A symplectic proof of the Horn inequalities. (English) Zbl 1372.15011
Summary: In this paper, we give a symplectic proof of the Horn inequalities on eigenvalues of a sum of two Hermitian matrices with given spectra. Our method is a combination of tropical calculus for matrix eigenvalues, combinatorics of planar networks, and estimates for the Liouville volume. As a corollary, we give a tropical description of the Duistermaat-Heckman measure on the Horn polytope.
MSC:
| 15A42 | Inequalities involving eigenvalues and eigenvectors |
| 15B57 | Hermitian, skew-Hermitian, and related matrices |
| 15A80 | Max-plus and related algebras |
| 14T05 | Tropical geometry (MSC2010) |
Keywords:
Horn inequalities; tropical approximation; Poisson-Lie groups; planar networks; eigenvalue; Liouville volume; Duistermaat-Heckman measure; Horn polytopeReferences:
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