Journal of Statistical Physics

, Volume 149, Issue 2, pp 246–258 | Cite as

On Eigenvalues of the Sum of Two Random Projections



We study the behavior of eigenvalues of matrix P N +Q N where P N and Q N are two N-by-N random orthogonal projections. We relate the joint eigenvalue distribution of this matrix to the Jacobi matrix ensemble and establish the universal behavior of eigenvalues for large N. The limiting local behavior of eigenvalues is governed by the sine kernel in the bulk and by either the Bessel or the Airy kernel at the edge depending on parameters. We also study an exceptional case when the local behavior of eigenvalues of P N +Q N is not universal in the usual sense.


Random matrices Random projections Eigenvalues Universality Tracy–Widom distribution Sine kernel Airy kernel 


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© Springer Science+Business Media, LLC 2012

Authors and Affiliations

  1. 1.Statistical Laboratory, Department of MathematicsUniversity of CambridgeCambridgeUK

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