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Found problems: 1

2022 SEEMOUS, 3
Tags: linear algebra , matrices , hermitian , inequality
Let $\alpha \in \mathbb{C}\setminus \{0\}$ and $A \in \mathcal{M}_n(\mathbb{C})$, $A \neq O_n$, be such that $$A^2 + (A^*)^2 = \alpha A\cdot A^*,$$ where $A^* = (\bar A)^T.$ Prove that $\alpha \in \mathbb{R}$, $|\alpha| \le 2$. and $A\cdot A^* = A^*\cdot A.$