Solvability and uniqueness criteria for generalized Sylvester-type equations

TL;DR

This paper establishes necessary and sufficient conditions for the unique solvability of a generalized Sylvester-type matrix equation with arbitrary coefficients, extending previous results to more general cases.

Contribution

It provides a comprehensive set of criteria for solvability and uniqueness of the generalized $ ext{ extasteriskcentered}$-Sylvester equation with arbitrary matrices, generalizing prior work.

Findings

Derived necessary and sufficient conditions for unique solutions.

Extended existing results to rectangular and non-square matrices.

Reviewed known solvability and uniqueness conditions for related equations.

Abstract

We provide necessary and sufficient conditions for the generalized $\star$-Sylvester matrix equation, $AXB + CX^\star D = E$, to have exactly one solution for any right-hand side E. These conditions are given for arbitrary coefficient matrices $A, B, C, D$ (either square or rectangular) and generalize existing results for the same equation with square coefficients. We also review the known results regarding the existence and uniqueness of solution for generalized Sylvester and $\star$-Sylvester equations.