Suppose $D = P^{-1} A P$. When is $P$ unitary?
In other words, what kind of a matrix $A$ should be, such that $D=P^{\dagger}AP$? i.e. what are the conditions a matrix must have to be able to diagonalize it using unitary matrices?
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2Normal. $A A^* = A^* A$. – copper.hat Dec 03 '14 at 22:44
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@copper.hat, is that an "iff" condition? – Sparkler Dec 03 '14 at 22:45
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1Yes indeed. The significant part of the result is that a normal matrix is unitarily diagonalisable, as if a matrix is unitarily diagonalisable, then it is straightforward to show that it is normal (since all diagonal matrices are normal). – copper.hat Dec 03 '14 at 22:48
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http://en.wikipedia.org/wiki/Spectral_theorem – Qiaochu Yuan Dec 03 '14 at 22:58
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Any normal matrix ($AA^{\dagger}=A^{\dagger}A$) is unitarily diagonalisable.
In particular, any hermitian matrix is unitarily diagonalisable, since $H=H^{\dagger}$.
Sparkler
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