Horn's problem and Harish-Chandra's integrals. Probability density functions

Abstract

Horn's problem – to find the support of the spectrum of eigenvalues of the sum $C=A+B$ of two $n$ by $n$ Hermitian matrices whose eigenvalues are known – has been solved by Klyachko and by Knutson and Tao. Here the probability distribution function (PDF) of the eigenvalues of $C$ is explicitly computed for low values of $n$, for $A$ and $B$ uniformly and independently distributed on their orbit, and confronted to numerical experiments. Similar considerations apply to skew-symmetric and symmetric real matrices under the action of the orthogonal group. In the latter case, where no analytic formula is known in general and we rely on numerical experiments, curious patterns of enhancement appear.