On Least Squares Solutions of Matrix Equation MZN=S

Abstract

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 .

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Zhang, Y. , Dong, C. and Song, J. (2013) On Least Squares Solutions of Matrix Equation MZN=S. Advances in Linear Algebra & Matrix Theory, 3, 47-49. doi: 10.4236/alamt.2013.34009.

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

(1)

where and are given matrices. For instance, Cvetković-Ilić [1] and Peng et al. [2] 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. [3] has considered generalized reflexive solutions of the matrix Equation (1); in addition, Herrero and Thome [4] 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 [5].

Lemma 1. (Theorem 3.1 in [5]) 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

(2)

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

(3)

where. Partitioning

(3) is equivalent to

Therefore,

(4)

Partition and denote

(5)

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).

Conflicts of Interest

The authors declare no conflicts of interest.

 [1] D. 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? [2] 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.2006.01.024 [3] Y. 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.2008.03.015 [4] A. 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.2011.01.039 [5] G. 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