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

TL;DR

This paper investigates the solvability of Sylvester-like matrix equations involving structured operators, providing conditions for unique solutions and deriving closed-form solutions using Kronecker products.

Contribution

It offers new criteria for solvability and explicit solutions of Sylvester-like matrix equations with structured operators, expanding existing theoretical frameworks.

Findings

Conditions for unique solvability are established.

Closed-form solutions are derived using previous results.

Kronecker product techniques are employed to analyze solvability.

Abstract

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.