On the Sylvester-like matrix equation $AX+f(X)B=C$
For researchers in applied mathematics and control theory, this offers theoretical insights into a specialized matrix equation, but the contribution is incremental.
The paper addresses the solvability of a Sylvester-like matrix equation with a structured operator f, providing conditions for unique solvability and closed-form solutions via an auxiliary Sylvester equation. No concrete numerical results are reported.
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.