Mathematical Problems in Engineering
Volume 2017 (2017), Article ID 9756035, 6 pages
https://doi.org/10.1155/2017/9756035

Research Article

The Least Squares Hermitian (Anti)reflexive Solution with the Least Norm to Matrix Equation

Xin Liu¹ and Qing-Wen Wang²

¹Faculty of Information Technology, Macau University of Science and Technology, Avenida Wai Long, Taipa 999078, Macau, China
²Department of Mathematics, Shanghai University, Shanghai 200444, China

Correspondence should be addressed to Xin Liu; xiliu@must.edu.mo

Received 28 June 2017; Accepted 20 July 2017; Published 29 August 2017

Academic Editor: Kishin Sadarangani

Copyright © 2017 Xin Liu and Qing-Wen Wang. 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.

For a given generalized reflection matrix , that is, , , where is the conjugate transpose matrix of , a matrix is called a Hermitian (anti)reflexive matrix with respect to if and By using the Kronecker product, we derive the explicit expression of least squares Hermitian (anti)reflexive solution with the least norm to matrix equation over complex field.

1. Introduction

Throughout this paper, we denote the set of all matrices over the real field and complex field by and , respectively. Let the symbols , , , , and stand for the set of all complex Hermitian matrices, the set of all real symmetric matrices, the set of all real skew symmetric matrices, the identity matrix, and the zero matrix with appropriate size, respectively. We also denote the transpose, the conjugate transpose, the Moore-Penrose inverse, and the Frobenius norm of by , , , and , respectively. Let ; we define , where is the -column of ,  .

In this paper, we consider the Hermitian (anti)reflexive solution problem of matrix equation Let us first recall the definition of those two kinds of matrices.

Definition 1. Given a generalized reflection matrix , that is, , , a matrix is called a Hermitian reflexive matrix with respect to if and The set of all Hermitian reflexive matrices with respect to is denoted by

Definition 2. Given a generalized reflection matrix , that is, , , a matrix is called a Hermitian antireflexive matrix with respect to if and The set of all Hermitian antireflexive matrices with respect to is denoted by

As an important kind of matrices, the reflexive matrices with different constraints were studied by many authors in recent years in literatures and references therein [1–29]. For instance, Zhao et al. [1] derived the general Hermitian reflexive solution to , while Yu and Shen [2] considered a more general case, that is, the Hermitian -(anti)-reflexive solution to . As an extension of matrix equation , the matrix system was considered by Zhou and Yang [3]. They gave the necessary and sufficient conditions for the existence of a Hermitian reflexive solution to the above matrix system. As we can see, many important associated works were widely studied. But finding the least squares Hermitian (anti)reflexive solution with the least norm to matrix equation is still a problem. Therefore, we in this paper will fill this gap.

We consider the following two problems.

Problem 3. Given matrices ,  , and , let

Find such that

Problem 4. Given matrices ,  , and , let

Find such that

Now, we start with some useful and fundamental results in the following.

Lemma 5. The least squares solution to the matrix equation with and is given by , where is an arbitrary vector. And the least squares solution with the least norm is

Kronecker product method plays an important role in this paper. We use this method to get some important vectorizing formulas.

For any , , where , The real representation matrix of is given by , which has the following properties:

(1)

(2)

(3) For , we define Then

Analogously to the method of Yuan et al. (see [30]), we can get a fundamental result as below.

Lemma 6. Let , , and be complex matrices with appropriate sizes; then

Proof. We denote , , and Then

By the rule of

we have

The structure of Hermitian (anti)reflexive matrix promises the following decompositions, which would simplify our problems.

Lemma 7. Let be the generalized reflection matrix. Then there exists an unitary matrix such that

Lemma 8 (see [18]). (1) Let and let the spectral decomposition of the reflection matrix be given as (7). Then if and only if

where and
(2) See [4]. Let and let the spectral decomposition of the reflection matrix be given as (7). Then if and only if

where

2. The Solution of Problem 3

In this section, we provide the least squares Hermitian reflexive solution with the least norm to matrix equation . There are the vectorizing formulas of Hermitian matrix, which are very useful in what comes below.

Definition 9. Let ; we define

where , ,  , and

Definition 10. Let ; we define

where , , and

Lemma 11 (see [30]). Suppose that Then if and only if , where is given by (10), where

Obviously, is the -column of , , and .

Lemma 12. Suppose that Then if and only if , where is given by (11), where

Obviously, is the -column of , , and .

