The eigenvalues and eigenvectors of finite-rank normal perturbations of large rotationally invariant non-Hermitian matrices
Pierre Bousseyroux, Marc Potters
TL;DR
The paper addresses the BBP-type outlier phenomenon for non-Hermitian matrices by studying $M = A + T$ with $T = \sum_{k=1}^n s_k u_k u_k^*$ and $A$ a large rotationally invariant non-Hermitian matrix. It develops a unified approach using the Stieltjes transform $g_A(z)$ and $R$-transforms to derive explicit outlier conditions $g_A(z) = 1/s_k$ and outlier locations $z = s_k + R_{2,A}(0, 1/s_k)$, together with eigenvector overlaps $|\langle u_k, \varphi_k\rangle|^2 = 1 - \frac{\partial_\alpha R_{1,A}(0,1/s_k)}{|s_k|^2}$. Fluctuations of the outliers are Gaussian with variance $\sigma^2/N$, where $\sigma^2$ is given by $\frac{\partial_{\alpha} R_{1,A}(0, 1/s_k) |s_k^2 - \partial_{\beta} R_{2,A}(0, 1/s_k)|^2}{|s_k|^4 - \partial_{\alpha} R_{1,A}(0, 1/s_k) |s_k|^2}$. The framework recovers Hermitian and bi-invariant cases and extends to complex perturbations, providing a unified treatment of outliers and eigenvectors for a broad class of non-Hermitian random matrices.
Abstract
We study finite-rank normal deformations of rotationally invariant non-Hermitian random matrices. Extending the classical Baik-Ben Arous-Péché (BBP) framework, we characterize the emergence and fluctuations of outlier eigenvalues in models of the form $\mathbf{A} + \mathbf{T}$, where $\mathbf{A}$ is a large rotationally invariant non-Hermitian random matrix and $\mathbf{T}$ is a finite-rank normal perturbation. We also describe the corresponding eigenvector behavior. Our results provide a unified framework encompassing both Hermitian and non-Hermitian settings, thereby generalizing several known cases.