#### 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 ${\alpha }_{1},\dots ,{\alpha }_{n}$ of $A$ are known, as well as the eigenvalues ${\beta }_{1},\dots ,{\beta }_{n}$ 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 $\left\{{\alpha }_{1},\dots ,{\alpha }_{n}\right\}$ is an orbit ${\mathsc{O}}_{\alpha }$ for the natural action of the unitary group $U\left(n\right)$ on the space of $n×n$ Hermitian matrices. Assume that the random Hermitian matrix $X$ is uniformly distributed on the orbit ${\mathsc{O}}_{\alpha }$ and, independently, the random Hermitian matrix $Y$ is uniformly distributed on ${\mathsc{O}}_{\beta }$. 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.

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##### 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