Eigenvector Under Random Perturbation: A Nonasymptotic Rayleigh-Schrödinger Theory

Abstract

Rayleigh-Schrödinger perturbation theory is a well-known theory in quantum mechanics and it offers useful characterization of eigenvectors of a perturbed matrix. Suppose $A$ and perturbation $E$ are both Hermitian matrices, $A^t=A+tE$, $\{{\lambda }_j{\}}_{j=1}^n$ are eigenvalues of $A$ in descending order, and $u_1,u_1^t$ are leading eigenvectors of $A$ and $A^t$. Rayleigh-Schrödinger theory shows asymptotically, $⟨u_1^t,u_j⟩\propto t/({\lambda }_1-{\lambda }_j)$ where $t=o(1)$. However, the asymptotic theory does not apply to larger $t$; in particular, it fails when $t\Vert E{\Vert }_2>{\lambda }_1-{\lambda }_2$. In this paper, we present a nonasymptotic theory with $E$ being a random matrix. We prove that, when $t=1$ and $E$ has independent and centered subgaussian entries above its diagonal, with high probability, \begin{equation*} | \langle u^1_1, u_j \rangle | = O(\sqrt{\log n} / (λ_1 - λ_j)), \end{equation*} for all $j>1$ simultaneously, under a condition on eigenvalues of $A$ that involves all gaps ${\lambda }_1-{\lambda }_j$. This bound is valid, even in cases where $\Vert E{\Vert }_2\gg {\lambda }_1-{\lambda }_2$. The result is optimal, except for a log term. It also leads to an improvement of Davis-Kahan theorem.

36 pages

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