In my research I came across the following problem. Let $A$ be a symmetric and $\Gamma$ be a diagonal $n\times n$ matrices. The eigenvalues of $A$ are known $\lambda_1,\ldots\lambda_n$. The traces $\mathrm{Tr}(A^k\,\Gamma)=t_k$, $k\in\mathbb{N}$ are also known, $\Gamma$ is given. Can $A$ be found based on this information? If yes, how? While I am curious about the general $n$ case, information on $n=3$ would be most valuable. Any reference would be greatly appreciated.

P.S.

- From the answer of Francesco Polizzi it is known that the reconstruction is not possible when there is an orthogonal matrix $M$ that commutes with $\Gamma$. Fortunately, in the case of interest, this situation can be excluded. Specifically,
*it is known that diagonal entries of $\Gamma$ are all distinct and positive.*By this answer, $\Gamma$ then commutes only with diagonal matrices. - user44191 suggested that $B=M^{-1}AM$ ($M$-nonsingular) will have the same traces as $A$. However, apart from a trivial $\pm1$ possibility, which can be excluded by a proper sign convention, the new matrix $B$ is no longer symmetric. Thus, it leaves the question open.