Review on Solving Sylvester-Type Equations

The solution theory of Sylvester-type equations finds wide applications in control theory, robotics, and image processing. This paper systematically surveys, classifies and summarizes the existing research results of three classes of Sylvester-type equations: matrix equations, tensor equations, and operator equations. It extracts nine mainstream research methods and clarifies the internal correlations among these methods as well as their applicable equation types. This work establishes a complete framework for solving Sylvester-type equations and, together with four prior review articles, forms a comprehensive framework for linear equations. It not only provides a systematic theoretical foundation and a clear research thread for subsequent researchers, but also offers valuable methodological insights for further investigations in related fields.