Tag Archives: trace

2013-01 Inequality involving eigenvalues and traces

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

\[ \sum_{i=1}^N |\lambda_i^A – \lambda_i^B|^2 \leq Tr (A-B)^2 \]

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