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

Qing-Wen Wang (), Lv-Ming Xie and Zi-Han Gao
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Qing-Wen Wang: Department of Mathematics and Newtouch Center for Mathematics, Shanghai University, 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

Mathematics, 2025, vol. 13, issue 3, 1-50

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.

Keywords: matrix equation; general solution; special solution; numerical algorithm; color image (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2025
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