We generalise the inference procedure for eigenvectors of symmetrizable matrices of Tyler (1981) to that of invariant and singular subspaces of non-diagonalizable matrices. Wald tests for invariant vectors and $t$-tests for their individual coefficients perform well in simulations, despite the matrix being not symmetric. Using these results, it is now possible to perform inference on network statistics that depend on eigenvectors of non-symmetric adjacency matrices as they arise in empirical applications from directed networks. Further, we find that statisticians only need control over the first-order Davis-Kahan bound to control convergence rates of invariant subspace estimators to higher-orders. For general invariant subspaces, the minimal eigenvalue separation dominates the first-order bound potentially slowing convergence rates considerably. In an example, we find that accounting for uncertainty in network estimates changes empirical conclusions about the ranking of nodes' popularity.

翻译：本文将Tyler (1981)关于可对称化矩阵特征向量的推断方法，推广至非对角化矩阵的不变子空间与奇异子空间。针对不变向量的Wald检验及其系数的$t$检验在模拟中表现良好，即使矩阵非对称。基于这些结果，现可对依赖于非对称邻接矩阵特征向量的网络统计量进行推断，此类问题常见于有向网络的实际应用。此外，我们发现统计学家仅需控制一阶Davis-Kahan界，即可控制不变子空间估计量至高阶的收敛速率。对于一般不变子空间，最小特征值间距主导一阶界，可能显著降低收敛速率。通过示例分析，我们发现考虑网络估计中的不确定性会改变关于节点流行度排序的实证结论。