Sylvester matrix and common factors in polynomial matrices

With the coefficient matrices of the polynomial matrices replacing the scalar coefficients in the standard Sylvester matrix, common factors exist if and only if this (generalized) Sylvester matrix is singular and the coefficient matrices commute. If the coefficient matrices do not commute, a necessary and sufficient condition for a common factor to exist is that a submatrix of the ratio (transfer) coefficient matrices is of less than full row rank. Whether coefficient matrices commute or not, a nonsingular (generalized) Sylvester matrix is always a sufficient condition for no common factors to exist. These conditions hold whether common factors are unimodular or not unimodular. These results follow from requiring that in the potential alternative pair of polynomial matrices with the same matrix ratio, i.e. with the same transfer function, all coefficient matrices beyond the given integers p and q are null matrices. Algebraically these requirements take the form of linear equations in the coefficient matrices of the inverse of the potential common factor. Lower block triangular Toeplitz matrices appear in these equations and the sequential inverse of these matrices generates sequentially the coefficient matrices of the inverse of the common factor. The conclusions follow from the properties of infinite dimensional diagonally dominant matrices.