Rational Krylov matrices and QR steps on Hermitian diagonal-plus-semiseparable matrices

摘要

We prove that the unitary factor appearing in the QR factorization of a suitably defined rational Krylov matrix transforms a Hermitian matrix having pairwise distinct eigenvalues into a diagonal-plus-semiseparable form with prescribed diagonal term. This transformation is essentially uniquely defined by its first column. Furthermore, we prove that the set of Hermitian diagonal-plus-semiseparable matrices is invariant under QR iteration. These and other results are shown to be the rational counterpart of known facts involving structured matrices related to polynomial computations. Copyright (c) 2005 John Wiley & Sons, Ltd.