Additive/multiplicative free subordination property and limiting eigenvectors of spiked additive deformations of Wigner matrices and spiked sample covariance matrices

2011/03/15 by Capitaine, Mireille, Mireille Capitaine
Computer Science · Mathematics · #Advanced Algebra and Geometry #Advanced Combinatorial Mathematics #FOS: Mathematics #Matrix Theory and Algorithms #Probability (math.PR) #Quantum Information and Cryptography #Random Matrices and Applications #math.PR

paper · pdf · doi:10.48550/arxiv.1103.2899

openalex publication_date 2011/03/15 · arxiv created 2012/02/23 · arxiv updated 2012/02/24 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

When some eigenvalues of a spiked multiplicative resp. additive deformation model of a Hermitian Wigner matrix resp. a sample covariance matrix separate from the bulk, we study how the corresponding eigenvectors project onto those of the perturbation. We point out that the inverse of the subordination function relative to the free additive resp. multiplicative convolution plays an important part in the asymptotic behavior.

Citations

• On the distribution of the largest eigenvalue in principal components analysis
• Non-white Wishart ensembles
• Analysis of the Limiting Spectral Distribution of Large Dimensional Random Matrices
• On the Empirical Distribution of Eigenvalues of a Class of Large Dimensional Random Matrices
• Limit laws for Random matrices and free products
• On the Distribution of the Roots of Certain Symmetric Matrices

Related

• Free convolution with a semi-circular distribution and eigenvalues of spiked deformations of Wigner matrices
• Spectral Measures of Spiked Random Matrices
• Central limit theorems for eigenvalues of deformations of Wigner matrices
• The Isotropic Semicircle Law and Deformation of Wigner Matrices
• Eigenvectors of Deformed Wigner Random Matrices