Hypothesis testing on invariant subspaces of non-symmetric matrices with applications to network statistics

J\'er\^ome R. Simons

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Abstract: 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 $\nu \in \mathbb{R}^{p \times r}$ is an element of an invariant subspace of $M \in \mathbb{R}^{p \times p}$. 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.

Date: 2023-03, Revised 2025-05
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