Parametric solutions to rectangular high-order Sylvester equations—Case of F arbitrary

In this paper a class of high-order rectangular Sylvester equations are considered. It is first shown that this type of Sylvester equations cover many such types of equations appeared in the literature. Then, based on Smith form reduction of polynomial matrices, a general complete parametric solution to the proposed type of Sylvester equations are proposed, which possesses a very neat explicit closed form. The primary feature of this solution is that the matrix F does not need to be in any canonical form, or may be even unknown a priori, and thus may be set undetermined and used as degrees of freedom beyond the completely free parameter matrix Z. For the nonhomogeneous case, the matrix R is also allowed to be set undetermined and used as a part of the degrees of freedom.