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Let $\mathcal{N}$ be the space of all $n \times n$ matrices that are similar to some nonnegative matrix with zero diagonal and let $f: \mathcal{N} \to \mathbb{R}$ be a continuously differentiable function. What are the lightest assumptions I can impose on $f$ to obtain that $f(M)$ it depends only on eigenvalues on $M$? Of course I can require that $f(M)$ be invariant to similarity transformations of $M$. But I would like something lighter. Ideally, I'd like to know that it suffices to assume only that $f(D M D^{-1})=f(M)$ for any diagonal, nonnegative matrix $D$.

Any leads?

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Well, ANY similarity transformation doesn't change eigenvalues, so if your function feels them in any way, you have no chance. Clearly, there are functions that feel them but do not feel diagonal similarity transformations, so no, you cannot get as much as you want... –  fedja Nov 9 '12 at 15:55
    
I wonder If continuity of $f$ and thinking about nearby matrices gets any additional traction, though. –  Ben Golub Nov 9 '12 at 15:59
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Clearly not. Conjugating by a diagonal matrix does not change the diagonal entries of $M$. The diagonal entries are trivially continuous functions. They are not similarity invariant. –  Will Sawin Nov 9 '12 at 16:14
    
Not really. For the particular case you mentioned just consider the product of matrix elements. I guess we can try to do some general construction for an arbitrary proper closed subgroup but I cannot think right now. :) –  fedja Nov 9 '12 at 17:03
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