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Hypothesis testing on invariant subspaces of non-symmetric matrices with applications to network statistics
Jérôme R. Simons
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 ν∈ℝ^(p ×r) is an element of an invariant subspace of M ∈ℝ^(p ×p). 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.
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