Horn's problem deals with the following question: what can be said about the spectrum of eigenvalues of the sum C=A+B of two Hermitian matrices of given spectrum ? The support of the spectrum of C is well understood, after a long series of works from Weyl (1912)  to Klyachko (1998) and Knutson and Tao (1999).

 

In this talk, after a short review of the problem,I show how to compute the probability distribution function of the eigenvalues of C, when A and B areindependently and uniformly distributed on their orbit under the action of the unitary group. Comparison with the similar problem for real symmetric matrices and the action of the orthogonal group reveals unexpected differences…