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For questions involving the trace of a square matrix, i.e. the sum of the elements on the main diagonal.
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Reconstruct a matrix from its traces
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? … 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 $A$. …