A real representation method for special least squares solutions of the quaternion matrix equation $  (AXB, DXE) = (C, F)  $

Fengxia Zhang; Ying Li; Jianli Zhao; Fengxia Zhang; Ying Li; Jianli Zhao

doi:10.3934/math.2022803

AIMS Mathematics

2022, Volume 7, Issue 8: 14595-14613. doi: 10.3934/math.2022803

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Research article

A real representation method for special least squares solutions of the quaternion matrix equation $ (AXB, DXE) = (C, F) $

College of Mathematical Science, Liaocheng University, Liaocheng 252000, China

Received: 27 December 2021 Revised: 18 May 2022 Accepted: 27 May 2022 Published: 07 June 2022
MSC : 15A06, 15A24, 65F35

In this article, our interest is the quaternion matrix equation $ (AXB, DXE) = (C, F) $, and we study its minimal norm centrohermitian least squares solution and skew centrohermitian least squares solution. By applying of the real representation matrices of quaternion matrices and relative properties, we convert the quaternion least squares problems with constrained variables into the corresponding real least squares problems with free variables, and then we obtain the solutions of corresponding problems. The final results can be expressed only by real matrices and vectors, and thus the corresponding algorithms only involves real operations and avoid complex quaternion operations. Therefore, they are portable and convenient. In the end, we give two examples to verify the effectiveness of the purposed algorithms.
- quaternion matrix equation,
- real representation,
- quaternion centrohermitian matrix,
- quaternion skew centrohermitian matrix,
- least squares solution
Citation: Fengxia Zhang, Ying Li, Jianli Zhao. A real representation method for special least squares solutions of the quaternion matrix equation $ (AXB, DXE) = (C, F) $[J]. AIMS Mathematics, 2022, 7(8): 14595-14613. doi: 10.3934/math.2022803

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Abstract

In this article, our interest is the quaternion matrix equation $ (AXB, DXE) = (C, F) $, and we study its minimal norm centrohermitian least squares solution and skew centrohermitian least squares solution. By applying of the real representation matrices of quaternion matrices and relative properties, we convert the quaternion least squares problems with constrained variables into the corresponding real least squares problems with free variables, and then we obtain the solutions of corresponding problems. The final results can be expressed only by real matrices and vectors, and thus the corresponding algorithms only involves real operations and avoid complex quaternion operations. Therefore, they are portable and convenient. In the end, we give two examples to verify the effectiveness of the purposed algorithms.

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