Quaternion modified conjugate gradient algorithm to solve Sylvester-type quaternion matrix equations with generalized coupled form as well as application

Sylvester-type matrix equations have a wide range of applications in various fields, including stability analysis, control theory, system theory, image and signal processing, and optimization problems. In this study, we aim to address the necessity of employing mathematical approaches to solve a category of Sylvester-type quaternion matrix equations with generalized coupled form. Firstly, we establish a sufficient and necessary condition to ensure that the solution set of this category of Sylvester-type quaternion matrix equations is nonempty. This involves utilizing the real representation operator, vectorization operator, and Kronecker product on the real field. Secondly, we develop the conjugate linear operator on the quaternion algebra by utilizing the real inner product defined between two quaternion matrices. Thirdly, we introduce a quaternion modified conjugate gradient algorithm to find a general solution of this class quaternion matrix equations, along with the theoretical analysis results of the proposed algorithm. Finally, we propose a novel framework for simultaneously encrypting and decrypting multi-color images through the quaternion matrix equations. Additionally, several examples are presented to elucidate the main results.