  Given a matrix $A$ and iteration step $k$, we study a best possible
attainable upper bound on the GMRES residual norm that does not depend on the
initial vector $b$. This quantity is called the worst-case GMRES approximation.
We show that the worst case behavior of GMRES for the matrices $A$ and $A^T$ is
the same, and we analyze properties of initial vectors for which the worst-case
residual norm is attained. In particular, we show that such vectors satisfy a
certain "cross equality", and we characterize them as right singular vectors of
the corresponding GMRES residual matrix. We show that the worst-case GMRES
polynomial may not be uniquely determined, and we consider the relation between
the worst-case and the ideal GMRES approximations, giving new examples in which
the inequality between the two quantities is sharp at all iteration steps
$k\geq 3$. Finally, we give a complete characterization of how the values of
the approximation problems in the context of worst-case and ideal GMRES for a
real matrix change, when one considers complex (rather than real) polynomials
and initial vectors in these problems.
