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Let $F$ be a continuous function on the space of $N\times N$ hermitian matrices $\mathcal{H}_N$ taking values in $\mathbb{C}$ which satisfies $F(UMU^*)=F(M)$ for all $M\in \mathcal{H}_N$ and $U\in U(N)$, where the later stands for the unitary group of $\mathbb{C}^N$.

The first examples coming in mind are the Trace, and the Determinant. Do you know is there exists a way to express $F$ in terms of these two functions ? Or maybe under stronger conditions than continuity for $F$ ?

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Every symmetric function of the eigenvalues has this property. The elementary symmetric polynomials in the eigenvalues (the coefficients of the characteristic polynomial) freely generate the ring of symmetric polynomials, and these are dense in the space of functions you're interested in.

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    Thanks Qiaochu, I was looking for an answer like that. But how can you write the elementary symmetric polynomials in the eigenvalues in terms of Trace and Det ? (except the first and last one of course), maybe using minors ?2011-11-25
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    Something else, my function can be defined on an unbounded set, right ? I thus don't understand the density argument.2011-11-25
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    @Student: you can't. Their values aren't determined by the values of the trace and determinant. I don't understand the second question.2011-11-25
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    But you can write the elementary symmetric polynomial as a trace of an appropriate exterior power of the matrix (and perhaps also using a sum of determinants using Cauchy-Binet type of stuff)2011-11-25