The Equation $\boldsymbol{AX-XB=C}$ Without a Unique Solution: the Ambiguity Which Benefits Applications

Bogdan D. Djordjević

This paper is a survey of the author's results regarding the equation $AX-XB=C$, in the case when it is without a unique solution. Sufficient conditions for the existence of infinitely many solutions are re-derived; methods for obtaining infinitely many solutions are revisited; some characterizations of the solution set are provided; the results are demonstrated on exact examples which require singularity of the initial equation.