Let \(H_{n}\) be the set of all \(n\times n\) Hermitian matrices. Let \[ \alpha =(\alpha _{1},\alpha _{2},\dots ,\alpha _{n}) \] be a \(n\)-tuple of eigenvalues of \(A\in H_{n}\), where \(\alpha _{1}\geq \alpha_{2}\geq \dots \geq \alpha _{n}\), and let \(\underline{\alpha}\) be the diagonal matrix \(\underline{\alpha}=\mathrm{diag}(\alpha _{1},\alpha _{2},\dots,\alpha _{n})\). By \(\Omega _{\alpha }\) we will denote the orbit of \(\underline{\alpha}\), i.e., the set of all matrices in \(H_{n}\) with eigenvalues \(\alpha _{1},\alpha _{2},\ldots ,\alpha _{n}\).
Let \(\alpha \) and \(\beta \) be two \(n\)-tuples ordered as above and let \(A\in \Omega _{\alpha }\) and \(B\in \Omega _{\beta }\). Horn’s conjecture about the set of necessary and sufficient inequalities to be satisfied to assure that \(\gamma \) belongs to the spectrum of the matrix \(C=A+B\) was proved by A. Knutson and T. Tao [J. Am. Math. Soc. 12, No. 4, 1055–1090 (1999; Zbl 0944.05097); Notices Am. Math. Soc. 48, No. 2, 175–186 (2001; Zbl 1047.15006)].
Here, the authors present a formula for the probability density function of the eigenvalues of \(C=A+B\) where \(A\) and \(B\) are random matrices of \(H_{n}\), uniformly and independently distributed on their orbits. As noted by the authors, this formula may be known in the literature. Therefore the main original results of the paper are mostly calculations for some low-dimension cases, i.e., \(n=2,3,4,5\).
In the second part of the paper, the authors restrict their attention to real symmetric matrices and real skew-symmetric matrices.