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 ?