Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

I have a simple question. Is the inner product $\langle x | z\rangle$ always real or does this hold only for an inner product with itself $\langle x | x \rangle$?

share|cite|improve this question
    
$(x,z) \mapsto \overline{x} z$ (possibly conjugate, depending on how you like your inner product) is an inner product on $\mathbb{C}$. Hence it is not necessarily real. – copper.hat May 20 '13 at 14:49

Hint: Any inner product $\langle {-} | {-} \rangle$ on a complex vector space satisfies $\langle \lambda x | y \rangle = \lambda^* \langle x | y \rangle$ for all $\lambda \in \mathbb{C}$.

You're right in saying that $\langle x|x \rangle$ is always real when the field is defined over the real numbers: in general, $\langle x|y \rangle = \overline{\langle y|x \rangle}$, so $\langle x|x \rangle = \overline{\langle x|x \rangle}$, so $\langle x|x \rangle$ is real. (It's also always positive.)

share|cite|improve this answer
    
I forgot to mention i need info for $\mathbb{C}$. – 71GA May 20 '13 at 14:43
    
What do you mean? – Clive Newstead May 20 '13 at 16:35

In real vector spaces, both are real, and the latter is also positive.

In complex vector spaces, the first can be anything (e.g. $\left< x|(i | x\right>)$, but the latter is still real and positive.

share|cite|improve this answer

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.