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A Survey on Solving the Matrix Equation AXB = C with Applications

Author

Listed:
  • Qing-Wen Wang

    (Department of Mathematics and Newtouch Center for Mathematics, Shanghai University, Shanghai 200444, China
    Collaborative Innovation Center for the Marine Artificial Intelligence, Shanghai 200444, China)

  • Lv-Ming Xie

    (Department of Mathematics and Newtouch Center for Mathematics, Shanghai University, Shanghai 200444, China)

  • Zi-Han Gao

    (Department of Mathematics and Newtouch Center for Mathematics, Shanghai University, Shanghai 200444, China)

Abstract

This survey provides a comprehensive overview of the solutions to the matrix equation A X B = C over real numbers, complex numbers, quaternions, dual quaternions, dual split quaternions, and dual generalized commutative quaternions, including various special solutions. Additionally, we summarize the numerical algorithms for these special solutions. This matrix equation plays an important role in solving linear systems and control theory. We specifically explore the application of this matrix equation in color image processing, highlighting its unique value in this field. Taking the dual quaternion matrix equation A X B = C as an example, we design a scheme for simultaneously encrypting and decrypting two color images. The experimental results demonstrate that this scheme is highly feasible.

Suggested Citation

  • Qing-Wen Wang & Lv-Ming Xie & Zi-Han Gao, 2025. "A Survey on Solving the Matrix Equation AXB = C with Applications," Mathematics, MDPI, vol. 13(3), pages 1-50, January.
    • Handle: RePEc:gam:jmathe:v:13:y:2025:i:3:p:450-:d:1579399
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    References listed on IDEAS

    as
    1. Ruopeng Liao & Xin Liu & Sujuan Long & Yang Zhang, 2024. "( R , S )-(Skew) Symmetric Solutions to Matrix Equation AXB = C over Quaternions," Mathematics, MDPI, vol. 12(2), pages 1-12, January.
    2. Tian, Zhaolu & Li, Xiaojing & Dong, Yinghui & Liu, Zhongyun, 2021. "Some relaxed iteration methods for solving matrix equation AXB=C," Applied Mathematics and Computation, Elsevier, vol. 403(C).
    3. Xin Liu & Qing-Wen Wang, 2017. "The Least Squares Hermitian (Anti)reflexive Solution with the Least Norm to Matrix Equation," Mathematical Problems in Engineering, Hindawi, vol. 2017, pages 1-6, August.
    4. Liu, Xin, 2018. "The η-anti-Hermitian solution to some classic matrix equations," Applied Mathematics and Computation, Elsevier, vol. 320(C), pages 264-270.
    5. Liu, Zhongyun & Li, Zhen & Ferreira, Carla & Zhang, Yulin, 2020. "Stationary splitting iterative methods for the matrix equation AXB=C," Applied Mathematics and Computation, Elsevier, vol. 378(C).
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