Solving Quaternion Linear System Based on Semi-Tensor Product of Quaternion Matrices
Abstract
:1. Introduction
2. Preliminaries
2.1. Quaternion and Quaternion Matrices
2.2. Semi-Tensor Product of Quaternion Matrices
- (1)
- (Associative rule)
- (2)
- (Distributive rule)
- (3)
- (Conjugate Transpose)
3. Main Conclusions
3.1. Vector Representation of Quaternion Matrices
3.2. -Representation of Quaternion Matrices
4. -Representation of Quaternion Matrices
5. The Solutions of Problem 1 and Problem 2
6. Algorithms and Numerical Examples
Algorithm 1 Calculate the minimal norm centrosymmetric solution of quaternion matrix Equation (1). |
Input:
Quaternion matrix , , , ; Output:
Output the minimal norm centrosymmetric solution of quaternion matrix Equation (1) according to (5);
|
Algorithm 2 Calculate the minimal norm anti-centrosymmetric solution of quaternion matrix Equation (1). |
Input:
Quaternion matrix , , , ; Output:
Output the minimal norm centrosymmetric solution of quaternion matrix Equation (1) according to (9);
|
Algorithm 3 Calculate the minimal norm centrosymmetric solution of quaternion matrix Equation (1) according to the method of reference [43]. |
Input:
Quaternion matrix , , , ; Output:
Output the minimal norm centrosymmetric solution of quaternion matrix Equation (1);
|
Algorithm 4 Calculate the minimal norm centrosymmetric solution of quaternion matrix Equation (1) according to the method of reference [44]. |
Input:
Quaternion matrix , , , ; Output:
Output the minimal norm centrosymmetric solution of quaternion matrix Equation (1);
|
7. Application in Color Digital Image Restoration
Algorithm 5 Calculate the minimal norm least squares pure imaginary centrosymmetric solution of color digital image model . |
Output: Output the minimal norm least squares pure imaginary centrosymmetric solution of quaternion matrix equation ;
|
8. Conclusions
- Notes:
- The images used are from the MATLAB image processing toolbox or USC-SIPI image database image library of the University of Southern California (http://sipi.usc.edu/database/, accessed on 1 June 2022).
- All computations are performed on an Intel(R) core(TM) i9-10940U @3.30 GHz/64 GB computer using MATLAB R2019b software.
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Conflicts of Interest
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Fan, X.; Li, Y.; Liu, Z.; Zhao, J. Solving Quaternion Linear System Based on Semi-Tensor Product of Quaternion Matrices. Symmetry 2022, 14, 1359. https://doi.org/10.3390/sym14071359
Fan X, Li Y, Liu Z, Zhao J. Solving Quaternion Linear System Based on Semi-Tensor Product of Quaternion Matrices. Symmetry. 2022; 14(7):1359. https://doi.org/10.3390/sym14071359
Chicago/Turabian StyleFan, Xueling, Ying Li, Zhihong Liu, and Jianli Zhao. 2022. "Solving Quaternion Linear System Based on Semi-Tensor Product of Quaternion Matrices" Symmetry 14, no. 7: 1359. https://doi.org/10.3390/sym14071359
APA StyleFan, X., Li, Y., Liu, Z., & Zhao, J. (2022). Solving Quaternion Linear System Based on Semi-Tensor Product of Quaternion Matrices. Symmetry, 14(7), 1359. https://doi.org/10.3390/sym14071359