Journal of Applied Mathematics

Volume 2014 (2014), Article ID 412094, 4 pages

http://dx.doi.org/10.1155/2014/412094

Research Article

The Solutions to Matrix Equation with Some Constraints

Chang-Zhou Dong and Yu-Ping Zhang

School of Mathematics and Science, Shijiazhuang University of Economics, Shijiazhuang 050031, China

Received 14 November 2013; Accepted 15 March 2014; Published 7 April 2014

Academic Editor: Morteza Rafei

Copyright © 2014 Chang-Zhou Dong and Yu-Ping Zhang. 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.

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 constraints and . A similar problem of this matrix equation with generalized constrained is also discussed.

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.

It is a very active research topic to study solutions to various matrix equations [1–4]. There are many authors who have investigated the classical matrix equation

with different constraints such as symmetric, reflexive, Hermitian-generalized Hamiltonian, and repositive definite [5–9]. By special matrix decompositions such as singular value decompositions (SVDs) and CS decompositions [10–12], Hu and his collaborators [13–15], Dai [16], and Don [17] have presented the existence conditions and detailed representations of constrained solutions for (1) with corresponding constraints, respectively. For instance, Peng and Hu [18] presented the eigenvectors-involved solutions to (1) with reflexive and antireflexive constraints; Wang and Yu [19] derived the bi(skew-)symmetric solutions and the bi(skew-)symmetric least squares solutions with the minimum norm to this matrix equation; Qiu and Wang [20] proposed an eigenvectors-free method to (1) with and constraints, where is a Hermitian involutory matrix and .

Inspired by the work mentioned above, we focus on the matrix equation (1) with and constraints, which can be described as follows: find such that

Moreover, we also discuss the least squares solutions of (1) with and constraints, where is a given unitary matrix of order .

In Section 2, we present the least squares solutions to the matrix equation (1) with the constraints and . In Section 3, we derive the least squares solutions to the matrix equation (1) with the constraints and . In Section 4, we give an algorithm and a numerical example to illustrate our results.

2. Least Squares Solutions to the Matrix Equation (1) with the Constraints   and

It is required to transform the constrained problem to unconstrained one. To this end, let

be the eigenvalue decomposition of the Hermitian matrix with unitary matrix . Obviously, holds if and only if

where . Partitioning

(4) is equivalent to

Therefore,

The constraint is equivalent to

with .

Partition and denote

then assume that the singular value decomposition of and is as follows:

where , and are unitary matrices, ,  ,  ,  ,  ,   and  .

Theorem 1. Given . Then the least squares solutions to the matrix equation (1) with the constraints and can be expressed as

where and are arbitrary matrix.

Proof. According to (8) and the unitary invariance of Frobenius norm

By (9), the least squares problem is equivalent to

We get

According to (10), the least squares problem is equivalent to

Assume that

Then we have

Hence

is solvable if and only if there exist such that

It follows from (19) that

Substituting (20) into (16) and then into (8), we can get that the form of is (11).

3. Least Squares Solutions to the Matrix Equation (1) with the Constraints and

In this section, we generalize the constraints to , where is a given unitary matrix of order . Obviously, the constraint is equal to

Notice that (1) can be equivalently rewritten in

Denoting by and setting , the equation becomes

with the constraints and .

Therefore, the least squares solutions to matrix equation (1) with the constraints and can be solved similar to Theorem 1.

Theorem 2. Given . Then the least squares solutions to the matrix equation (1) with the constraints and can be expressed as

where and are arbitrary matrix.

4. An Algorithm and Numerical Examples

Based on the main results of this paper, we in this section propose an algorithm for finding the least squares solutions to the matrix equation with the constraints and . All the tests are performed by MATLAB 6.5 which has a machine precision of around .

Algorithm 3. Input and compute ,  ,   by the eigenvalue decomposition to .

Compute according to (9).

Compute by the singular value decomposition of .

Compute according to (16).

Compute by Theorem 1.

Example 4. Suppose

Applying Algorithm 3, we obtain the following:

Conflict of Interests

The authors declare that they have no conflict of interests regarding the publication of this paper.

Acknowledgments

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).

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