On the use of certain matrix algebras associated with trigonometric transforms in matrix displacement decomposition

The authors extend some recent results of Di Fiore and Zellini [Linear Algebra Appl., to appear], obtaining new classes of formulas for the displacement operator-based decomposition of matrices. It is shown how an arbitrary matrix can be expressed as the sum of products of matrices belonging to matrix algebras associated with certain versions of sine and cosine transforms. Applications to the representation of the inverse of a Toeplitz and a Toeplitz plus Hankel matrix, with and without symmetry, are presented. Implications on the computation of the product of these matrices by a vector are discussed.