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Inventiones mathematicae

, Volume 190, Issue 3, pp 647–697 | Cite as

Eigenvectors and eigenvalues in a random subspace of a tensor product

  • Serban Belinschi
  • Benoît Collins
  • Ion Nechita
Article

Abstract

Given two positive integers n and k and a parameter t∈(0,1), we choose at random a vector subspace V n ⊂ℂ k ⊗ℂ n of dimension Ntnk. We show that the set of k-tuples of singular values of all unit vectors in V n fills asymptotically (as n tends to infinity) a deterministic convex set K k,t that we describe using a new norm in ℝ k .

Our proof relies on free probability, random matrix theory, complex analysis and matrix analysis techniques. The main result comes together with a law of large numbers for the singular value decomposition of the eigenvectors corresponding to large eigenvalues of a random truncation of a matrix with high eigenvalue degeneracy.

Mathematics Subject Classification (2000)

15A52 52A22 46L54 

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Copyright information

© Springer-Verlag 2012

Authors and Affiliations

  • Serban Belinschi
    • 1
    • 2
  • Benoît Collins
    • 3
    • 4
  • Ion Nechita
    • 3
  1. 1.Institute of Mathematics “Simion Stoilow” of the Romanian AcademyBucharestRomania
  2. 2.Department of Mathematics & StatisticsUniversity of SaskatchewanSaskatoonCanada
  3. 3.Département de Mathématique et StatistiqueUniversité d’OttawaOttawaCanada
  4. 4.CNRSInstitut Camille Jordan Université Lyon 1VilleurbanneFrance

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