Weyl and Lidskiĭ inequalities for general hyperbolic polynomials
In this paper the author focuses on some linear inequalities satisfied by the roots of hyperbolic polynomials (and by eigenvalues of Hermitian matrices as well). The whole set of such inequalities was first supposed to be true for the case of Hermitian matrices by \textit{A. Horn} [Pac. J. Math. 12, 225--241 (1962; Zbl 0112.01501)]. Horn's conjecture was later proved with some advanced tools, such as actions of Lie groups, Schubert calculus and representations of the group \(U(n)\). The author proposes a new proof for a part of Horn's inequalities, the so-called Weyl and Lidskiĭ ones. This proof is elementary and elegant, it relies on a combinatorial criterion and it is free from representation theory and Schubert calculus arguments, as well as the theory of hyperbolic partial differential equations.
scientific article; zbMATH DE number 5652266
scientific article