On the distribution of the spectrum of the sum of two hermitian or real symmetric matrices

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Abstract

Let A, B and C = U1 A U1* + U2 B U2* be hermitian (or real symmetric) matrices, where U1 and U2 are unitary (respectively orthogonal). Our goal is to develop as concrete as possible expression to the probability distribution of the spectrum of C where Ui are drawn from the unitary (respectively orthogonal) group with respect to the Haar measure. While we do this for the unitary group, using representation theory of Un, we can only accomplish the same for the real symmetric case where B has rank 1, by performing explicit calculations. Here is what we do in the unitary case. For a given n by n matrix A over the field of complex numbers we study the operator EAN = ∫ u A u*⊗ N d u, where the integration is taken over the unitarian group with respect to the Haar measure. When A has nonnegative spectrum, we show that EAN, as N tends to infinity, is concentrated around some simple SN × Un submodule of (Cn)⊗ N determined only by the spectra of A. Along the proof we reprove a generalization of Heckman to the convexity theorem of Horn. The Schur-Weyl duality and the technique of Bernstein polynomials approximations are our tools. Using this we compute the distribution, induced by a sum of two hermitian orbits on the set of hermitian orbits in terms of asymptotic Littlewood-Richardson coefficients times an asymptotic version of the Weyl dimensional formula. More precisely, translation of the lattice permutation and semistandard Young tableaux rules to a set of linear inequalities, enable us to express the density of the distribution above as multiplication of volumes of two concrete polytopes.

Original languageEnglish
Pages (from-to)268-286
Number of pages19
JournalAdvances in Applied Mathematics
Volume37
Issue number2
DOIs
StatePublished - Aug 2006

Keywords

  • Eigenvalues
  • Hermitian matrices
  • Measure
  • Unitary representations
  • Young tableaux

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