A. Horn’s inequalities [Pac. J. Math. 12, 225-241 (1962; Zbl 0112.01501)] are about the sum of some of the eigenvalues of \(A\) and \(B\) and some of \(C\) where \(C\) is the sum of two Hermitian \(n\times n\) matrices \(A\) and \(B\). This interesting expository article describes the development from the 1960s till the end of the century where a conjecture of Horn [loc. cit.] were solved by A. A. Klyachko [Sel. Math., New Ser. 4, No. 3, 419-445 (1998; Zbl 0915.14010) and by A. Knutson and T. Tao [J. Am. Math. Soc. 12, No. 4, 1055-1090 (1999; Zbl 0944.05097)].
It is a consequence of the additivity of the trace that the sum of all the eigenvalues of \(C\) and the sum of those of \(A\) and \(B\) are the same. Also by inspection of the corresponding quadratic forms one observes that the largest eigenvalue of \(C\) is dominated by the sum of the largest eigenvalues of \(A\) and \(B\).
Now arrange the eigenvalues of \(A\), \(B\) and \(C\) in a decreasing order. Horn’s inequalities deal with the question: For which triplets \(I\), \(J\) and \(K\) consisting of equally many natural numbers between 1 and \(n\) is true that the sum of the eigenvalues of \(C\) with indices from \(I\) is dominated by the sum of the eigenvalues of \(A\) with indices from \(J\) and the eigenvalues of \(B\) with indices from \(K\).
In a very lucid style the reader is taken on a route from the elementary facts mentioned above through Weyl’s inequalities, Schur’s and Fan’s inequalities, Lidskii and Wielandt’s inequalities and finally to Horn’s inequalities and the corresponding conjecture. The solution is beautiful: Fix \(A\) and \(B\). The eigenvalues of \(A+U^*BU\) as \(U\) varies over the unitary matrices is a convex polyhedron. This convex polyhedron is described by Horn’s inequalities. These inequalities can be obtained by an inductive procedure.
Finally the solution of the Horn conjecture and its connections to quantum Schubert calculus in algebraic geometry and irreducible representations of \(\text{GL}(n)\) are described.