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Eigenvectors of deformed Wigner random matrices

Haddadi, F ; Sharif University of Technology | 2020

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  1. Type of Document: Article
  2. DOI: 10.1109/TIT.2020.3039173
  3. Publisher: Institute of Electrical and Electronics Engineers Inc , 2020
  4. Abstract:
  5. We investigate eigenvectors of rank-one deformations of random matrices B = A + θuu* in which A ∈ RN×N is a Wigner real symmetric random matrix, θ ∈ R+, and u is uniformly distributed on the unit sphere. It is well known that for θ > 1 the eigenvector associated with the largest eigenvalue of B closely estimates u asymptotically, while for θ < 1 the eigenvectors of B are uninformative about u. We examine O(1/N) correlation of eigenvectors with u before phase transition and show that eigenvectors with larger eigenvalue exhibit stronger alignment with deforming vector through an explicit inverse law 1/θ*-x with θ* := θ + 1/θ. This distribution function will be shown to be the ordinary generating function of Chebyshev polynomials of the second kind. These polynomials form an orthogonal set with respect to the semicircle weighting function. This law is an increasing function in the support of semicircle law for eigenvalues (-2, +2). Therefore, most of energy of the unknown deforming vector is concentrated in a cN-dimensional (c < 1) known subspace of B. We use a combinatorial approach to prove the result. We also extend the result to constant rank-r deformations. IEEE
  6. Keywords:
  7. Catalan number ; Chebychev polynomial ; Eigenvalues and eigenfunctions ; Eigenvector ; Linear matrix inequalities ; Matrix completion ; Matrix decomposition ; Perturbation methods ; Phase transition ; Random matrix ; Rank-one deformation ; Strain ; Symmetric matrices ; Transforms ; Wigner matrix ; Deformation ; Distribution functions ; Matrix algebra ; Orthogonal functions ; Polynomials ; Chebyshev polynomials of the second kind ; Combinatorial approach ; Generating functions ; Increasing functions ; Largest eigenvalues ; Orthogonal sets ; Random matrices ; Weighting functions
  8. Source: IEEE Transactions on Information Theory ; 18 November , 2020
  9. URL: https://ieeexplore.ieee.org/document/9262963