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Many applications in applied mathematics and control theory give rise to theunique solution of a Sylvester-like matrix equation associated with anunderlying structured matrix operator $f$. In this paper, we will discuss thesolvability of the Sylvester-like matrix equation through an auxiliary standard(or generalized) Sylvester matrix equation. We also show that when thisSylvester-like matrix equation is uniquely solvable, the closed-form solutionscan be found by using previous result. In addition, with the aid of theKronecker product some useful results of the solvability of this matrixequation are provided.

On the Sylvester-like matrix equation $AX+f(X)B=C$