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Eigenvalues & Eigenvectorshard
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Let AAA and BBB be n×nn \times nn×n Hermitian matrices. Let the eigenvalues of A,BA, BA,B, and A+BA+BA+B be sorted in descending order: α1≥α2≥⋯≥αn\alpha_1 \ge \alpha_2 \ge \dots \ge \alpha_nα1​≥α2​≥⋯≥αn​, β1≥β2≥⋯≥βn\beta_1 \ge \beta_2 \ge \dots \ge \beta_nβ1​≥β2​≥⋯≥βn​, and γ1≥γ2≥⋯≥γn\gamma_1 \ge \gamma_2 \ge \dots \ge \gamma_nγ1​≥γ2​≥⋯≥γn​. Which inequality is guaranteed to hold for all 1≤i≤n1 \le i \le n1≤i≤n?