Title: Dynamics of a rank-one perturbation of a Hermitian matrix Show Chinese title

Title: 一个厄米矩阵的一阶扰动的动力学

Abstract: We study the eigenvalue trajectories of a time dependent matrix $ G_t = H+i t vv^*$ for $t \geq 0$, 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. Show Chinese abstract

Abstract: 我们研究时间依赖矩阵$ G_t = H+i t vv^*$的特征值轨迹，对于$t \geq 0$，其中$H$是一个$N \times N$厄米随机矩阵，$v$是一个单位向量。 特别是，我们证明在高概率下，可以在所有时间$t>1+N^{-1/3+\epsilon}$识别出一个异常值，对于任何$\epsilon>0$。 对这一自然过程的研究结合了厄米和非厄米分析的元素，并展示了（即使是很弱的）非厄米矩阵内在不稳定性的一些方面。