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)中的对称化条件并非建立特征向量渐近推断程序的必要条件。我们分别为全向量假设和单个系数假设建立了Wald检验和t检验的分布理论。该检验统计量源自非对称矩阵的特征投影。将投影视为从基础矩阵到其谱数据的映射，我们通过解析摄动理论求得导数。这些结果证明了Sun (1991)的解析摄动理论在多变量统计中的实用性，并具有独立研究价值。作为应用实例，我们为有向图邻接矩阵估计的Bonacich中心性构建了置信集。