Dynamics of a rank-one perturbation of a Hermitian matrix

Abstract

We study the eigenvalue trajectories of a time dependent matrix $G_t=H+itv{v}^{\ast }$ for $t\ge 0$, where $H$ is an $N\times N$ Hermitian random matrix and $v$ is a unit vector. In particular, we establish that with high probability, an outlier can be distinguished at all times $t>1+N^{-1/3+\epsilon }$, for any $\epsilon >0$. The study of this natural process combines elements of Hermitian and non-Hermitian analysis, and illustrates some aspects of the intrinsic instability of (even weakly) non-Hermitian matrices.

arXiv comment14 pages, 2 figures

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Cited by in corpus (3)

• Extreme eigenvalues and the emerging outlier in rank-one non-Hermitian deformations of the Gaussian Unitary Ensemble
• Multiplicative non-Hermitian perturbations of classical $\beta$-ensembles
• Dynamics of a rank-one multiplicative perturbation of a unitary matrix