Let , , Then and By Lemmas 11 and 12, one has

If , then, by Lemma 8, there exists a unitary matrix such that

where , and

Set

where is given by (7). , are given by (12) with the sizes , , respectively. , are given by (13) with the sizes , , respectively. Then one has the following result.

Theorem 13. Let , , and . Then the solution to matrix equation is given by

where is given by (7), , and is an arbitrary vector. And the solution with the least norm is given by

Proof. Since

it follows from (14) that

Now, by Lemmas 6 and 11 and (23), we have

where is given by (17); . Then, by Lemma 5,

is the required least squares solution, where is an arbitrary vector. Recall that

Therefore

Then the least squares Hermitian reflexive solution is given by (18). Obviously, the least squares Hermitian reflexive solution with the least norm is given by (20).

3. The Solution of Problem 4

In this section, we provide the least squares Hermitian antireflexive solution with the least norm to matrix equation

We first derive the relation between and .

Lemma 14 (see [31]). Let be any matrix. Then

where

with , , , being an matrix with the element at position being and the others being . Moreover, is an orthogonal matrix, which is uniquely determined by the integers

Let ; then, by Lemma 8, there exists a unitary matrix such that

where and is given by (7). Since ,

By Lemma 14, one has

Set

and is given by (29). Then one has the following results.

Theorem 15. Let , , and Then on has the following:
(1) The solution to matrix equation is given by

where

is given by (7), , and is an arbitrary vector.
(2) The solution with the least norm is given by

where is given by (7) and , .

Proof. It follows from Lemma 6 that

Combining (32) and (37), we obtain

Then

where and
By Lemma 5, the least squares solution is given by

where is an arbitrary vector.
The least squares solution with least norm is
So

Now, it is easy to see that our required solutions are given by (34) and (36).

4. Conclusion

By exploiting the decomposition of Hermitian (anti)reflexive matrix and the Kronecker product, we derive the expression of the least squares Hermitian (anti)reflexive solution with the least norm to This method overcomes the difficulty of finding the structured least squares solution with the least norm but creates a large-scale linear system Therefore, it fits well with the small-scale coefficient matrix, but, for the large-scale case, it may not work so well.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Authors’ Contributions

Both authors read and approved the final manuscript.

Acknowledgments

This research was supported by Macau Science and Technology Development Fund (no. 003/2015/A1) and a grant from the National Natural Science Foundation of China (11571220).

References

L. Zhao, X. Hu, and L. Zhang, “Linear restriction problem of Hermitian reflexive matrices and its approximation,” Applied Mathematics and Computation, vol. 200, no. 1, pp. 341–351, 2008.

J. Yu and S.-q. Shen, “The Hermitian -(anti-)reflexive solutions of a linear matrix equation,” Computers & Mathematics with Applications, vol. 71, no. 12, pp. 2513–2523, 2016.

S. Zhou and S. Yang, “The Hermitian reflexive solutions and the anti-Hermitian reflexive solutions for a class of matrix equations ,” Energy Procedia, vol. 17, part B, pp. 1591–1597, 2012.

Z.-Y. Peng, “The inverse eigenvalue problem for Hermitian anti-reflexive matrices and its approximation,” Applied Mathematics and Computation, vol. 162, no. 3, pp. 1377–1389, 2005.

L. Lebtahi, O. Romero, and N. Thome, “Characterizations of -potent matrices and applications,” Linear Algebra and its Applications, vol. 436, no. 2, pp. 293–306, 2012.

L. Lebtahi and N. Thome, “A note on k-generalized projections,” Linear Algebra and its Applications, vol. 420, no. 2-3, pp. 572–575, 2007.

L. Lebtahi, Ó. Romero, and N. Thome, “Relations between -potent matrices and different classes of complex matrices,” Linear Algebra and its Applications, vol. 438, no. 4, pp. 1517–1531, 2013.

L. Lebtahi, Ó. Romero, and N. Thome, “Algorithms for -potent matrix constructions,” Journal of Computational and Applied Mathematics, vol. 249, pp. 157–162, 2013.

S. Gigola, L. Lebtahi, and N. Thome, “The inverse eigenvalue problem for a Hermitian reflexive matrix and the optimization problem,” Journal of Computational and Applied Mathematics, vol. 291, pp. 449–457, 2016.

M. Dehghan and M. Hajarian, “Two algorithms for finding the Hermitian reflexive and skew-Hermitian solutions of Sylvester matrix equations,” Applied Mathematics Letters, vol. 24, no. 4, pp. 444–449, 2011.

