This paper argues that the symmetrisability condition in Tyler(1981) is not necessary to establish asymptotic inference procedures for eigenvectors. We establish distribution theory for a Wald and t-test for full-vector and individual coefficient hypotheses, respectively. Our test statistics originate from eigenprojections of non-symmetric matrices. Representing projections as a mapping from the underlying matrix to its spectral data, we find derivatives through analytic perturbation theory. These results demonstrate how the analytic perturbation theory of Sun(1991) is a useful tool in multivariate statistics and are of independent interest. As an application, we define confidence sets for Bonacich centralities estimated from adjacency matrices induced by directed graphs.

翻译：本文论证了Tyler(1981)中的对称化条件对于建立特征向量的渐近推断程序并非必要。我们分别为全向量假设和单个系数假设建立了Wald检验与t检验的分布理论。我们的检验统计量源于非对称矩阵的特征投影。通过将投影表示为从底层矩阵到其谱数据的映射，我们利用解析摄动理论求出了其导数。这些结果揭示了Sun(1991)的解析摄动理论作为多元统计学的有用工具，并具有独立的理论价值。作为应用，我们为有向图邻接矩阵估计的Bonacich中心性定义了置信集。