Skip to content
DraconiticResearch
Search Sources API
203,203 studies31 sourcesUpdated just now
Peer-reviewedCC BY-NC-NDCommunications of the American Mathematical Society

Comparison theorems for the minimum eigenvalue of a random positive-semidefinite matrix

  • Joel A. Tropp
Open at sourceDOICite

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.

Related

  1. Eigenvalue comparison theorems for the Witten-Laplacian and the weighted p-Laplacian on complete manifolds with a modified Ricci curvature bounded from below
    Ruifeng Chen, Qiyue Ma, Jing Mao
    arXivPreprintarXiv licenceFull textGeometry & topologyarXiv
  2. On Completely Positive and Semidefinite Programs for Moment Bounds on Maximum of Random Quadratics
    Buse Unluturk, Divya Padmanabhan, Karthik Natarajan
    arXivPreprintCC BY 4.0Full textOptimisation & controlarXiv
  3. New Pareto Z-eigenvalue localization theorems for tensors with applications
    Licai Shao, Xingong Zhang, Yisheng Song
    arXivPreprintCC BY 4.0Full textOptimisation & controlarXiv
  4. Minimum-Area Enclosing Rectangles and the First Neumann Eigenvalue of Planar Convex Domains
    Xiaolong Li, Jin Sun, Kui Wang
    arXivPreprintCC0Full textAnalysis & PDEsarXiv
  5. Optimal Condition Numbers in Low-Rank Positive Semidefinite Matrix Sensing
    Mingxuan Sun, Zhiqiang Xu
    arXivPreprintarXiv licenceFull textAnalysis & PDEsNumerical analysis & scientific computingarXiv

Cite

BibTeXRIS
BibTeX
@article{tropp2026comparison,
  title = {{Comparison theorems for the minimum eigenvalue of a random positive-semidefinite matrix}},
  author = {Joel A. Tropp},
  journal = {Communications of the American Mathematical Society},
  year = {2026},
  month = jul,
  doi = {10.1090/cams/72},
  url = {https://doi.org/10.1090/cams/72},
  note = {CC BY-NC-ND},
}
RIS
TY  - JOUR
TI  - Comparison theorems for the minimum eigenvalue of a random positive-semidefinite matrix
AU  - Joel A. Tropp
PY  - 2026
DA  - 2026/07/07
JO  - Communications of the American Mathematical Society
DO  - 10.1090/cams/72
UR  - https://doi.org/10.1090/cams/72
AB  - 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.
SN  - 2692-3688
ER  -