A refinement of Horn’s conjecture

The study of the relationships between the eigenvalues of two matrices and the eigenvalues of the sum of these two matrices is a classical problem appearing, for example, in physics or numerical analysis. In the course of the 20th century, a certain number of inequalities were exhibited to characterise the eigenvalues of the sum of two Hermitian matrices and, in 1962, A. Horn put forward the hypothesis that all these relationships could be described recursively. We will look at some examples of such relations, a theorem answering Horn's conjecture, the idea of its proof and two improvements on this result.