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Trudy Matematicheskogo Instituta imeni V.A. Steklova, 2003, Volume 241, Pages 68–89
(Mi tm388)
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This article is cited in 11 scientific papers (total in 11 papers)
Discrete Convexity and Hermitian Matrices
V. I. Danilov, G. A. Koshevoy Central Economics and Mathematics Institute, RAS
Abstract:
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. Klyachko. 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.
Received in November 2002
Citation:
V. I. Danilov, G. A. Koshevoy, “Discrete Convexity and Hermitian Matrices”, Number theory, algebra, and algebraic geometry, Collected papers. Dedicated to the 80th birthday of academician Igor' Rostislavovich Shafarevich, Trudy Mat. Inst. Steklova, 241, Nauka, MAIK «Nauka/Inteperiodika», M., 2003, 68–89; Proc. Steklov Inst. Math., 241 (2003), 58–78
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https://www.mathnet.ru/eng/tm388 https://www.mathnet.ru/eng/tm/v241/p68
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Abstract page: | 949 | Full-text PDF : | 343 | References: | 71 |
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