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Exercise 2.32 #15
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P^2 = P is just the definition of a Projector |
Your concern is 100% valid, I had the same issue with the given proofs I found online. Or for a more complex example (pun intended) In both cases, P is not a projector. For one they are not Hermitian, which all projectors are. |
The proof here for Exercise 2.32 appears to assume P^2=P implies P is a projector. I am not certain this is true (the converse is certainly true, a projector satisfies P^2=P as indicated by Exercise 2.16), but could there not be other non-projectors that also satisfy P^2=P ?
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