Solvability and uniqueness criteria for generalized Sylvester-type equations
Provides a theoretical foundation for solving a class of matrix equations, relevant to researchers in linear algebra and control theory.
The paper establishes necessary and sufficient conditions for the generalized *-Sylvester matrix equation to have a unique solution for any right-hand side, extending previous results to arbitrary rectangular coefficient matrices.
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.