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

## TL;DR

This paper computes the probability density functions of eigenvalues for sums of random Hermitian matrices, providing explicit results for small sizes and exploring patterns in symmetric and skew-symmetric cases through numerical experiments.

## Contribution

It explicitly derives eigenvalue PDFs for sums of random Hermitian matrices for small sizes and investigates numerical patterns in symmetric and skew-symmetric cases.

## Key findings

- Explicit eigenvalue PDFs for small n Hermitian matrices.
- Numerical patterns of enhancement in symmetric and skew-symmetric cases.
- Comparison of theoretical results with numerical experiments.

## 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 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.