Hypothesis Testing on Invariant Subspaces of Non-Symmetric Matrices with Applications to Network Statistics

We extend the inference procedure for eigenvectors of Tyler (1981), which assumes symmetrizable matrices to generic invariant and singular subspaces of non-diagonalisable matrices to test whether {code} is an element of an invariant subspace of {code}. Our results include a Wald test for full-vector hypotheses and a t-test for coefficient-wise hypotheses. We employ perturbation expansions of invariant subspaces from Sun (1991) and singular subspaces from Liu et al. (2007). Based on the former, we extend the popular Davis-Kahan bound to estimations of its higher-order polynomials and study how the bound simplifies for eigenspaces but attains complexity for generic invariant subspaces. We apply our methods to obtain standard errors for subspace-based network statistics and degree centrality scores, when links between nodes are measured with error. We further derive convergence rates of these statistics when a network estimator has known {code}. We also derive the convergence rate for both node-wise and global clustering coefficients. Finally, we establish a formula for the density of network centrality scores based on finite-rank approximations of graphons.