Solution: 2013-20 Eigenvalues of Hermitian matrices

Let $$A, B, C = A+B$$ be $$N \times N$$ Hermitian matrices. Let $$\alpha_1 \geq \cdots \geq \alpha_N$$, $$\beta_1 \geq \cdots \geq \beta_N$$, $$\gamma_1 \geq \cdots \geq \gamma_N$$ be the eigenvalues of $$A, B, C$$, respectively. For any $$1 \leq i, j \leq N$$ with $$i+j -1 \leq N$$, prove that
$\gamma_{i+j-1} \leq \alpha_i + \beta_j$

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