  We investigate how invariant subspaces will change when a matrix with a
single eigenvalue is perturbed. We focus on the case when an invariant subspace
corresponds to the eigenvalues perturbed from those associated with the same
order Jordan blocks. An invariant subspace can be expressed as the range of a
full column matrix. We characterize the perturbations in terms of fractional
orders for the blocks of such a matrix. We also provide the formulas for the
coefficient matrices associated with the zero and first fractional orders. The
results generalize the existing standard invariant subspace perturbation
theory.
