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

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 © 2005 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]