Vol. 1, No. 4, 2019

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Horn's problem and Fourier analysis

Jacques Faraut

Vol. 1 (2019), No. 4, 585–606
Abstract

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.

Keywords
eigenvalue, Horn's problem, Heckman's measure
Mathematical Subject Classification 2010
Primary: 15A18, 15A42
Secondary: 22E15, 42B37
Milestones
Received: 16 April 2018
Revised: 5 September 2018
Accepted: 19 September 2018
Published: 14 December 2018
Authors
Jacques Faraut
Institut de Mathematiques de Jussieu
Sobronne Université
Paris
France