2015/09/30 by Sean O’Rourke, Sean O'Rourke, O'Rourke, Sean +2
Mathematics · #60B20 #Advanced Algebra and Geometry #Advanced Combinatorial Mathematics #FOS: Mathematics #Probability (math.PR) #Random Matrices and Applications #math.PR #msc:60B20
paper · pdf · doi:10.48550/arxiv.1510.00039
49 pages, 5 figures
openalex publication_date 2015/09/30 · arxiv created 2016/04/20 · arxiv updated 2016/04/21 · openalex created_date 2016/06/24 · openalex updated_date 2026/07/28
We consider the eigenvalues and eigenvectors of matrices of the form M + P, where M is an n by n Wigner random matrix and P is an arbitrary n by n deterministic matrix with low rank. In general, we show that none of the eigenvalues of M + P need be real, even when P has rank one. We also show that, except for a few outlier eigenvalues, most of the eigenvalues of M + P are within 1/n of the real line, up to small order corrections. We also prove a new result quantifying the outlier eigenvalues for multiplicative perturbations of the form S ( I + P ), where S is a sample covariance matrix and I is the identity matrix. We extend our result showing all eigenvalues except the outliers are close to the real line to this case as well. As an application, we study the critical points of the characteristic polynomials of nearly Hermitian random matrices.