Journal of Applied Mathematics

Volume 2014, Article ID 857081, 9 pages

http://dx.doi.org/10.1155/2014/857081

Research Article

## Least Squares Pure Imaginary Solution and Real Solution of the Quaternion Matrix Equation with the Least Norm

School of Mathematics and Computational Science, Wuyi University, Jiangmen 529020, China

Received 5 December 2013; Accepted 3 February 2014; Published 15 April 2014

Academic Editor: Qing-Wen Wang

Copyright © 2014 Shi-Fang Yuan. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Linked References

- F. Z. Zhang, “Quaternions and matrices of quaternions,”
*Linear Algebra and Its Applications*, vol. 251, pp. 21–57, 1997. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - 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. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - 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. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - 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. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - 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. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - 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. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - 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. View at Publisher · View at Google Scholar · View at MathSciNet - 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. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - 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. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - 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. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - 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. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - 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. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - 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. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - 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. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - 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. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - Z.-Y. Peng, “Solutions of symmetry-constrained least-squares problems,”
*Numerical Linear Algebra with Applications*, vol. 15, no. 4, pp. 373–389, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - 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 $,”
*Linear Algebra and Its Applications*, vol. 375, pp. 135–145, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - Y. X. Yuan, “Two classes of best approximation problems of matrices,”
*Mathematica Numerica Sinica*, vol. 23, no. 4, pp. 429–436, 2001. View at Google Scholar · View at MathSciNet - Y. X. Yuan, “The minimum norm solutions of two classes of matrix equations,”
*Numerical Mathematics*, vol. 24, no. 2, pp. 127–134, 2002. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - Y. X. Yuan, “The optimal solution of linear matrix equations by using matrix decompositions,”
*Mathematica Numerica Sinica*, vol. 24, no. 2, pp. 165–176, 2002 (Chinese). View at Google Scholar · View at MathSciNet - T. S. Jiang, Y. H. Liu, and M. S. Wei, “Quaternion generalized singular value decomposition and its applications,”
*Applied Mathematics*, vol. 21, no. 1, pp. 113–118, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - T. S. Jiang and M. S. Wei, “Real representations of quaternion matrices and quaternion matrix equations,”
*Acta Mathematica Scientia A*, vol. 26, no. 4, pp. 578–584, 2006 (Chinese). View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - Q.-W. Wang, “The general solution to a system of real quaternion matrix equations,”
*Computers & Mathematics with Applications*, vol. 49, no. 5-6, pp. 665–675, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - Q.-W. Wang, H.-X. Chang, and C.-Y. Lin, “$P$-(skew)symmetric common solutions to a pair of quaternion matrix equations,”
*Applied Mathematics and Computation*, vol. 195, no. 2, pp. 721–732, 2008. View at Publisher · View at Google Scholar · View at MathSciNet - 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. View at Publisher · View at Google Scholar · View at MathSciNet - Q. Wang and Z. He, “A system of matrix equations and its applications,”
*Science China Mathematics*, vol. 56, no. 9, pp. 1795–1820, 2013. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - 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. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - 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. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - 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. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - 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. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - 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. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - 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. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - J. R. Magnus, “$L$-structured matrices and linear matrix equations,”
*Linear and Multilinear Algebra*, vol. 14, no. 1, pp. 67–88, 1983. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - 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. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - 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. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - 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. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - 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. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - 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. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - S. K. Mitra, “The matrix equation $\mathrm{AXB}+\mathrm{CXD}=E$,”
*SIAM Journal on Applied Mathematics*, vol. 32, no. 4, pp. 823–825, 1977. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - L. P. Huang, “The matrix equation $AXB-GXD=E$ over the quaternion field,”
*Linear Algebra and Its Applications*, vol. 234, pp. 197–208, 1996. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - 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. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - 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. View at Publisher · View at Google Scholar · View at MathSciNet - A. Ben-Israel and T. N. E. Greville,
*Generalized Inverses: Theory and Applications*, John Wiley & Sons, New York, NY, USA, 1974. View at MathSciNet