M. Dehghan and M. Hajarian, “An iterative algorithm for the reflexive solutions of the generalized coupled Sylvester matrix equations and its optimal approximation,” Applied Mathematics and Computation, vol. 202, no. 2, pp. 571–588, 2008.

M. Dehghan and M. Hajarian, “On the reflexive and anti-reflexive solutions of the generalised coupled Sylvester matrix equations,” International Journal of Systems Science. Principles and Applications of Systems and Integration, vol. 41, no. 6, pp. 607–625, 2010.

M. Dehghan and M. Hajarian, “Finite iterative algorithms for the reflexive and anti-reflexive solutions of the matrix equation ,” Mathematical and Computer Modelling, vol. 49, no. 9-10, pp. 1937–1959, 2009.

M. Dehghan and M. Hajarian, “On the reflexive solutions of the matrix equation ,” Bulletin of the Korean Mathematical Society, vol. 46, no. 3, pp. 511–519, 2009.

M. Dehghan and M. Hajarian, “The reflexive and anti-reflexive solutions of a linear matrix equation and systems of matrix equations,” The Rocky Mountain Journal of Mathematics, vol. 40, no. 3, pp. 825–848, 2010.

F. L. Li, X. Y. Hu, and L. Zhang, “The generalized reflexive solutions for a class of matrix equations ,” Acta Mathematica Scientia, vol. 28B, no. 1, pp. 185–193, 2008.

F.-L. Li, X.-Y. Hu, and L. Zhang, “The generalized anti-reflexive solutions for a class of matrix equations ,” Computational & Applied Mathematics, vol. 27, no. 1, pp. 31–46, 2008.

Z.-y. Peng and X.-y. Hu, “The reflexive and anti-reflexive solutions of the matrix equation ,” Linear Algebra and Its Applications, vol. 375, pp. 147–155, 2003.

H.-C. Chen, “Generalized reflexive matrices: special properties and applications,” SIAM Journal on Matrix Analysis and Applications, vol. 19, no. 1, pp. 140–153, 1998.

D. S. Cvetković-Iliíc, “The reflexive solutions of the matrix equation ,” Computers & Mathematics with Applications, vol. 51, no. 6-7, pp. 897–902, 2006.

H.-x. Chang and Q.-w. Wang, “Reflexive solution to a system of matrix equations,” Journal of Shanghai University, vol. 11, no. 4, pp. 355–358, 2007.

Q. W. Wang and F. Zhang, “The reflexive re-nonnegative definite solution to a quaternion matrix equation,” Electronic Journal of Linear Algebra, vol. 17, pp. 88–101, 2008.

X.-y. Peng, X.-y. Hu, and L. Zhang, “The reflexive and anti-reflexive solutions of the matrix equation ,” Journal of Computational and Applied Mathematics, vol. 200, no. 2, pp. 749–760, 2007.

Z. Y. Peng, Y. B. Deng, and J. W. Liu, “Least-squares solution of inverse problem for Hermitian anti-reflexive matrices and its approximation,” Acta Mathematica Sinica (English Series), vol. 22, no. 2, pp. 477–484, 2006.

Z. Peng and H. Xin, “The reflexive least squares solutions of the general coupled matrix equations with a submatrix constraint,” Applied Mathematics and Computation, vol. 225, pp. 425–445, 2013.

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, no. 7, pp. 1130–1141, 2011.

M.-l. Liang, L.-f. Dai, and Y.-f. Yang, “The -reflexive solution of matrix equation ,” Journal of Applied Mathematics and Computing, vol. 42, no. 1-2, pp. 339–350, 2013.

F. P. Beik and M. M. Moghadam, “The general coupled linear matrix equations with conjugate and transpose unknowns over the mixed groups of generalized reflexive and anti-reflexive matrices,” Computational & Applied Mathematics, vol. 33, no. 3, pp. 795–820, 2014.

J.-C. Zhang, S.-Z. Zhou, and X.-Y. Hu, “The generalized reflexive and anti-reflexive solutions of the matrix equation ,” Applied Mathematics and Computation, vol. 209, no. 2, pp. 254–258, 2009.

S. Yuan, A. Liao, and Y. Lei, “Least squares Hermitian solution of the matrix equation with the least norm over the skew field of quaternion,” Mathematical and Computer Modelling, vol. 48, no. 1-2, pp. 91–100, 2008.

J. L. Chen and X. H. Chen, Special Matrices, Tsinghua University Press, Beijing, China, 2002.