# 2015-5 trace and matrices

Determine all $$n\times n$$ matrices A such that $$\operatorname{tr}(AXY)=\operatorname{tr}(AYX)$$ for all $$n\times n$$ matrices $$X$$ and $$Y$$.

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Let $$A, B$$ be $$N \times N$$ symmetric matrices with eigenvalues $$\lambda_1^A \leq \lambda_2^A \leq \cdots \leq \lambda_N^A$$ and $$\lambda_1^B \leq \lambda_2^B \leq \cdots \leq \lambda_N^B$$. Prove that
$\sum_{i=1}^N |\lambda_i^A – \lambda_i^B|^2 \leq Tr (A-B)^2$