Matrix decompositions using displacement rank and classes of commutative matrix algebras

Using the notion of displacement rank, we look for a unifying approach to representations of a matrix A as sums of products of matrices belonging to commutative matrix algebras. These representations are then considered in case A is the inverse of a Toeplitz or a Toeplitz plus Hankel matrix. Some well-known decomposition formulas for A (Gohberg-Semencul or Kailath et al., Gader, Bini-Pan, and Gohberg-Olshevsky) turn out to be special cases of the above representations. New formulas for A in terms of algebras of symmetric matrices are studied, and their computational aspects are discussed.