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 ; for example, H. Weyl gave some examples as early as 1912. In 1962, A. Horn put forward the hypothesis that all these relationships could be described recursively. The Horn conjecture was solved by A. Klyachko and by A. Knutson and T. Tao at the end of the last century. We will look at some examples of such relations between eigenvalues, a theorem answering Horn's conjecture, the idea of its proof and two refinements of this result.
