (Chinese).
• T. S. Jiang, Y. H. Liu, and M. S. Wei, “Quaternion generalized singular value decomposition and its applications,” Applied Mathematics, vol. 21, no. 1, pp. 113–118, 2006.
• T. S. Jiang and M. S. Wei, “Real representations of quaternion matrices and quaternion matrix equations,” Acta Mathematica Scientia A, vol. 26, no. 4, pp. 578–584, 2006 (Chinese).
• Q.-W. Wang, “The general solution to a system of real quaternion matrix equations,” Computers & Mathematics with Applications, vol. 49, no. 5-6, pp. 665–675, 2005.
• Q.-W. Wang, H.-X. Chang, and C.-Y. Lin, “$P$-(skew)symmetric common solutions to a pair of quaternion matrix equations,” Applied Mathematics and Computation, vol. 195, no. 2, pp. 721–732, 2008.
• Q.-W. Wang and Z.-H. He, “Solvability conditions and general solution for mixed Sylvester equations,” Automatica, vol. 49, no. 9, pp. 2713–2719, 2013.
• Q. Wang and Z. He, “A system of matrix equations and its applications,” Science China Mathematics, vol. 56, no. 9, pp. 1795–1820, 2013.
• Q.-W. Wang, J. W. van der Woude, and H.-X. Chang, “A system of real quaternion matrix equations with applications,” Linear Algebra and Its Applications, vol. 431, no. 12, pp. 2291–2303, 2009.
• Q. Wang, J. W. van der Woude, and S. W. Yu, “An equivalence canonical form of a matrix triplet over an arbitrary division ring with applications,” Science China Mathematics, vol. 54, no. 5, pp. 907–924, 2011.
• Q.-W. Wang, S.-W. Yu, and C.-Y. Lin, “Extreme ranks of a linear quaternion matrix expression subject to triple quaternion matrix equations with applications,” Applied Mathematics and Computation, vol. 195, no. 2, pp. 733–744, 2008.
• Q.-W. Wang and F. Zhang, “The reflexive re-nonnegative definite solution to a quaternion matrix equation,” Electronic Journal of Linear Algebra, vol. 17, pp. 88–101, 2008.
• Q. W. Wang, H.-S. Zhang, and G.-J. Song, “A new solvable condition for a pair of generalized Sylvester equations,” Electronic Journal of Linear Algebra, vol. 18, pp. 289–301, 2009.
• Q.-W. Wang, X. Zhang, and J. W. van der Woude, “A new simultaneous decomposition of a matrix quaternity over an arbitrary division ring with applications,” Communications in Algebra, vol. 40, no. 7, pp. 2309–2342, 2012.
• J. R. Magnus, “$L$-structured matrices and linear matrix equations,” Linear and Multilinear Algebra, vol. 14, no. 1, pp. 67–88, 1983.
• S. F. Yuan, A. P. Liao, and Y. Lei, “Least squares Hermitian solution of the matrix equation $(AXB,CXD)=(E,F)$ with the least norm over the skew field of quaternions,” Mathematical and Computer Modelling, vol. 48, no. 1-2, pp. 91–100, 2008.
• S.-F. Yuan, Q.-W. Wang, and X. Zhang, “Least-squares problem for the quaternion matrix equation $AXB+CYD=E$ over different constrained matrices,” International Journal of Computer Mathematics, vol. 90, no. 3, pp. 565–576, 2013.
• A.-G. Wu, F. Zhu, G.-R. Duan, and Y. Zhang, “Solving the generalized Sylvester matrix equation $AV+BW=EVF$ via a Kronecker map,” Applied Mathematics Letters, vol. 21, no. 10, pp. 1069–1073, 2008.
• S. F. Yuan and A. P. Liao, “Least squares solution of the quaternion matrix equation $X-A\widehat{X}B=C$ with the least norm,” Linear and Multilinear Algebra, vol. 59, no. 9, pp. 985–998, 2011.
• V. Hernández and M. Gassó, “Explicit solution of the matrix equation $AXB-CXD=E$,” Linear Algebra and Its Applications, vol. 121, pp. 333–344, 1989.
• S. K. Mitra, “The matrix equation $AXB+CXD=E$,” SIAM Journal on Applied Mathematics, vol. 32, no. 4, pp. 823–825, 1977.
• L. P. Huang, “The matrix equation $AXB-GXD=E$ over the quaternion field,” Linear Algebra and Its Applications, vol. 234, pp. 197–208, 1996.
• Q. W. Wang, S. W. Yu, and W. Xie, “Extreme ranks of real matrices in solution of the quaternion matrix equation $AXB=C$ with applications,” Algebra Colloquium, vol. 17, no. 2, pp. 345–360, 2010.
• S.-F. Yuan, Q.-W. Wang, and X.-F. Duan, “On solutions of the quaternion matrix equation $AX=B$ and their applications in color image restoration,” Applied Mathematics and Computation, vol. 221, pp. 10–20, 2013.
• A. Ben-Israel and T. N. E. Greville, Generalized Inverses: Theory and Applications, John Wiley & Sons, New York, NY, USA, 1974. \endinput