Invariant subspace perturbation of a matrix with Jordan blocks

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.