Inference on eigenvectors of non-symmetric matrices

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.