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I'm reading Hoffman and Kunze's linear algebra book and I'm a bit stuck with it's definition of inner product. One of the properties of the inner product is said to be that the product of any vector with itself is positive if the vector is nonzero, but it also says that the inner product is defined on a complex or real vector space. My question is, when is a complex number positive? I thought there was no way of determining wether a complex number is positive or negative in a consistent way since $\mathbb C$ is not an ordered field. I'm thinking that maybe it refers to both the real and imaginary parts being positive but I don't seem to be able to find any confirmation.

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  • $\begingroup$ @hardmath: Done. $\endgroup$ Jun 1 '14 at 14:43
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When one says that a complex number is "positive", one always means that it is "real and positive".

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  • $\begingroup$ Alright, thanks! $\endgroup$ Jun 1 '14 at 14:47

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