Short recurrences for computing extended Krylov bases for Hermitian and unitary matrices

It is well known that the projection of a matrix A onto a Krylov subspace span { h, Ah, Ah, . . . , Ak−1h } results in a matrix of Hessenberg form. We show that the projection of the same matrix A onto an extended Krylov subspace, which is of the form span { A−krh, . . . , A−2h, A−1h,h, Ah, Ah . . . , A`h } , is a matrix of so-called extended Hessenberg form which can be characterized uniquely by its QR-factorization. In case A is a Hermitian or unitary matrix, this extended Hessenberg matrix is banded, resulting in short recurrence relations. For the unitary case, coupled two term recurrence relations are derived of which the coefficients capture all information necessary for a sparse factorization of the corresponding extended Hessenberg matrix. This generalizes the approach used by Watkins to retrieve the CMV-form for unitary matrices.