Optimal expansion of subspaces for eigenvector approximations

We consider the problem of how to expand a given subspace for approximating an eigenvalue and eigenvector of a matrix A. Specifically, we consider which vector in the subspace, after multiplied by A, provides optimal expansion of the existing subspace for the eigenvalue problem. We determine the optimal vector, when the quality of subspace for approximation is measured by the angle between the subspace and the eigenvector. We have also derived some characterization of the angle that might lead to more practically useful choice of the expansion vector.