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](https://math.stackexchange.com/questions/1697991/do-diagonal-matrices-always-commute), $\Gamma$ 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.



Reconstruct a matrix from its traces
Unfortunately even the generic situation is bad. Since we know the eigenvalues, we should search for the orthonormal system of eigenvectors $v\_i$ of $A$. We have ($e\_i$ is the standard basis)
$$
Tr(A^k\Gamma)=\sum\_i\left[\sum\_j\gamma\_j \langle v\_i,e\_j\rangle^2\right]\lambda\_i^k
$$
so, in effect, you have the knowledge of $\sum\_j\gamma\_j \langle v\_i,e\_j\rangle^2$ ($i=1,2,3$). However, these three values are not independent: since the matrix $(\langle v\_i,e\_j\rangle^2)\_{ij}$ is bistochastic, their sum is just $Tr \Gamma$, so you have only $2$ independent equations, while the orthogonal matrices form a 3D manifold. Thus in the generic case you should expect a continuous 1-parametric family of solutions.


