Tunisian Journal of Mathematics
- Tunisian J. Math.
- Volume 1, Number 4 (2019), 585-606.
Horn's problem and Fourier analysis
Let and be two Hermitian matrices. Assume that the eigenvalues of are known, as well as the eigenvalues of . What can be said about the eigenvalues of the sum ? This is Horn’s problem. We revisit this question from a probabilistic viewpoint. The set of Hermitian matrices with spectrum is an orbit for the natural action of the unitary group on the space of Hermitian matrices. Assume that the random Hermitian matrix is uniformly distributed on the orbit and, independently, the random Hermitian matrix is uniformly distributed on . We establish a formula for the joint distribution of the eigenvalues of the sum . The proof involves orbital measures with their Fourier transforms, and Heckman’s measures.
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
Permanent link to this document
Digital Object Identifier
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]
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