  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.
