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If we're given all the eigenvalues of a square matrix $A$ and the corresponding eigenvectors of each eigenvalue, then in what case(s) is it possible theoretically to recover $A$ from this much information? And how exactly when it is possible?

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In general you can't. This is not nearly enough information, unless there are so many eigenvectors that they span the whole space (in which case, of course, you can). – Andrés E. Caicedo Feb 9 '13 at 5:46

In general, this will be possible only if the matrix is diagonalizable. In that case, we simply form the matrix $P$ of eigenvectors and the diagonal matrix $D$ ordered corresponding to $P$. Then $$A = PDP^{-1}$$ You essentially need a basis and you need to know how the matrix acts on the basis to determine the mapping. The eigenvectors and the eigenvalues for a diagonalizable matrix gives you precisely the needed amount of information, but for non-diagonalizable matrices there will be non-trivial Jordan blocks of the matrix which are not recoverable (even if you are given the generalized eigenvectors).

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