复矩阵方程AXB = C的最小二乘Hermite解
Least Squares Hermitian Solution of Complex Matrix Equation AXB = C

作者: 王 鹏 * , 陈剑波 :五邑大学数学与计算科学学院,广东 江门;

关键词: 矩阵方程最小二乘解Moore-Penrose广义逆Hermitian解Matrix Equation Least-Square Solution Moore-Penrose Inverse Generalized Hermitian Solution

摘要:

本文利用Moore-Penrose广义逆的方法,探讨了复矩阵方程的最小二乘Hermitian解,推到出了该类方程最小范数约束的最小二乘Hermitian解的解析形式。

Based on Moore-Penrose generalized inverse, by making use of matrix-vector production, an analytical expression of the least-squares Hermitian solution with the minimum-norm of complex matrix equation AXB = C is derived.

文章引用: 王 鹏 , 陈剑波 (2016) 复矩阵方程AXB = C的最小二乘Hermite解。 理论数学, 6, 42-49. doi: 10.12677/PM.2016.61007

参考文献

[1] Liao, A.-P. and Lei, Y. (2005) Least Squares Solution with the Mininum-Norm for the Matrix Equation (AXB, GXH) = (C, D). Computers & Mathematics with Applications, 50, 539-549.
http://dx.doi.org/10.1016/j.camwa.2005.02.011

[2] Yuan, Y.-X. (2001) On the Two Classes of Best Approximation Problems. Mathematica Numerica Sinica, 23, 429- 436.

[3] Yuan, Y.-X. (2002) The Optimal Solution of Linear Matrix Equation by Matrix Decompositions. Mathematica Numerica Sinica, 24, 165-176.

[4] Yuan, S.-F., Liao, A.-P. and Lei, Y. (2008) 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, 48, 91-100.
http://dx.doi.org/10.1016/j.mcm.2007.08.009

[5] Ben-Israel, A. and Greville, T.N.E. (1974) Generalized Inverses: Theory and Applications. John Wiley and Sons, New York.

[6] Dehghan, M. and Hajarian, M. (2012) The generalized Sylvester Matrix Equations over the Generalized Bisymmetric and Skew-Symmetric Matrices. International Journal of Systems Science, 43, 1580-1590.
http://dx.doi.org/10.1080/00207721.2010.549584

[7] Krishnaswamy, D. (2011) The Skew-Symmetric Ortho-Symmetric Solutions of the Matrix Equations A^* XA=D. International Journal of Algebra, 5, 1489-1504.

[8] Sheng, X.-P. and Chen, G.-L. (2010) An Iterative Method for the Symmetric and Skewsymmetric Solutions of a Linear Matrix Equation AXB + CYD = E. Journal of Computational and Applied Mathematics, 233, 3030-3040.
http://dx.doi.org/10.1016/j.cam.2009.11.052

[9] Wang, Q.-W. and He, Z.-H. (2013) Solvability Conditions and General Solution for Mixed Sylvester Equations. Automatica, 49, 2713-2719.
http://dx.doi.org/10.1016/j.automatica.2013.06.009

[10] Xiao, Q.F. (2012) The Hermitian R-Symmetric Solutions of the Matrix Equation AXA^*=B. International Journal of Algebra, 6, 903-911.

[11] Farid, F.O., Moslehian, M.S., Wang, Q.W. and Wu, Z.C. (2012) On the Hermitian Solutions to a System of Adjointable Operator Equations. Linear Algebra and Its Applications, 437, 1854-1891.
http://dx.doi.org/10.1016/j.laa.2012.05.012

[12] Dong, C.Z., Wang, Q.W. and Zhang, Y.P. (2012) The Common Positive Solution to Adjointable Operators Equations with an Application. Journal of Mathematical Analysis and Applications, 396, 670-679.
http://dx.doi.org/10.1016/j.jmaa.2012.07.001

分享
Top