Let $A$ and $B$ be $n\times n$ 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. The set of matrices $X$ with spectrum ${\alpha_1, \dots , \alpha_n}$ is an orbit $\mathcal O_\alpha$ for the action of the unitary group $U(n)$ on the space of $n \times n$ Hermitian matrices. Assume that the random matrix $X$ is uniformly distributed on $\mathcal O_\alpha$, and, independently, the random matrix $Y$ is uniformly distributed on $\mathcal O_\beta$. The probabilistic Horn’s problem is to determine the joint distribution of the eigenvalues of the sum $Z = X + Y$ . This joint distribution involves the projection of orbital measures on the subspace of diagonal matrices. It is also related to the Littlewood-Richardson coefficients.
De Salem Ben Said
De Patrick Delorme
De Salma Nasrin
De Genkai Zhang
De Andreas Juhl
De Reiko Miyaoka
De Ali Baklouti
De Yoshiki Oshima
De Dan Barbasch
De Kazuki Kannaka
De Jan Frahm
De Angela Pasquale
De Yves André
De Vasudevan Srinivas
