The arnoldi process, short recursions, and displacement ranks Original Research Article

We introduce two new methods for generating an orthonormal basis of a Krylov subspace spanned by the columns of Kn(A, r) = (r, Ar, …, An − 1r) with A an arbitrary matrix. We show that both of the methods produce short recursions when A−H = p(A), where p(λ) is a polynomial of low degree. This property of A also leads to the low-displacement-rank structure of the cross-product matrix Kn(A, r)HKn(A, r).