Discrete convexity and Hermitian matrices. (English. Russian original) Zbl 1071.15019
Proc. Steklov Inst. Math. 241, 58-78 (2003); translation from Tr. Mat. Inst. Im. V. A. Steklova 241, 68-89 (2003).
Summary: The question (Horn problem) about the spectrum of the sum of two real symmetric (or complex Hermitian) matrices with given spectra is considered. This problem was solved by A. A. Klyachko [Sel. Math., New Ser. 4, No. 3, 419–445 (1998; Zbl 0915.14010)]. We suggest a different formulation of the solution to the Horn problem with a significantly more elementary proof. Our solution is that the existence of the required triple of matrices \((A,B,C)\) for given spectra \((\alpha,\beta,\gamma)\) is equivalent to the existence of a so-called discrete concave function on the triangular grid \(\Delta(n)\) with boundary increments \(\alpha\), \(\beta\) and \(\gamma\).
In addition, we propose a hypothetical explanation for the relation between Hermitian matrices and discrete concave functions. Namely, for a pair \((A, B)\) of Hermitian matrices, we construct a certain function \(\phi(A,B;\cdot)\) on the grid \(\Delta(n)\). Our conjecture is that this function is discrete concave, which is confirmed in several special cases.
For the entire collection see [Zbl 1059.11002].
In addition, we propose a hypothetical explanation for the relation between Hermitian matrices and discrete concave functions. Namely, for a pair \((A, B)\) of Hermitian matrices, we construct a certain function \(\phi(A,B;\cdot)\) on the grid \(\Delta(n)\). Our conjecture is that this function is discrete concave, which is confirmed in several special cases.
For the entire collection see [Zbl 1059.11002].
MSC:
15A42 | Inequalities involving eigenvalues and eigenvectors |
15B57 | Hermitian, skew-Hermitian, and related matrices |
52A05 | Convex sets without dimension restrictions (aspects of convex geometry) |
52B12 | Special polytopes (linear programming, centrally symmetric, etc.) |