  Matrices whose adjoint is a low rank perturbation of a rational function of
the matrix naturally arise when trying to extend the well known
Faber-Manteuffel theorem, which provides necessary and sufficient conditions
for the existence of a short Arnoldi recurrence. We show that an orthonormal
Krylov basis for this class of matrices can be generated by a short recurrence
relation based on GMRES residual vectors. These residual vectors are computed
by means of an updating formula. Furthermore, the underlying Hessenberg matrix
has an accompanying low rank structure, which we will investigate closely.
