Eigenvectors of deformed wigner random matrices

Haddadi, F ; Sharif University of Technology | 2021

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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 , 2021
  4. Abstract:
  5. We investigate eigenvectors of rank-one deformations of random matrices boldsymbol B = boldsymbol A + theta boldsymbol {uu}{} in which boldsymbol A in mathbb R{N times N} is a Wigner real symmetric random matrix, theta in mathbb R{+} , and boldsymbol u is uniformly distributed on the unit sphere. It is well known that for theta > 1 the eigenvector associated with the largest eigenvalue of boldsymbol B closely estimates boldsymbol u asymptotically, while for theta < 1 the eigenvectors of boldsymbol B are uninformative about boldsymbol u. We examine mathcal O({1}/{N}) correlation of eigenvectors with boldsymbol u before phase transition and show that eigenvectors with larger eigenvalue exhibit stronger alignment with deforming vector through an explicit inverse law {1}/{theta {}-x} with theta {}:= theta + ({1}/{theta }). 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 boldsymbol B. We use a combinatorial approach to prove the result. We also extend the result to constant rank-r deformations. © 1963-2012 IEEE
  6. Keywords:
  7. 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 ; Eigenvalues and eigenfunctions
  8. Source: IEEE Transactions on Information Theory ; Volume 67, Issue 2 , 2021 , Pages 1069-1079 ; 00189448 (ISSN)
  9. URL: https://ieeexplore.ieee.org/document/9262963