$ \mathcal{H} $-representation method for solving reduced biquaternion matrix equation

Xueling Fan; Ying Li; Wenxv Ding; Jianli Zhao; Xueling Fan; Ying Li; Wenxv Ding; Jianli Zhao

doi:10.3934/mmc.2022008

Mathematical Modelling and Control

2022, Volume 2, Issue 2: 65-74. doi: 10.3934/mmc.2022008

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Research article

$ \mathcal{H} $-representation method for solving reduced biquaternion matrix equation

1.
College of Mathematical Sciences, Liaocheng University, Liaocheng, 252000, China
2.
Research Center of Semi-tensor Product of Matrices: Theory and Applications, Liaocheng, 252000, China

Received: 15 December 2021 Revised: 03 March 2022 Accepted: 15 March 2022 Published: 27 June 2022

In this paper, we study the Hankel and Toeplitz solutions of reduced biquaternion matrix equation (1.1). Using semi-tensor product of matrices, the reduced biquaternion matrix equation (1.1) can be transformed into a general matrix equation of the form $ AX = B $. Then, due to the special structure of Hankel matrix and Toeplitz matrix, the independent elements of Hankel matrix or Toeplitz matrix can be extracted by combing the $ \mathcal{H} $-representation method of matrix, so as to reduce the elements involved in the operation in the process of solving matrix equation and reduce the complexity of the problem. Finally, by using Moore-Penrose generalized inverse, the necessary and sufficient conditions for the existence of solutions of reduced biquaternion matrix equation (1.1) are given, and the corresponding numerical examples are given.
- reduced biquaternion matrix equation,
- semi-tensor product of matrices,
- $ \mathcal{H} $-representation,
- real representation,
- Hankel matrix,
- Toeplitz matrix
Citation: Xueling Fan, Ying Li, Wenxv Ding, Jianli Zhao. $ \mathcal{H} $-representation method for solving reduced biquaternion matrix equation[J]. Mathematical Modelling and Control, 2022, 2(2): 65-74. doi: 10.3934/mmc.2022008

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Abstract

In this paper, we study the Hankel and Toeplitz solutions of reduced biquaternion matrix equation (1.1). Using semi-tensor product of matrices, the reduced biquaternion matrix equation (1.1) can be transformed into a general matrix equation of the form $ AX = B $. Then, due to the special structure of Hankel matrix and Toeplitz matrix, the independent elements of Hankel matrix or Toeplitz matrix can be extracted by combing the $ \mathcal{H} $-representation method of matrix, so as to reduce the elements involved in the operation in the process of solving matrix equation and reduce the complexity of the problem. Finally, by using Moore-Penrose generalized inverse, the necessary and sufficient conditions for the existence of solutions of reduced biquaternion matrix equation (1.1) are given, and the corresponding numerical examples are given.

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