An efficient method for least-squares problem of the quaternion matrix equation X - AX̂B = C

ABSTRACT

In this paper, we consider the quaternion matrix equation $X-A\hat{X}B=C$, and study its minimal norm least squares solution, j-self-conjugate least-squares solution and anti-j-self-conjugate least-squares solution. By the real representation matrices of quaternion matrices, their particular structure and the properties of Frobenius norm, we convert above least-squares problems into corresponding problems of real matrix equations. The final results of the expressions only involve real matrices, and thus, the corresponding algorithms only involve real operations. Compared with the existing results, they are more convenient and efficient, which are also illustrated by the last two numerical examples.

KEYWORDS:

• Quaternion matrix equation
• least-squares solution
• real representation matrix
• j-self-conjugate matrix
• anti-j-self-conjugate matrix

2010 MATHEMATICS SUBJECT CLASSIFICATIONS:

• 65F05
• 65H10
• 15B33

Acknowledgments

The study was 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 was supported by the National Natural Science Foundation of China [No. 11801249] and the Scientific Research Foundation of Liaocheng University [No. 318011921].