Abstract. Given an n by n nonsingular matrix A and an n-vector v, we consider the spaces of the form AKk (A, v), k = 1, ... , n, where Kk(A, v) is the kth Krylov space, equal to span{ v, Av, ... , A k -1 v}. We characterize the set of matrices B that, with the given vector v, generate the same spaces; i.e., those matrices B for which BKk (B,v) = AKk(A,v), for all k = 1, ... ,n. It is shown that any such sequence of spaces can be generated by a unitary matrix. If zero is outside the field of values of A, then there is a Hermitian positive definite matrix that generates the same spaces, and, moreover, if A is close to Hermitian then there is a nearby Hermitian matrix that generates the same spaces. It is also shown that any such sequence of spaces can be generated by a matrix having any desired eigenvalues.Implications about the convergence rate of the GMRES method are discussed. A new proof is given that if zero is outside the field of values of A, then convergence of the GMRES algorithm is strictly monotonic. It is shown that if A is close to Hermitian, then the eigenvalues of A essentially determine the behavior of the GMRES iteration but that, in general, eigenvalue information alone is never sufficient to ensure rapid convergence of the GMRES algorithm.