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I have encoutered several times the following claim :

Let $A$ and $B$ be two real symmetric matrices (of dimension $n\times n$) with nonnegative coefficients and such that their eigenvalues are nonnegative. Let $C$ be their pointwise product, i.e. the $n\times n$ matrix with coefficients $C_{ij}=A_{ij}B_{ij}$. Then the matrix $C$ has nonnegative eigenvalues.

It seems to be true, but I can't find a proof. Does anybody know how to show this ?

Also, this result implies that if we denote by $A^{[n]}$ the matrix with coefficients $A^{[n]}_{ij}=(A_{ij})^n$ with $n\in \mathbb{N}$, then $A^{[n]}$ has nonnegative eigenvalues. Does this result still hold if we only suppose that $n\in\mathbb{R}^+$ ? If it doesn't hold anymore, is there a sufficient condition weaker than $n\in\mathbb{N}$ under which the result holds ?

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    I don't have the time to write an answer right now, but you should try to show that $S_n^{+}$ (ie. symmetric positive semi-definite matrices) is stable by this product (call it $\otimes$), one way to see that is to decompose the two matrices in their spectral projectors (with the eigenvalues as coefficients) and show that two orthogonal projections behave well under $\otimes$.2012-02-15
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    Here's the reference given on [the Wikipedia page](http://en.wikipedia.org/wiki/Hadamard_product_%28matrices%29): http://buzzard.ups.edu/courses/2007spring/projects/million-paper.pdf2012-02-15

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