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 $\nu \in \mathbb{R}^{p \times r}$ 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.

@article{simons2025_2303.18233,
  title={ Hypothesis testing on invariant subspaces of non-symmetric matrices with applications to network statistics },
  author={ Jérôme R. Simons },
  journal={arXiv preprint arXiv:2303.18233},
  year={ 2025 }
}