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Corollary: If two normal operators M and N are similar, then they are unitarily equivalent.
This proof is seriously flawed. Consider for instance any diagonal matrix M=N and any positive definite diagonal P. Let S be the positive square root of P. Then it is clear that MS=SM=SN. In the proof it is claimed that this implies that P is the identity on the range of M, but this is clearly false in general. I suspect one might fix the proof by considering the polar decomposition of S, but this would not yield a unitary in general. Isdatmaths (talk) 08:18, 9 December 2014 (UTC)[reply]
