A number of random matrix ensembles permitting exact determination of their eigenvalue and eigenvector statistics maintain this property under a rank $1$ perturbation. Considered in this review are the additive rank $1$ perturbation of the Hermitian Gaussian ensembles, the multiplicative rank $1$ perturbation of the Wishart ensembles, and rank $1$ perturbations of Hermitian and unitary matrices giving rise to a two-dimensional support for the eigenvalues. The focus throughout is on exact formulas, which are typically the result of various integrable structures. The simplest is that of a determinantal point process, with others relating to partial differential equations implied by a formulation in terms of certain random tridiagonal matrices. Attention is also given to eigenvector overlaps in the setting of a rank $1$ perturbation.

许多随机矩阵系综允许精确确定其特征值和特征向量统计量，从而在一个等级下保持此属性$1$扰动。本次审查考虑的是加性排名$1$埃尔米特高斯系综的扰动，乘法阶$1$Wishart 系综的扰动和等级$1$埃尔米特矩阵和酉矩阵的扰动产生了特征值的二维支持。整个重点是精确的公式，这些公式通常是各种可积结构的结果。最简单的是行列式点过程，其他过程则与某些随机三对角矩阵的公式所隐含的偏微分方程有关。还应注意排名设置中的特征向量重叠$1$扰动。