Solvability and uniqueness of solution of generalized ★‐Sylvester equations with arbitrary coefficients

Abstract We analyze the consistency and uniqueness of solution of the generalized ‐Sylvester equation , with , and being complex matrices (and being either the transpose or the conjugate transpose). In particular, we obtain characterizations for the equation to have at most one solution and to be consistent for any right‐hand side. Such characterizations are given in terms of spectral properties of the matrix pencils and , respectively. This approach deals with matrices whose size is of the same order as that of , and , contrary to the naive procedure that addresses the equation as a linear system, whose coefficient matrix can be much larger. The characterizations are valid in the most general setting, namely for all coefficient matrices , and for which the equation is well‐defined, and generalize the known characterizations for the case where they are all square. As a corollary, we obtain necessary and sufficient conditions for the ‐Sylvester equation and the ‐Stein equation to have at most one solution or to be consistent, for any right‐hand side .