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Eigenvalues of sums of Hermitian matrices [after A. Klyachko]. (English) Zbl 0929.15006
Séminaire Bourbaki. Volume 1997/98. Exposés 835–849. Paris: Société Mathématique de France, Astérisque. 252, 255-269, Exp. No. 845 (1998).
In the simplest form, the problem solved by A. A. Klyachko [Stable bundles, representation theory and Hermitian operators, Institute Mittag-Leffler Preprint (1996-97)] is to describe the eigenvalues of the sum $$A+B$$ of two Hermitian $$n{\times}n$$ matrices in terms of the eigenvalues of $$A$$ and $$B$$. List the eigenvalues of a Hermitian matrix $$H$$ in decreasing order $$\lambda(H): \lambda_1(H) \geq \lambda_2(H) \geq \cdots \geq \lambda_n(H)$$. Write $$\lambda(A): \alpha_1 \geq \alpha_2 \geq \cdots \geq \alpha_n$$, $$\lambda(B): \beta_1 \geq \beta_2 \geq \cdots \geq \beta_n$$; let $$C = A + B$$, and $$\lambda(C): \gamma_1 \geq \gamma_2 \geq \cdots \geq \gamma_n$$ Of course one always has the trace identity $$\sum{\gamma_i} = \sum{\alpha_i} + \sum{\beta_i}$$. The problem arose from questions in solid mechanics, where eigenvalues of symmetric matrices determine shapes of ellipsoids. The inequality $$\gamma_1 \leq \alpha_1 + \beta_1$$ was known more then a century ago. For $$n = 2$$ the inequalities $$\gamma_1 \leq \alpha_1 + \beta_1$$, $$\gamma_2 \leq \alpha_1 + \beta_2$$ and $$\gamma_2 \leq \alpha_2 + \beta_1$$ are easily seen to be sufficient to characterize the possible eigenvalues of the sum.
In this paper the author rephrases and proves some of Klyachko’s theorems which generalize to sums of more than two Hermitian matrices, and to tensor products of more than two representations. His methods and discussions are based on the Schubert calculus. The final section of the paper contains some applications, and a brief discussion of some more recent works on eigenvalues of unitary matrices.
For the entire collection see [Zbl 0911.00019].

##### MSC:
 15A18 Eigenvalues, singular values, and eigenvectors 14N15 Classical problems, Schubert calculus 14N35 Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects) 22E46 Semisimple Lie groups and their representations 15A42 Inequalities involving eigenvalues and eigenvectors 14M15 Grassmannians, Schubert varieties, flag manifolds
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