$AX - XB = C$, Resultants and Generalized Inverses
Robert E. Hartwig · SIAM Journal on Applied Mathematics · 1975
The theory of generalized inverses is used to derive the solutions to the matrix equation $AX - XB = C$ for the singular case over an arbitrary field $\mathcal{F}$. The results are based on the Smith normal form substitution method and the generalized inversion of certain triangular block matrices. The most important methods of solution for the nonsingular case are extended to a commutative regular ring with unity and tested for generalized invertibility.