Research Article | Open Access

Chang-Zhou Dong, Yu-Ping Zhang, "The Solutions to Matrix Equation with Some Constraints", *Journal of Applied Mathematics*, vol. 2014, Article ID 412094, 4 pages, 2014. https://doi.org/10.1155/2014/412094

# The Solutions to Matrix Equation with Some Constraints

**Academic Editor:**Morteza Rafei

#### 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).

#### References

- Q.-W. Wang and Z.-H. He, “A system of matrix equations and its applications,”
*Science China Mathematics*, vol. 56, no. 9, pp. 1795–1820, 2013. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - Q.-W. Wang and Z.-H. He, “Solvability conditions and general solution for mixed Sylvester equations,”
*Automatica*, vol. 49, no. 9, pp. 2713–2719, 2013. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - Q.-W. Wang and Z.-H. He, “Some matrix equations with applications,”
*Linear and Multilinear Algebra*, vol. 60, no. 11-12, pp. 1327–1353, 2012. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - S.-F. Yuan, Q.-W. Wang, and X.-F. Duan, “On solutions of the quaternion matrix equation $AX=B$ and their applications in color image restoration,”
*Applied Mathematics and Computation*, vol. 221, pp. 10–20, 2013. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - K.-E. Chu, “Symmetric solutions of linear matrix equations by matrix decompositions,”
*Linear Algebra and Its Applications*, vol. 119, pp. 35–50, 1989. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - Z.-H. He and Q.-W. Wang, “A real quaternion matrix equation with applications,”
*Linear and Multilinear Algebra*, vol. 61, no. 6, pp. 725–740, 2013. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - I. Kyrchei, “Explicit representation formulas for the minimum norm least squares solutions of some quaternion matrix equations,”
*Linear Algebra and Its Applications*, vol. 438, no. 1, pp. 136–152, 2013. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - Y. Li, Y. Gao, and W.-B. Guo, “A Hermitian least squares solution of the matrix equation $AXB=C$ subject to inequality restrictions,”
*Computers & Mathematics with Applications*, vol. 64, no. 6, pp. 1752–1760, 2012. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - L. Wu, “The re-positive definite solutions to the matrix inverse problem $AX=B$,”
*Linear Algebra and Its Applications*, vol. 174, pp. 145–151, 1992. View at: Google Scholar | Zentralblatt MATH | MathSciNet - K.-E. Chu, “Singular value and generalized singular value decompositions and the solution of linear matrix equations,”
*Linear Algebra and Its Applications*, vol. 88-89, pp. 83–98, 1987. View at: Publisher Site | Google Scholar | MathSciNet - C.-C. Paige, “Computing the generalized singular value decomposition,”
*SIAM Journal on Scientific and Statistical Computing*, vol. 7, no. 4, pp. 1126–1146, 1986. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - C.-C. Paige and M.-A. Saunders, “Towards a generalized singular value decomposition,”
*SIAM Journal on Numerical Analysis*, vol. 18, no. 3, pp. 398–405, 1981. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - C.-J. Meng, X.-Y. Hu, and L. Zhang, “The skew-symmetric orthogonal solutions of the matrix equation $AX=B$,”
*Linear Algebra and Its Applications*, vol. 402, pp. 303–318, 2005. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - C.-J. Meng and X.-Y. Hu, “An inverse problem for symmetric orthogonal matrices and its optimal approximation,”
*Mathematica Numerica Sinica*, vol. 28, no. 3, pp. 269–280, 2006. View at: Google Scholar | Zentralblatt MATH | MathSciNet - Z.-Z. Zhang, X.-Y. Hu, and L. Zhang, “On the Hermitian-generalized Hamiltonian solutions of linear matrix equations,”
*SIAM Journal on Matrix Analysis and Applications*, vol. 27, no. 1, pp. 294–303, 2005. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - H. Dai, “On the symmetric solutions of linear matrix equations,”
*Linear Algebra and Its Applications*, vol. 131, pp. 1–7, 1990. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - F.-J. Henk Don, “On the symmetric solutions of a linear matrix equation,”
*Linear Algebra and Its Applications*, vol. 93, pp. 1–7, 1987. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - Z.-Y. Peng and X.-Y. Hu, “The reflexive and anti-reflexive solutions of the matrix equation $AX=B$,”
*Linear Algebra and Its Applications*, vol. 375, pp. 147–155, 2003. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - Q.-W. Wang and J. Yu, “On the generalized bi (skew-) symmetric solutions of a linear matrix equation and its procrust problems,”
*Applied Mathematics and Computation*, vol. 219, no. 19, pp. 9872–9884, 2013. View at: Publisher Site | Google Scholar | MathSciNet - Y.-Y. Qiu and A.-D. Wang, “Eigenvector-free solutions to $AX=B$ with $PX=XP$ and ${X}^{H}=sX$ constraints,”
*Applied Mathematics and Computation*, vol. 217, no. 12, pp. 5650–5657, 2011. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet

#### Copyright

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.