Horn’s problem and Fourier analysis

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.