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Let $A = \left( {a_{ij} } \right)$ be a matrix over $\mathbb R$, of size $n \times n$. Let $\left\{ \lambda _k \right\}_{k = 1}^n$ be the $n$ eigenvalues of the matrix. Prove the following inequality:

$$ \sum_{k = 1}^n \left| {\lambda _k } \right|^2 \leqslant \sum_{i = 1}^n \sum\limits_{j = 1}^n \left| {a_{ij} } \right|^2 . $$

I can't prove it, and I'm not sure if in the problem one assumes that there exist $n$ real eigenvalues, or it's also true for complex eigenvalues. D:

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    I'm fairly certain this follows from Jordan form, for example, and that one has an equality iff $A$ is diagonal over $\mathbf{C}$. Will come back to it if I can...2011-11-14
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    Jordan form won't quite work because RHS isn't invariant under general conjugations.2011-11-14
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    Yeah, that is why I hesitated writing an answer. Glad someone wrote it out.2011-11-14
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    Right hand side is equal to $\operatorname{Tr}(AA^*)$ and trace is equal to the sum of the eigenvalues of the matrix. You might want to continue from there.2011-11-14

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