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.
