The QR iteration method for quasiseparable matrices

Let {ak}, k = 1, . . . , N be a family of matrices of sizes rk × rk−1. For positive integers i, j, i > j define the operation aij as follows: a × ij = ai−1 · · ·aj+1 for i > j + 1, aj+1,j = Irj . Let {bk}, k = 1, . . . , N be a family of matrices of sizes rk−1 × rk. For positive integers i, j, j > i define the operation bij as follows: b × ij = bi+1 · · · bj−1 for j > i+ 1, bi,i+1 = Iri . It is easy to see that aik = a × ija × j+1,k, i > j ≥ k (2.1)