ALAMTAdvances in Linear Algebra & Matrix Theory2165-333XScientific Research Publishing10.4236/alamt.2013.34009ALAMT-40530ArticlesPhysics&Mathematics On Least Squares Solutions of Matrix Equation MZN=S upingZhang1*ChangzhouDong1*JianminSong1School of Mathematics and Science, Shijiazhuang University of Economics, Shijiazhuang, China* E-mail:yuping.zh@163.com(UZ);dongchangzh@sina.com(CD);0612201303044749September 29, 2013October 31, 2013 November 10, 2013© Copyright 2014 by authors and Scientific Research Publishing Inc. 2014This work is licensed under the Creative Commons Attribution International License (CC BY). http://creativecommons.org/licenses/by/4.0/

Let be a given Hermitian matrix satisfying . Using the eigenvalue decomposition of , we consider the least squares solutions to the matrix equation , with the constraint .

Matrix Equation; Eigenvalue Decomposition; Canonical Correlation Decomposition; Reflexive Matrix; Least Squares Solution
1. Introduction

Throughout we denote the complex matrix space by . The symbols and stand for the identity matrix with the appropriate size, the conjugate transpose, the inverse, and the Frobenius norm of , respectively.

The reflexive matrices have extensive applications in engineering and scientific computation. It is a very active research topic to investigate the reflexive solution to the linear matrix equation

where and are given matrices. For instance, Cvetković-Ilić  and Peng et al.  have given the necessary and sufficient conditions for the existence and the expressions of the reflexive solutions for the matrix Equation (1) by using the structure properties of matrices in required subset of and the generalized singular value decomposition (GSVD); Different from [1,2], Ref.  has considered generalized reflexive solutions of the matrix Equation (1); in addition, Herrero and Thome  have found the reflexive (with respect to a generalized —reflection matrix ) solutions of the matrix Equation (1) by the (GSVD) and the lifting technique combined with the Kronecker product.

2. The Reflexive Least Squares Solutions to Matrix Equation (1)

We begin this section with the following lemma, which can be deduced from .

Lemma 1. (Theorem 3.1 in ) Let the canonical correlation decomposition of matrix pair and with . rank , rank , rank , rank be given as  where with the same row partitioning, and ,    with the same row partitioning, and    and let Then general forms of least squares solutions of matrix equation are as follows:  where and are arbitrary matrices.

Theorem 2. Given   . Then the reflexive least squares solutions to the matrix Equation (1) can be expressed as  where and are arbitrary matrices.

Proof. It is required to transform the constrained problem to unconstrained ones. To this end, let be the eigenvalue decomposition of the Hermitian matrix with unitary matrix . Obviously, holds if and only if

where . Partitioning (3) is equivalent to Therefore,

Partition and denote

According to (4), (5) and the unitary invariance of Frobenius norm Applying Lemma 2.1, we derive the reflexive least squares solutions to matrix Equation (1) with the constraint which can be expressed as (2).

3. Acknowledgements

This research was supported by the Natural Science Foundation of Hebei province (A2012403013), the Natural Science Foundation of Hebei province (A2012205028) and the Education Department Foundation of Hebei province (Z2013110).

REFERENCESReferencesD. S. Cvetkovic-Ilic, “The Reflexive Solutions of the Matrix Equation ,” Computers & Mathematics with Applications, Vol. 51, 2006, pp. 897-902. http://dx.doi.org/10.1016/j.camwa.2005.11.032? X. Y. Peng, X. Y. Hu and L. Zhang, “The Reflexive and Anti-Reflexive Solutions of the Matrix Equation ,” The Journal of Computational and Applied Mathematics, Vol. 200, 2007, pp. 749-760. http://dx.doi.org/10.1016/j.cam.2006.01.024Y. X. Yuan and H. Dai, “Generalized Reflexive Solutions of the Matrix Equation and an Associated Optimal Approximation Problem,” Computers & Mathematics with Applications, Vol. 56, 2008, pp. 1643-1649. http://dx.doi.org/10.1016/j.camwa.2008.03.015A. Herrero and N. Thome, “Using the GSVD and the Lifting Technique to Find Reflexive and Anti-Reflexive Solutions of ,” Applied Mathematics Letters, Vol. 24, 2011, pp. 1130-1141. http://dx.doi.org/10.1016/j.aml.2011.01.039G. P. Xu, M. S. Wei and D. S. Zheng, “On Solutions of Matrix Equation ,” Linear Algebra and Its Applications, Vol. 279, 1998, pp. 93-100. http://dx.doi.org/10.1016/S0024-3795(97)10099-4