  It is well-established that any non-increasing convergence curve is possible
for GMRES and a family of pairs $(A,b)$ can be constructed for which GMRES
exhibits a given convergence curve with $A$ having arbitrary spectrum. No
analog of this result has been established for block GMRES, wherein multiple
right-hand sides are considered. By reframing the problem as a single linear
system over a ring of square matrices, we develop convergence results for block
Arnoldi and block GMRES. In particular, we show what convergence behavior is
admissible for block GMRES and how the matrices and right-hand sides producing
any admissible behavior can be constructed. Moreover, we show that the
convergence of the block Arnoldi method for eigenvalue approximation can be
almost fully independent of the convergence of block GMRES for the same
coefficient matrix and the same starting vectors.
