How to Diagonalize a Matrix

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It is not possible to diagonalize a matrix that has an insufficient number of eigenvalues (when counted using multiplicities). If we don't take into account complex numbers, this is the only scenario in which it is possible for it to occur (see points 1 and 2). To be more specific, a matrix that does not include any real eigenvalues does not qualify as a diagonalizable matrix (in the field of real numbers).
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