Horn's problem and Fourier analysis

Abstract

Let $A$ and $B$ be two $n\times n$ 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 ${\mathcal{O}}_{\alpha }$ for the natural action of the unitary group $U\left(n\right)$ on the space of $n\times n$ Hermitian matrices. Assume that the random Hermitian matrix $X$ is uniformly distributed on the orbit ${\mathcal{O}}_{\alpha }$ and, independently, the random Hermitian matrix $Y$ is uniformly distributed on ${\mathcal{O}}_{\beta }$. 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.