Rank 1 perturbations in random matrix theory — A review of exact results

A number of random matrix ensembles permitting exact determination of their eigenvalue and eigenvector statistics maintain this property under a rank [Formula: see text] perturbation. Considered in this review are the additive rank [Formula: see text] perturbation of the Hermitian Gaussian ensembles, the multiplicative rank [Formula: see text] perturbation of the Wishart ensembles, and rank [Formula: see text] 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 [Formula: see text] perturbation.