Spectral clustering is one of the most popular methods for data clustering, and its performance is determined by the quality of the eigenvectors of the related graph Laplacian. Generally, graph Laplacian is constructed using the full features, which will degrade the quality of the related eigenvectors when there are a large number of noisy or irrelevant features in datasets. To solve this problem, we propose a novel unsupervised feature selection method inspired by perturbation analysis theory, which discusses the relationship between the perturbation of the eigenvectors of a matrix and its elements' perturbation. We evaluate the importance of each feature based on the average L1 norm of the perturbation of the first k eigenvectors of graph Laplacian corresponding to the k smallest positive eigenvalues, with respect to the feature's perturbation. Extensive experiments on several high-dimensional multi-class datasets demonstrate the good performance of our method compared with some state-of-the-art unsupervised feature selection methods.