We study the Smith forms of matrices form f ( C g ) where t , ∈ R [ ] is an elementary divisor domain and companion matrix (monic) polynomial . Prominent examples such are circulant matrices, skew-circulant triangular Toeplitz matrices. In particular, we reduce calculation to that F G quotients by some common divisor. This allows us express last non-zero determinantal as a resultant. A key tool...