Least Squares Symmetrizable Solutions for a Class of Matrix Equations

Fanliang Li

School of Sciences, Institute of Mathematics and Physics, Central South University of Forestry and Technology, Changsha, China.

DOI: 10.4236/am.2013.45102                

Abstract

In this paper, we discuss least squares symmetrizable solutions of matrix equations (AX = B, XC = D) and its optimal approximation solution. With the matrix row stacking, Kronecker product and special relations between two linear subspaces are topological isomorphism, and we derive the general solutions of least squares problem. With the invariance of the Frobenius norm under orthogonal transformations, we obtain the unique solution of optimal approximation problem. In addition, we present an algorithm and numerical experiment to obtain the optimal approximation solution.

Keywords

Matrix Equations; Matrix Row Stacking; Topological Isomorphism; Least Squares Solution; Optimal Approximation

Conflicts of Interest

The authors declare no conflicts of interest.

References

[1]	A. Dajic and J. J. Koliha, “Equations ax = c and xb = b in Rings and Rings with Involution with Applications to Hilbert Space Operators,” Linear Algebra and Its Applications, Vol. 429, No. 7, 2008, pp. 1779-1809. doi:10.1016/j.laa.2008.05.012
[2]	S. K. Mitra, “The Matrix Equations AX = C, XB = D,” Linear Algebra and Its Applications, Vol. 59, 1984, pp. 171-181. doi:10.1016/0024-3795(84)90166-6
[3]	K. W. E. Chu, “Singular Value and Generalized Singular Value Decomposition and the Solution of Linear Matrix Equations,” Linear Algebra and Its Applications, Vol. 88-89, 1987, pp. 83-98. doi:10.1016/0024-3795(87)90104-2
[4]	S. K. Mitra, “A Pair of Simultaneous Linear Matrix Equations A1XB1 = C1, A2XB2 = C2 and a Matrix Programming Problem,” Linear Algebra and Its Applications, Vol. 131, 1990, pp. 107-123. doi:10.1016/0024-3795(90)90377-O
[5]	A. Dajic 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, 2007, pp. 567-576. doi:10.1016/j.jmaa.2006.11.016
[6]	Q. X. Xu, “Common Hermitian and Positive Solutions to the Adjointable Operator Equations AX = C, XB = D,” Linear Algebra and Its Applications, Vol. 429, No. 1, 2008, pp. 1-11. doi:10.1016/j.laa.2008.01.030
[7]	Q. W. Wang, “Bisymmetric and Centrosymmetric Solutions to Systems of Real Quaternion Matrix Equations,” Computers and Mathematics with Applications, Vol. 49, No. 5-6, 2005, pp. 641-650. doi:10.1016/j.camwa.2005.01.014
[8]	Y. Qiu and A. Wang, “Least Squares Solutions to the Equations AX = B, XC = D with Some Constraints,” Applied Mathematics and Computation, Vol. 204, No. 2, 2008, pp. 872-880. doi:10.1016/j.amc.2008.07.035
[9]	F. L. Li, X. Y. Hu and L. Zhang, “The Generalized Reflexive Solution for a Class of Matrix Equations (AX = B, XC = D),” Acta Mathematica Scientia Series B, Vol. 1, No. 28, 2008, pp. 185-193.
[10]	F. L. Li, X. Y. Hu and L. Zhang, “The Generalized Anti-Reflexive Solution for a Class of Matrix Equations (BX = C, XD = E),” Computational & Applied Mathematics, Vol. 1, No. 27, 2008, pp. 31-46.
[11]	O. Taussky, “The Role of Symmetric Matrices in the Study of General Matrices,” Linear Algebra and Its Applications, Vol. 51, 1972, pp. 13-18.
[12]	S. J. Chang, “On Positive Symmetrizable Matrices and Pre-Symmetry Iteration Algorithms,” Mathematica Numerica Sinica, Vol. 3, No. 22, 2000, pp. 379-384.
[13]	Z. Y. Peng, “The Least-Squares Solution of Inverse Problem for One Kind of Symmetrizable Matrices,” Chinese Journal of Numerical Mathematics and Applications, No. 3, 2004, pp. 219-224.
[14]	D. W. Fausett and C. T. Fulton, “Large Least Squares Problems Involving Kronecker Products,” SIAM Journal on Matrix Analysis and Applications, Vol. 15, No. 1, 1994, pp. 219-227. doi:10.1137/S0895479891222106