Comparison theorems for the minimum eigenvalue of a random positive-semidefinite matrix
Abstract
This paper establishes a new comparison principle for the minimum eigenvalue of a sum of independent random positive-semidefinite matrices. The principle states that the minimum eigenvalue of the matrix sum is controlled by the minimum eigenvalue of a Gaussian random matrix whose statistics are determined by the summands. This methodology is powerful because of the vast arsenal of tools for treating Gaussian random matrices. As applications, the paper presents short, conceptual proofs of some old and new results in high-dimensional statistics. It also settles a long-standing open question in computational linear algebra about the injectivity properties of very sparse random matrices.
Fields of Study & Subjects
Random Matrices and Applications
Mathematical Inequalities and Applications
Point processes and geometric inequalities
Publication details
- Journal / source
- Communications of the American Mathematical Society
- Volume
- 6
- Issue
- 16
- Pages
- 844–895
- Publication date
- 7 July 2026
- Language
- EN