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.

翻译：本文研究当具有单一特征值的矩阵受到扰动时，其不变子空间的变化情况。我们重点关注与相同阶若尔当块对应的特征值发生扰动时的不变子空间。不变子空间可表示为满列矩阵的值域。我们利用分式阶次刻画了此类矩阵中各分块对应的扰动特征，并给出了零阶和一阶分式阶次对应的系数矩阵计算公式。该结果推广了现有的标准不变子空间扰动理论。