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An operator is Hermitian if and only if it has real eigenvalues.

An operator is Hermitian if and only if it has real eigenvalues.

I find it questionable because I thought that non-Hermitian operators can sometimes have real eigenvalues. We can correctly say that Hermitian operators can only have real eigenvalues but that does not define the operator, right? Is it some kind of convention or is it just plain wrong? Alas the physicists often don't understand the difference between an implication and equivalence.