Tunisian Journal of Mathematics

Horn's problem and Fourier analysis

Jacques Faraut

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Let A and B be two n×n Hermitian matrices. Assume that the eigenvalues α1,,αn of A are known, as well as the eigenvalues β1,,βn of B. What can be said about the eigenvalues of the sum C=A+B? This is Horn’s problem. We revisit this question from a probabilistic viewpoint. The set of Hermitian matrices with spectrum {α1,,αn} is an orbit Oα for the natural action of the unitary group U(n) on the space of n×n Hermitian matrices. Assume that the random Hermitian matrix X is uniformly distributed on the orbit Oα and, independently, the random Hermitian matrix Y is uniformly distributed on Oβ. We establish a formula for the joint distribution of the eigenvalues of the sum Z=X+Y. The proof involves orbital measures with their Fourier transforms, and Heckman’s measures.

Article information

Tunisian J. Math., Volume 1, Number 4 (2019), 585-606.

Received: 16 April 2018
Revised: 5 September 2018
Accepted: 19 September 2018
First available in Project Euclid: 18 December 2018

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 15A18: Eigenvalues, singular values, and eigenvectors 15A42: Inequalities involving eigenvalues and eigenvectors
Secondary: 22E15: General properties and structure of real Lie groups 42B37: Harmonic analysis and PDE [See also 35-XX]

eigenvalue Horn's problem Heckman's measure


Faraut, Jacques. Horn's problem and Fourier analysis. Tunisian J. Math. 1 (2019), no. 4, 585--606. doi:10.2140/tunis.2019.1.585. https://projecteuclid.org/euclid.tunis/1545102024

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