# 2019-15 Singular matrix

Let $$A, B$$ be $$n \times n$$ Hermitian matrices. Find all positive integer $$n$$ such that the following statement holds:

“If $$AB – BA$$ is singular, then $$A$$ and $$B$$ have a common eigenvector.”

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# 2017-01 Eigenvalues of Hermitian matrices

Let $$A, B, C$$ be $$N \times N$$ Hermitian matrices with $$C = A+B$$. Let $$\alpha_1 \geq \dots \geq \alpha_N$$, $$\beta_1 \geq \dots \geq \beta_N$$, $$\gamma_1 \geq \dots \geq \gamma_N$$ be the eigenvalues of $$A, B, C$$, respectively. For any $$1 \leq k \leq N$$, prove that
$\gamma_1 + \gamma_2 + \dots + \gamma_k \leq (\alpha_1 + \alpha_2 + \dots + \alpha_k) + (\beta_1 + \beta_2 + \dots + \beta_k)$

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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
$\gamma_{i+j-1} \leq \alpha_i + \beta_j$