An efficient real representation method for least squares problem of the quaternion constrained matrix equation AXB + CY D = E

Abstract

Let ${\eta \mathbb{H}\mathbb{Q}}^{k\times k}$ and ${\eta \mathbb{A}\mathbb{Q}}^{k\times k}$ represent the sets of all $k\times k$ η-Hermitian quaternion matrices and η-anti-Hermitian quaternion matrices, respectively. On the basis of the real representation matrix of a quaternion matrix and its particular structure, we convert the least squares problem of the quaternion matrix equation AXB + CY D = E over $X\in {\eta \mathbb{H}\mathbb{Q}}^{n\times n},Y\in {\eta \mathbb{A}\mathbb{Q}}^{k\times k}$ into the corresponding problem of the real matrix equation over free variables, and then we establish its unique minimal norm least squares solution. Our resulting expressions are expressed only by real matrices, and the algorithm only includes real operations. Consequently, they are very simple and convenient. Compared with the existing method [S.F. Yuan, Q.W. Wang, and X. Zhang, Least-squares problem for the quaternion matrix equation AXB + CYD = E over different constrained matrices, Int. J. Comput. Math. 90 (2013), pp. 565–576], the final two examples show that our method is more efficient and superior.

Keywords:

• Quaternion matrix equation
• least squares solution
• real representation matrix
• η-Hermitian quaternion matrix
• η-anti-Hermitian quaternion matrix

2010 AMS Subject Classifications:

• 65F05
• 65H10
• 15B33

Acknowledgments

The authors are grateful to the anonymous referees for valuable comments and suggestions, which greatly improve the original manuscript of this paper. The study is supported by the National Natural Science Foundation of China (No. 11801249), and the Scientific Research Foundation of Liaocheng University (No. 318011921).

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

The study is supported by the National Natural Science Foundation of China [grant number 11801249], and the Scientific Research Foundation of Liaocheng University [grant number 318011921].