Parameterized solution to generalized Sylvester matrix equation

The paper considers the parameterized solution to generalized Sylvester matrix equation. A set of matrix computation formulas have been developed. Using them, formulas for converting generalized Sylvester matrix mappings into conventional linear mappings are obtained. Based on this equation, an easily computable numerical algorithm for complete parameterized solutions of generalized Sylvester matrix equation is provided. The standard Sylvester matrix equation AX − EXF = BY and its dual equation XA − FXE = YC are considered as its special cases and then the corresponding solutions can be produced easily. Some further properties are also investigated. Comparing with existing algorithms, new approach simplified the computation significantly.