On the Sylvester-like matrix equation $AX+f(X)B=C$
  Many applications in applied mathematics and control theory give rise to the unique solution of a Sylvester-like matrix equation associated with an underlying structured matrix operator $f$. In this paper, we will discuss the solvability of the Sylvester-like matrix equation through an auxiliary standard (or generalized) Sylvester matrix equation. We also show that when this Sylvester-like matrix equation is uniquely solvable, the closed-form solutions can be found by using previous result. In addition, with the aid of the Kronecker product some useful results of the solvability of this matrix equation are provided. 
Chun-Yueh Chiang
math.NA