  This work considers the convergence of GMRES for non-singular problems. GMRES
is interpreted as the GCR method which allows for simple proofs of the
convergence estimates. Preconditioning and weighted norms within GMRES are
considered. The objective is to provide a way of choosing the preconditioner
and GMRES norm that ensure fast convergence. The main focus of the article is
on Hermitian preconditioning (even for non-Hermitian problems). It is proposed
to choose a Hermitian preconditioner H and to apply GMRES in the inner product
induced by H. If moreover, the problem matrix A is positive definite, then a
new convergence bound is proved that depends only on how well H preconditions
the Hermitian part of A, and on how non-Hermitian A is. In particular, if a
scalable preconditioner is known for the Hermitian part of A, then the proposed
method is also scalable. This result is illustrated numerically.
