Journal of Applied Mathematics

Volume 2012, Article ID 398085, 14 pages

http://dx.doi.org/10.1155/2012/398085

Research Article

## On the Hermitian -Conjugate Solution of a System of Matrix Equations

^{1}School of Mathematics and Science, Shijiazhuang University of Economics, Shijiazhuang, Hebei 050031, China^{2}Department of Mathematics, Shanghai University, Shanghai, Shanghai 200444, China^{3}Department of Mathematics, Ordnance Engineering College, Shijiazhuang, Hebei 050003, China

Received 3 October 2012; Accepted 26 November 2012

Academic Editor: Yang Zhang

Copyright © 2012 Chang-Zhou Dong et al. 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

- R. D. Hill, R. G. Bates, and S. R. Waters, “On centro-Hermitian matrices,”
*SIAM Journal on Matrix Analysis and Applications*, vol. 11, no. 1, pp. 128–133, 1990. View at Publisher · View at Google Scholar - R. Kouassi, P. Gouton, and M. Paindavoine, “Approximation of the Karhunen-Loeve tranformation and Its application to colour images,”
*Signal Processing: Image Communication*, vol. 16, pp. 541–551, 2001. View at Google Scholar - A. Lee, “Centro-Hermitian and skew-centro-Hermitian matrices,”
*Linear Algebra and Its Applications*, vol. 29, pp. 205–210, 1980. View at Publisher · View at Google Scholar - Z.-Y. Liu, H.-D. Cao, and H.-J. Chen, “A note on computing matrix-vector products with generalized centrosymmetric (centrohermitian) matrices,”
*Applied Mathematics and Computation*, vol. 169, no. 2, pp. 1332–1345, 2005. View at Publisher · View at Google Scholar - W. F. Trench, “Characterization and properties of matrices with generalized symmetry or skew symmetry,”
*Linear Algebra and Its Applications*, vol. 377, pp. 207–218, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - K.-W. E. Chu, “Symmetric solutions of linear matrix equations by matrix decompositions,”
*Linear Algebra and Its Applications*, vol. 119, pp. 35–50, 1989. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - H. Dai, “On the symmetric solutions of linear matrix equations,”
*Linear Algebra and Its Applications*, vol. 131, pp. 1–7, 1990. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - F. J. H. Don, “On the symmetric solutions of a linear matrix equation,”
*Linear Algebra and Its Applications*, vol. 93, pp. 1–7, 1987. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - I. Kyrchei, “Explicit representation formulas for the minimum norm least squares solutions of some quaternion matrix equations,”
*Linear Algebra and Its Applications*, vol. 438, no. 1, pp. 136–152, 2013. View at Publisher · View at Google Scholar - Y. Li, Y. Gao, and W. Guo, “A Hermitian least squares solution of the matrix equation $AXB=C$ subject to inequality restrictions,”
*Computers & Mathematics with Applications*, vol. 64, no. 6, pp. 1752–1760, 2012. View at Publisher · View at Google Scholar - Z.-Y. Peng and X.-Y. Hu, “The reflexive and anti-reflexive solutions of the matrix equation $AX=B$,”
*Linear Algebra and Its Applications*, vol. 375, pp. 147–155, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - L. Wu, “The re-positive definite solutions to the matrix inverse problem $AX=B$,”
*Linear Algebra and Its Applications*, vol. 174, pp. 145–151, 1992. View at Google Scholar · View at Zentralblatt MATH - S. F. Yuan, Q. W. Wang, and X. Zhang, “Least-squares problem for the quaternion matrix equation $AXB+CYD=E$ over different constrained matrices,” Article ID 722626,
*International Journal of Computer Mathematics*. In press. View at Publisher · View at Google Scholar - Z.-Z. Zhang, X.-Y. Hu, and L. Zhang, “On the Hermitian-generalized Hamiltonian solutions of linear matrix equations,”
*SIAM Journal on Matrix Analysis and Applications*, vol. 27, no. 1, pp. 294–303, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - F. Cecioni, “Sopra operazioni algebriche,”
*Annali della Scuola Normale Superiore di Pisa*, vol. 11, pp. 17–20, 1910. View at Google Scholar - 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 - S. K. Mitra, “A pair of simultaneous linear matrix equations ${A}_{1}X{B}_{1}={C}_{1},{A}_{2}X{B}_{2}={C}_{2}$ and a matrix programming problem,”
*Linear Algebra and Its Applications*, vol. 131, pp. 107–123, 1990. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - H.-X. Chang and Q.-W. Wang, “Reflexive solution to a system of matrix equations,”
*Journal of Shanghai University*, vol. 11, no. 4, pp. 355–358, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - A. Dajić and J. J. Koliha, “Equations $ax=c$ and $xb=d$ in rings and rings with involution with applications to Hilbert space operators,”
*Linear Algebra and Its Applications*, vol. 429, no. 7, pp. 1779–1809, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - A. Dajić and J. J. Koliha, “Positive solutions to the equations $AX=C$ and $XB=D$ for Hilbert space operators,”
*Journal of Mathematical Analysis and Applications*, vol. 333, no. 2, pp. 567–576, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - C.-Z. Dong, Q.-W. Wang, and Y.-P. Zhang, “The common positive solution to adjointable operator equations with an application,”
*Journal of Mathematical Analysis and Applications*, vol. 396, no. 2, pp. 670–679, 2012. View at Publisher · View at Google Scholar - X. Fang, J. Yu, and H. Yao, “Solutions to operator equations on Hilbert ${C}^{*}$-modules,”
*Linear Algebra and Its Applications*, vol. 431, no. 11, pp. 2142–2153, 2009. View at Publisher · View at Google Scholar - C. G. Khatri and S. K. Mitra, “Hermitian and nonnegative definite solutions of linear matrix equations,”
*SIAM Journal on Applied Mathematics*, vol. 31, no. 4, pp. 579–585, 1976. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - I. Kyrchei, “Analogs of Cramer's rule for the minimum norm least squares solutions of some matrix equations,”
*Applied Mathematics and Computation*, vol. 218, no. 11, pp. 6375–6384, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - F. L. Li, X. Y. Hu, and L. Zhang, “The generalized reflexive solution for a class of matrix equations ($AX=C,XB=D$),”
*Acta Mathematica Scientia*, vol. 28B, no. 1, pp. 185–193, 2008. View at Google Scholar - Q.-W. Wang and C.-Z. Dong, “Positive solutions to a system of adjointable operator equations over Hilbert ${C}^{*}$-modules,”
*Linear Algebra and Its Applications*, vol. 433, no. 7, pp. 1481–1489, 2010. View at Publisher · View at Google Scholar - Q. Xu, “Common Hermitian and positive solutions to the adjointable operator equations $AX=C$, $XB=D$,”
*Linear Algebra and Its Applications*, vol. 429, no. 1, pp. 1–11, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - F. Zhang, Y. Li, and J. Zhao, “Common Hermitian least squares solutions of matrix equations ${A}_{1}X{A}_{1}^{*}={B}_{1}$ and ${A}_{2}X{A}_{2}^{*}={B}_{2}$ subject to inequality restrictions,”
*Computers & Mathematics with Applications*, vol. 62, no. 6, pp. 2424–2433, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - H.-X. Chang, Q.-W. Wang, and G.-J. Song, “$(R,S)$-conjugate solution to a pair of linear matrix equations,”
*Applied Mathematics and Computation*, vol. 217, no. 1, pp. 73–82, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - E. W. Cheney,
*Introduction to Approximation Theory*, McGraw-Hill, New York, NY, USA, 1966.