A Comprehensive Review on the Generalized Sylvester Equation AX-YB=C

Since Roth’s work on the generalized Sylvester equation AX −YB = C (GSE) in 1952, related research has consistently attracted significant attention. Building on this, this review systematically summarizes relevant research on GSE from five perspectives: research methods, constrained solutions, various generalizations, iterative algorithms, and applications. Furthermore, we provide comments on current research, put forward several intriguing questions, and offer prospects for future research trends. We hope this work can fill the gap in the review literature on GSE and offer some inspiration for subsequent studies in the field.